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Editorial BoardB. BOLLOB A S, W. FULTON, A. KATOK, F. KIRWAN,P. SARNAK, B. SIMON, B. TOTARO

Local Cohomology

This Second Edition of a successful graduate text provides a careful and detailedalgebraic introduction to Grothendieck’s local cohomology theory, including inmulti-graded situations, and provides many illustrations of the theory in commutativealgebra and in the geometry of quasi-affine and quasi-projective varieties. Topicscovered include Serre’s Affineness Criterion, the Lichtenbaum–Hartshorne VanishingTheorem, Grothendieck’s Finiteness Theorem and Faltings’ Annihilator Theorem,local duality and canonical modules, the Fulton–Hansen Connectedness Theorem forprojective varieties, and connections between local cohomology and both reductions ofideals and sheaf cohomology.

The book is designed for graduate students who have some experience of basiccommutative algebra and homological algebra, and also for experts in commutativealgebra and algebraic geometry. Over 300 exercises are interspersed among the text;these range in difficulty from routine to challenging, and hints are provided for someof the more difficult ones.

M. P. Brodmann is Emeritus Professor in the Institute of Mathematics at theUniversity of Zurich.

R. Y. Sharp is Emeritus Professor of Pure Mathematics at the University of Sheffield.

C A M B R I D G E S T U D I E S I N A D V A N C E D M A T H E M A T I C S

Editorial Board:B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro

All the titles listed below can be obtained from good booksellers or from Cambridge University Press.For a complete series listing visit: http://www.cambridge.org/mathematics.

Already published93 D. Applebaum Levy processes and stochastic calculus (1st Edition)94 B. Conrad Modular forms and the Ramanujan conjecture95 M. Schechter An introduction to nonlinear analysis96 R. Carter Lie algebras of finite and affine type97 H. L. Montgomery & R. C. Vaughan Multiplicative number theory, I98 I. Chavel Riemannian geometry (2nd Edition)99 D. Goldfeld Automorphic forms and L-functions for the group GL(n,R)

100 M. B. Marcus & J. Rosen Markov processes, Gaussian processes, and local times101 P. Gille & T. Szamuely Central simple algebras and Galois cohomology102 J. Bertoin Random fragmentation and coagulation processes103 E. Frenkel Langlands correspondence for loop groups104 A. Ambrosetti & A. Malchiodi Nonlinear analysis and semilinear elliptic problems105 T. Tao & V. H. Vu Additive combinatorics106 E. B. Davies Linear operators and their spectra107 K. Kodaira Complex analysis108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic analysis on finite groups109 H. Geiges An introduction to contact topology110 J. Faraut Analysis on Lie groups: An introduction111 E. Park Complex topological K-theory112 D. W. Stroock Partial differential equations for probabilists113 A. Kirillov, Jr An introduction to Lie groups and Lie algebras114 F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics116 D. Applebaum Levy processes and stochastic calculus (2nd Edition)117 T. Szamuely Galois groups and fundamental groups118 G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields120 P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths121 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups122 S. Kalikow & R. McCutcheon An outline of ergodic theory123 G. F. Lawler & V. Limic Random walk: A modern introduction124 K. Lux & H. Pahlings Representations of groups125 K. S. Kedlaya p-adic differential equations126 R. Beals & R. Wong Special functions127 E. de Faria & W. de Melo Mathematical aspects of quantum field theory128 A. Terras Zeta functions of graphs129 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I130 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II131 D. A. Craven The theory of fusion systems132 J. Vaananen Models and games133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type134 P. Li Geometric analysis135 F. Maggi Sets of finite perimeter and geometric variational problems136 M. P. Brodmann & R. Y. Sharp Local cohomology (2nd Edition)137 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I138 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, II139 B. Helffer Spectral theory and its applications

Local CohomologyAn Algebraic Introduction with

Geometric Applications

second edition

M. P. BRODMANNUniversitat Zurich

R. Y. SHARPUniversity of Sheffield

cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town,

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To Alice

from the second author

Contents

Preface to the First Edition page xiPreface to the Second Edition xviiNotation and conventions xxi

1 The local cohomology functors 11.1 Torsion functors 11.2 Local cohomology modules 31.3 Connected sequences of functors 10

2 Torsion modules and ideal transforms 162.1 Torsion modules 172.2 Ideal transforms and generalized ideal transforms 212.3 Geometrical significance 39

3 The Mayer–Vietoris sequence 473.1 Comparison of systems of ideals 483.2 Construction of the sequence 513.3 Arithmetic rank 553.4 Direct limits 59

4 Change of rings 654.1 Some acyclic modules 664.2 The Independence Theorem 704.3 The Flat Base Change Theorem 74

5 Other approaches 815.1 Use of Cech complexes 825.2 Use of Koszul complexes 945.3 Local cohomology in prime characteristic 101

6 Fundamental vanishing theorems 1066.1 Grothendieck’s Vanishing Theorem 107

viii Contents

6.2 Connections with grade 1126.3 Exactness of ideal transforms 1176.4 An Affineness Criterion due to Serre 1226.5 Applications to local algebra in prime characteristic 127

7 Artinian local cohomology modules 1357.1 Artinian modules 1357.2 Secondary representation 1397.3 The Non-vanishing Theorem again 143

8 The Lichtenbaum–Hartshorne Theorem 1478.1 Preparatory lemmas 1488.2 The main theorem 156

9 The Annihilator and Finiteness Theorems 1649.1 Finiteness dimensions 1649.2 Adjusted depths 1689.3 The first inequality 1719.4 The second inequality 1769.5 The main theorems 1839.6 Extensions 188

10 Matlis duality 19310.1 Indecomposable injective modules 19310.2 Matlis duality 199

11 Local duality 21111.1 Minimal injective resolutions 21211.2 Local Duality Theorems 216

12 Canonical modules 22312.1 Definition and basic properties 22412.2 The endomorphism ring 23812.3 S2-ifications 245

13 Foundations in the graded case 25113.1 Basic multi-graded commutative algebra 25313.2 *Injective modules 25713.3 The *restriction property 26113.4 The reconciliation 27113.5 Some examples and applications 274

14 Graded versions of basic theorems 28514.1 Fundamental theorems 28614.2 *Indecomposable *injective modules 29514.3 A graded version of the Annihilator Theorem 302

Contents ix

14.4 Graded local duality 30914.5 *Canonical modules 313

15 Links with projective varieties 33115.1 Affine algebraic cones 33115.2 Projective varieties 336

16 Castelnuovo regularity 34616.1 Finitely generated components 34616.2 The basics of Castelnuovo regularity 35116.3 Degrees of generators 358

17 Hilbert polynomials 36417.1 The characteristic function 36617.2 The significance of reg2 37317.3 Bounds on reg2 in terms of Hilbert coefficients 37817.4 Bounds on reg1 and reg0 383

18 Applications to reductions of ideals 38818.1 Reductions and integral closures 38818.2 The analytic spread 39318.3 Links with Castelnuovo regularity 397

19 Connectivity in algebraic varieties 40519.1 The connectedness dimension 40619.2 Complete local rings and connectivity 41019.3 Some local dimensions 41619.4 Connectivity of affine algebraic cones 42219.5 Connectivity of projective varieties 42419.6 Connectivity of intersections 42619.7 The projective spectrum and connectedness 432

20 Links with sheaf cohomology 43820.1 The Deligne Isomorphism 43920.2 The Graded Deligne Isomorphism 45220.3 Links with sheaf theory 45520.4 Applications to projective schemes 46520.5 Locally free sheaves 476References 480Index 485

Preface to the First Edition

One can take the view that local cohomology is an algebraic child of geomet-ric parents. J.-P. Serre’s fundamental paper ‘Faisceaux algebriques coherents’[77] represents a cornerstone of the development of cohomology as a tool inalgebraic geometry: it foreshadowed many crucial ideas of modern sheaf coho-mology. Serre’s paper, published in 1955, also has many hints of themes whichare central in local cohomology theory, and yet it was not until 1967 that thepublication of R. Hartshorne’s ‘Local cohomology’ Lecture Notes [25] (on A.Grothendieck’s 1961 Harvard University seminar) confirmed the effectivenessof local cohomology as a tool in local algebra.

Since the appearance of the Grothendieck–Hartshorne notes, local cohomol-ogy has become indispensable for many mathematicians working in the theoryof commutative Noetherian rings. But the Grothendieck–Hartshorne notes cer-tainly take a geometric viewpoint at the outset: they begin with the cohomologygroups of a topological spaceX with coefficients in an Abelian sheaf onX andsupports in a locally closed subspace.

In the light of this, we feel that there is a need for an algebraic introduc-tion to Grothendieck’s local cohomology theory, and this book is intended tomeet that need. Our book is designed primarily for graduate students who havesome experience of basic commutative algebra and homological algebra; fordefiniteness, we have assumed that our readers are familiar with many of thebasic sections of H. Matsumura’s [50] and J. J. Rotman’s [71]. Our approachis based on the fundamental ‘δ-functor’ techniques of homological algebra pi-oneered by Grothendieck, although we shall use the ‘connected sequence’ ter-minology of Rotman (see [71, pp. 212–214]).

However, we have not overlooked the geometric roots of the subject or thesignificance of the ideas for modern algebraic geometry. Indeed, the bookpresents several detailed examples designed to illustrate the geometrical sig-nificance of aspects of local cohomology; we have chosen examples which

xii Preface to the First Edition

require only basic ideas from algebraic geometry. In this spirit, there is oneparticular example, which we refer to as ‘Hartshorne’s Example’, to which wereturn several times in order to illustrate various points.

The geometric aspects are, in fact, nearer the surface of our treatment thanmight initially be realised, because we make much use of ideal transforms andtheir universal properties, but it is only in the final chapter that we exposethe fundamental links, expressed by means of the Deligne Correspondence,between the ideal transform functors and their right derived functors on theone hand, and section functors of sheaves and sheaf cohomology on the other.

We define the local cohomology functors to be the right derived functorsof the appropriate torsion functor, although we establish in the first chapterthat one can also construct local cohomology modules as direct limits of ‘Ext’modules; we also present alternative constructions of local cohomology mod-ules, one via cohomology of Cech complexes, and the other via direct limits ofhomology modules of Koszul complexes, in Chapter 5. (In fact, we do not usethis Koszul complex approach very much at all in this book.)

Chapters 2, 3 and 4 include fundamental ideas concerning ideal transformsand their universal properties, the Mayer–Vietoris Sequence for local cohomol-ogy and the Independence and Flat Base Change Theorems: we regard all ofthese as technical cornerstones of the subject, and we certainly use them overand over again.

The main purpose of Chapters 6 and 7 is the presentation of some of Groth-endieck’s important vanishing theorems for local cohomology, which relatesuch vanishing to the concepts of dimension and grade. This work is mainly‘algebraic’ in nature. In Chapter 8, we present another vanishing theorem forlocal cohomology modules, namely the local Lichtenbaum–Hartshorne Van-ishing Theorem: this has an ‘analytic’ flavour, in the sense that it is intimatelyrelated with ‘formal’ methods and techniques, that is, with passage to comple-tions of local rings and with the structure theory for complete local rings. TheLichtenbaum–Hartshorne Theorem has important geometric applications: forexample, we show in Chapter 19 how it can be used to obtain major resultsabout the connectivity of algebraic varieties.

Grothendieck’s Finiteness Theorem and G. Faltings’ Annihilator Theoremfor local cohomology are the main subjects of Chapter 9. These two theoremsalso have major geometric applications, including, for example, in Macaulay-fication of schemes. They also have significance for the theory of generalizedCohen–Macaulay modules and Buchsbaum modules, two concepts which fea-ture briefly in the exercises in Chapter 9.

We have delayed the introduction of duality (until Chapters 10 and 11) be-cause quite a lot can be achieved without it, and because, for our discussion

Preface to the First Edition xiii

of duality, we have had to assume (on account of limitations of space) that thereader is familiar with the Matlis–Gabriel decomposition theory for injectivemodules over a commutative Noetherian ring (although we have reviewed thattheory and provided some detailed proofs). We have not explicitly used dual-izing complexes or derived categories, as it seems to us that such technicalitiescould daunt youthful readers and are not essential for a presentation of themain ideas. After the introduction of local duality in Chapter 11, we show howthis duality can be used to derive some results established earlier in the bookby different means.

The many recent research papers involving local cohomology of gradedrings illustrate the importance of this aspect, and we have made some effortto develop the fundamentals of local cohomology in the graded case carefullyin Chapters 12 and 13: various representations of local cohomology modulesobtained in the earlier chapters inherit natural gradings when the ring, moduleand ideal concerned are all graded, and it seems to us that it is important toknow that there is really only one sensible way of grading local cohomologymodules in such circumstances. Our main aim in Chapter 12 has been to ad-dress this point. In Chapter 13, ‘graded frills’ are added to basic results provedearlier in the book.

The short Chapter 14 establishes some links between graded local cohomol-ogy and projective varieties; it has been included to provide a little geometricinsight, and in order to motivate the work on Castelnuovo–Mumford regularityin Chapters 15–17, and the connections between ideal transforms and sectionfunctors of sheaves presented in Chapter 20.

In Chapter 15, we study the graded local cohomology of a homogeneouspositively graded commutative Noetherian ringR with respect to the irrelevantideal. One of the most important invariants in this context is Castelnuovo–Mumford regularity. This concept has, in addition to fundamental significancein projective algebraic geometry, connections with the degrees of generators ofa finitely generated graded R-module M : it turns out that M can be generatedby homogeneous elements of degrees not exceeding reg(M), the Castelnuovo–Mumford regularity of M . In turn, this leads on to connections with the theoryof syzygies of finitely generated graded modules over polynomial rings over afield.

In certain circumstances, including when M is the vanishing ideal IPr (V )

of a projective variety V ⊂ Pr, the above-mentioned reg(M) coincides withreg2(M), the Castelnuovo–Mumford regularity of M at and above level 2.In Chapters 16 and 17, we present bounds for the invariant reg2(M). Chapter16 contains a priori bounds which apply whenever the underlying homogen-eous positively graded commutative Noetherian ring R has Artinian 0-th

xiv Preface to the First Edition

component. Chapter 17 is more specialized, and contains bounds expressedin terms of coefficients of Hilbert polynomials; our development of this theoryincludes a presentation of basic ideas concerning cohomological Hilbert poly-nomials. The motivation for our work in Chapter 17 comes from D. Mumford’sclassical work [54]: Mumford established the existence of bounds of the typewe present, but, in the spirit of this book, we have added some precision.

One could view Chapters 18 and 19 as propaganda for the effectiveness oflocal cohomology as a tool in algebra and geometry. Chapter 18 presents someapplications of Castelnuovo–Mumford regularities to reductions of ideals. Thisis a fast developing area, and we have not attempted to give an encyclopaedicaccount; instead, we have tried to present the basic ideas and a few recentresults to whet the reader’s appetite. The highlight of Chapter 18 is a theoremof L. T. Hoa; the statement of this theorem is satisfyingly simple, and makes nomention of local cohomology, and yet Hoa’s proof, which we present towardsthe end of the chapter, makes significant use of graded local cohomology.

Chapter 18 is a good advertisement for local cohomology as a ‘hidden tool’,and Chapter 19 continues this theme, although here the applications (to theconnectivity of algebraic varieties) are more geometrical in nature. The onlyappearances of local cohomology in Chapter 19 are in just two proofs, where afew central ideas (such as the Mayer–Vietoris Sequence and the Lichtenbaum–Hartshorne Vanishing Theorem) are used in crucial ways. No hypothesis orconclusion of any result in the chapter makes any mention of local cohomol-ogy, and yet we are able to show how the two results whose proofs use localcohomology can be developed into a theory which leads to proofs of major re-sults involving connectivity, such as Grothendieck’s Connectedness Theorem,the Bertini–Gothendieck Connectivity Theorem, the Connectedness Theoremfor Projective Varieties due to W. Barth, to W. Fulton and J. Hansen, and to G.Faltings, and a ring-theoretic version of Zariski’s Main Theorem. This chapteris certainly a good advertisement for the power of local cohomology as a toolin algebraic geometry!

Finally, in Chapter 20, we bring the subject ‘home to its roots’, so to speak,by presenting links between local cohomology and the cohomology of quasi-coherent sheaves over certain Noetherian schemes. (Chapter 20 is the only onefor which we have assumed that the reader has some basic knowledge aboutschemes and sheaves.)

Some parts of our presentation are fairly leisurely: this is deliberate, and hasbeen done with graduate students in mind, because we found several prepara-tory topics where either we knew of no suitable text-book account, or wefelt we had something to add to the existing accounts. Examples are the treat-ments of Matlis duality in Chapter 10, of *canonical modules in Chapter 13, of

Preface to the First Edition xv

reductions of ideals in Chapter 18, and of connectedness dimensions in Chap-ter 19; also, our presentation in Chapter 5 of some links between Koszul com-plexes and local cohomology is deliberately slow.

Our philosophy throughout has been to try to give a careful and accessiblepresentation of basic ideas and some important results, illustrating the ideaswith examples, to bring the reader to a level of expertise where he or she canapproach with some confidence recent research papers in local cohomology.To help with this, the book contains a large number of exercises, and we havesupplied hints for many of the more difficult ones.

We have tried out parts of the book, especially the earlier chapters, on someof our own graduate students, and their comments have influenced the finalversion. We are particularly grateful to Claudia Albertini, Carlo Matteotti,Francesco Mordasini, Henrike Petzl and Massoud Tousi for acting as ‘guineapigs’, so to speak. We should also like to express our gratitude to Peter Gabriel,John Greenlees, Martin Holland and Josef Rung for continual interest andencouragement, and to the Schweizerischer Nationalfonds zur Forderung derwissenschaftlichen Forschung, the Forschungsrat des Instituts fur Mathematikder Universitat Zurich, and the University of Sheffield Research Fund, for fi-nancial support to enable several visits for intense collaboration on the book totake place. Both authors would like to thank Alice Sharp: Markus Brodmannthanks her for kind hospitality during pleasant visits to Sheffield for discus-sions on the book; and Rodney Sharp thanks her for much sympathetic supportthrough the years during which this book was being written (as well as formany things which have nothing to do with local cohomology). Finally, weare very grateful to David Tranah and Roger Astley of Cambridge UniversityPress for their continual encouragement and assistance over many years, and,not least, for their cooperation over our request that the blue stripe on the coverof the book should match the blue of the Zurich trams!

Markus Brodmann Rodney SharpZurich Sheffield

April 1997

Preface to the Second Edition

In the fifteen years since we completed the First Edition of this book, we havehad opportunity to reflect on how we could change it in order to enhance itsusefulness to the graduate students at whom it is aimed. As a result, this Sec-ond Edition shows substantial differences from the First. The main ones aredescribed as follows.

One of the more dramatic changes is the introduction of a complete newchapter, Chapter 12, devoted to the study of canonical modules. The treatmentof canonical modules in the First Edition was brief and restricted to the casewhere the underlying ring is Cohen–Macaulay; we assumed that the reader wasfamiliar with the treatment in this case by W. Bruns and J. Herzog in their bookon Cohen–Macaulay rings (see [7]). In our new Chapter 12, we present someof the basic work of Y. Aoyama (see [1] and [2]) and follow M. Hochster andC. Huneke [39] in defining a canonical module over a (not necessarily Cohen–Macaulay) local ring (R,m) to be a finitely generated R-module whose Matlisdual is isomorphic to the ‘top’ local cohomology module HdimR

m (R). Thusthis topic is intimately related to local cohomology.

Canonical modules have connections with the theory of S2-ifications (here,the ‘S2’ refers to Serre’s condition), and we realised that the development ofS2-ifications can be facilitated by generalizations of arguments we had used tostudy ideal transforms in §2.2 of the First Edition. For this reason, instead ofdealing just with ideal transforms based on the sequence of powers of a fixedideal, we treat, in §2.2 of this Second Edition, a generalization based on a setB of ideals such that, whenever b, c ∈ B, there exists d ∈ B with d ⊆ bc.This represents a significant change to Chapter 2.

Another major change concerns our treatment of graded local cohomology.Our new Chapters 13 and 14 treat local cohomology in the situation where therings, ideals and modules concerned are graded by Zn, where n is a positiveinteger. In the First Edition we dealt only with the case where n = 1; in the

xviii Preface to the Second Edition

years since that edition was published, there have been more and more usesof local cohomology in multi-graded situations. It was not difficult for us toadapt the treatment of Z-graded local cohomology from the First Edition tothe multi-graded case. The main point of our Chapter 13 is to show that, eventhough there appear to be various possible approaches, there is really onlyone sensible way of grading local cohomology. Chapter 14 adds ‘(Zn-)gradedfrills’ to basic results proved earlier in the book. We illustrate this work withsome calculations over polynomial rings and Stanley–Reisner rings. Chapters13 and 14 are generalizations of Chapters 12 and 13 from the First Edition;however, we found it desirable to present some fundamental results from S.Goto’s and K.-i. Watanabe’s paper [22] about Zn-graded rings and modules;those results in the particular case when n = 1 are more readily available.

The last two decades have seen a surge in the use of local cohomology as atool in ‘characteristic p’ commutative algebra, that is, the study of commutativeNoetherian rings of prime characteristic p. The key to this is the fact that, foran ideal a of such a ring R, and any integer i ≥ 0, the i-th local cohomologymodule Hi

a(R) of R itself with respect to a has a so-called ‘Frobenius action’.In the new §5.3, we explain why this Frobenius action exists, and in the new§6.5, we use it to present Hochster’s proof of his Monomial Conjecture (incharacteristic p), and to give some examples of how local cohomology can beused as an effective tool in tight closure theory.

Another new section is §20.5 about locally free sheaves; here we proveSerre’s Cohomological Criterion for Local Freeness, Horrocks’ Splitting Cri-terion and Grothendieck’s Splitting Theorem. We have also expanded §20.4with an additional application to projective schemes: we now include a resultof Serre about the global generation of twisted coherent sheaves.

In order to include all this new material, we have had to omit some itemsthat were included in the First Edition but which, we now consider, no longercommand sufficiently compelling reasons for inclusion. The main topics thatfall under this heading are the old §11.3 containing some applications of localduality (one can take the view that the new Chapter 12, on canonical modules,represents a major application of local duality), and the a priori bounds of di-agonal type on Castelnuovo–Mumford regularity at and above level 2 that weretreated in Chapter 16 of the First Edition (Chapter 17 has been reorganized tosmooth over the omission, and expanded by the addition of further boundingresults which follow from our generalized version of Mumford’s bound onregularity at and above level 2).

There are also many minor changes, designed to improve the presentation orthe usefulness of the book. For example, the treatment of Faltings’ AnnihilatorTheorem in Chapter 9 now applies to two arbitrary ideals a and b, whereas in

Preface to the Second Edition xix

the First Edition we treated only the case where b ⊆ a; the syzygetic character-ization of Castelnuovo–Mumford regularity is given a full proof in the SecondEdition; and the graded Deligne Isomorphism in §20.2 is presented here for amulti-graded situation.

We should also point out that two comments made about the First Editionin its Preface do not apply to this Second Edition. Firstly, the example studiedin 2.3.7, 3.3.5, 4.3.7, . . . is not called ‘Hartshorne’s Example’ in this SecondEdition (but we do cite Hartshorne’s paper [28] when we first consider thisexample); secondly, there are a few more appearances of local cohomology inChapter 19 than there were in the First Edition (because we followed a sug-gestion of M. Varbaro that, in some formulas, the arithmetic rank of an ideal acould be replaced by the cohomological dimension of a). Nevertheless, Chap-ter 19 still contains several exciting examples of situations which represent‘hidden applications’ of local cohomology, in the sense that significant resultsthat do not mention local cohomology in either their hypotheses or their con-clusions have proofs in this book that depend on local cohomology. There areother examples of such ‘hidden applications’ in §6.5 and in Chapter 18.

We would like to add, to the list of people thanked in the Preface to the FirstEdition, several more of our students, namely Roberto Boldini, Stefan Fuma-soli, Simon Kurmann, Nicole Nossem and Fred Rohrer, who all contributedto this Second Edition, either by providing constructive criticism of the FirstEdition, or by trying out drafts of changed or new sections that we planned toinclude in the Second Edition. We are grateful to them all.

We thank the Schweizerischer Nationalfonds zur Forderung der wissensch-aftlichen Forschung, the Forschungsrat des Instituts fur Mathematik der Uni-versitat Zurich, and the Department of Pure Mathematics of the University ofSheffield, for financial support for visits for collaboration on this Second Edi-tion. We are also particularly grateful to the Scientific Council of the CentreInternational de Rencontres Mathematiques (CIRM) at Luminy, Marseille, fortheir award to us of a two-week ‘research in pairs’ in Spring 2011 that enabledus, in the excellent environment for mathematical research at CIRM, to pro-duce a complete draft. We again thank Alice Sharp for her continued supportand encouragement for our project. It is also a pleasure for us to record ourgratitude to Roger Astley and Clare Dennison of Cambridge University Pressfor their encouragement and support.

Markus Brodmann Rodney SharpZurich Sheffield

April 2012

Notation and conventions

All rings considered in this book will have identity elements.Throughout the book, R will always denote a non-trivial commutative Noe-

therian ring, and a will denote an ideal of R. We shall only assume that Rhas additional properties (such as being local) when these are explicitly stated;however, the phrase ‘(R,m) is a local ring’ will mean that R is a commutativeNoetherian quasi-local ring with unique maximal ideal m.

For an ideal c of R, we denote Supp(R/c) = {p ∈ Spec(R) : p ⊇ c} byVar(c), and refer to this as the variety of c.

By a multiplicatively closed subset of R, we shall mean a subset of R whichis closed under multiplication and contains 1. It should be noted (and this com-ment is relevant for the final chapter) that, if S is a non-empty subset of Rwhich is closed under multiplication, then, even if S does not contain 1, wecan form the commutative ring S−1R and, for an R-module M , the S−1R-module S−1M . In fact, S−1R ∼= (S ∪ {1})−1R, and, in S−1R, the elementsr/s, for r ∈ R and s ∈ S, is independent of the choice of such s; similarcomments apply to S−1M .

The symbol Z will always denote the ring of integers; in addition, N (respec-tively N0) will always denote the set of positive (respectively non-negative)integers. The field of rational (respectively real, complex) numbers will be de-noted by Q (respectively R, C).

The category of all modules and homomorphisms over a commutative ringR′ will be denoted by C(R′). When R′ is G-graded, where G is a finitelygenerated, torsion-free Abelian group, the category of all graded R′-modulesand homogeneousR′-homomorphisms will be denoted by *C(R′) (or *CG(R′)when it is desirable to indicate the grading group G).

The symbol ⊆ will stand for ‘is a subset of’; the symbol ⊂ will be reservedto denote strict inclusion. Thus, for sets A,B, the expression A ⊂ B meansthat A ⊆ B and A = B.

xxii Notation and conventions

The identity mapping on a set A will be denoted by IdA. If f : A → C isa mapping from the set A to the set C, and S ⊆ A, then f� S : S → C willdenote the restriction of f to S. Thus f� S(s) = f(s) for all s ∈ S.

Some of the exercises in the book are needed for the main developmentlater in the book, and these exercises are marked with a ‘�’; however, exerciseswhich are used later in the book but only in other exercises have not beenmarked with a ‘�’.

1

The local cohomology functors

The main objective of this chapter is to introduce the a-torsion functor Γa

(throughout the book, a always denotes an ideal in a (non-trivial) commutativeNoetherian ring R) and its right derived functors Hi

a (i ≥ 0), referred to as thelocal cohomology functors with respect to a. We shall see that Γa is naturallyequivalent to the functor lim

−→n∈N

HomR(R/an, • ) and, indeed, that Hi

a is natu-

rally equivalent to the functor lim−→n∈N

ExtiR(R/an, • ) for each i ≥ 0; moreover,

as Γa turns out to be left exact, the functors Γa andH0a are naturally equivalent.

This chapter also serves notice that our approach is based on fundamentaltechniques of homological commutative algebra, such as ones based on con-nected sequences of functors (see [71, pp. 212–214]): readers familiar withsuch ideas, and with the local cohomology functors, might like to just glancethrough this chapter and to move rapidly on to Chapter 2.

1.1 Torsion functors

1.1.1 Definition. For each R-module M , set Γa(M) =⋃

n∈N(0 :M an),

the set of elements of M which are annihilated by some power of a. Note thatΓa(M) is a submodule of M . For a homomorphism f : M −→ N of R-modules, we have f(Γa(M)) ⊆ Γa(N), and so there is a mapping Γa(f) :

Γa(M) −→ Γa(N) which agrees with f on each element of Γa(M).It is clear that, if g : M → N and h : N → L are further homomorphisms

of R-modules and r ∈ R, then Γa(h ◦ f) = Γa(h) ◦ Γa(f), Γa(f + g) =

Γa(f) + Γa(g), Γa(rf) = rΓa(f) and Γa(IdM ) = IdΓa(M). Thus, with theseassignments, Γa becomes a covariant, R-linear functor from C(R) to itself.(We say that a functor T : C(R) −→ C(R) is R-linear precisely when it is

2 The local cohomology functors

additive and T (rf) = rT (f) for all r ∈ R and all homomorphisms f ofR-modules.) We call Γa the a-torsion functor.

1.1.2 �Exercise. Let b be a second ideal of R. Show that

Γa(Γb(M)) = Γa+b(M)

for each R-module M .

1.1.3 �Exercise. Let b be a second ideal ofR. Show that Γa = Γb if and onlyif√a =

√b.

(The notation �, attached to some exercises, is explained in the section of‘Notation and conventions’ following the Preface to the Second Edition.)

1.1.4 Exercise. Suppose that the ideal b of R is a reduction of a; that is,b ⊆ a and there exists s ∈ N such that bas = as+1. Show that Γa = Γb.

1.1.5 Exercise. For a prime number p, find ΓpZ(Q/Z).

1.1.6 Lemma. The a-torsion functor Γa : C(R) −→ C(R) is left exact.

Proof. Let 0 −→ Lf−→ M

g−→ N −→ 0 be an exact sequence of R-modules and R-homomorphisms. We must show that

0 � � �Γa(L)Γa(f)

Γa(M)Γa(g)

Γa(N)

is still exact. It is clear that Γa(f) is a monomorphism and it follows immedi-ately from 1.1.1 that Γa(g) ◦ Γa(f) = 0, so that

Im(Γa(f)) ⊆ Ker(Γa(g)).

To prove the reverse inclusion, letm ∈ Ker(Γa(g)). Thusm ∈ Γa(M), so thatthere exists n ∈ N such that anm = 0, and g(m) = 0. Now there exists l ∈ Lsuch that f(l) = m, and our proof will be complete if we show that l ∈ Γa(L).To achieve this, note that, for each r ∈ an, we have f(rl) = rf(l) = rm = 0,so that rl = 0 because f is a monomorphism. Hence anl = 0.

The result of Lemma 1.1.6 will become transparent to many readers oncewe have covered a little more theory, and related the a-torsion functor Γa to afunctor defined in terms of direct limits of ‘Hom’ modules. However, beforewe proceed in that direction, we are going to introduce, at this early stage, thefundamental definition of the local cohomology modules of an R-module Mwith respect to a.

1.2 Local cohomology modules 3

1.2 Local cohomology modules

1.2.1 Definitions. For i ∈ N0, the i-th right derived functor of Γa is denotedby Hi

a and will be referred to as the i-th local cohomology functor with respectto a.

For anR-moduleM , we shall refer toHia(M), that is, the result of applying

the functor Hia to M , as the i-th local cohomology module of M with respect

to a, and to Γa(M) as the a-torsion submodule of M . We shall say that M isa-torsion-free precisely when Γa(M) = 0, and that M is a-torsion preciselywhen Γa(M) =M , that is, if and only if each element of M is annihilated bysome power of a.

It is probably appropriate for us to stress the implications of the above def-inition at this point, and list some basic properties of the local cohomologymodules.

1.2.2 Properties of local cohomology modules. Let M be an arbitrary R-module.

(i) To calculate Hia(M), one proceeds as follows. Take an injective reso-

lution

I• : 0d−1

−→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

of M , so that there is an R-homomorphism α : M −→ I0 such that thesequence

0 −→Mα−→ I0

d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

is exact. Apply the functor Γa to the complex I• to obtain

0 � � � � �Γa(d0)

Γa(I0) · · · Γa(I

i)Γa(d

i)Γa(I

i+1) · · ·

and take the i-th cohomology module of this complex; the result,

Ker(Γa(di))/ Im(Γa(d

i−1)),

which, by a standard fact of homological algebra, is independent (up to R-isomorphism) of the choice of injective resolution I• of M , is Hi

a(M).(ii) Since Γa is covariant and R-linear, it is automatic that each local co-

homology functor Hia (i ∈ N0) is again covariant and R-linear.

(iii) Since Γa is left exact, H0a is naturally equivalent to Γa. Thus, loosely,

we can use this natural equivalence to identify these two functors.(iv) The reader should be aware of the long exact sequence of local coho-

mology modules which results from a short exact sequence of R-modules andR-homomorphisms, and so we spell out the details here.


Let 0 −→ Lf−→M

g−→ N −→ 0 be an exact sequence of R-modules andR-homomorphisms. Then, for each i ∈ N0, there is a connecting homomor-phism Hi

a(N) → Hi+1a (L), and these connecting homomorphisms make the

resulting long sequence

0 H0a(L)

H0a(f)

H0a(M)

H0a(g)

H0a(N)

H1a(L)

H1a(f)

H1a(M)

H1a(g)

H1a(N)

· · · · · ·

Hia(L)

Hia(f)

Hia(M)

Hia(g)

Hia(N)

Hi+1a (L) · · ·

� � �

� � �

�

� � �

� �

exact. The reader should also be aware of the ‘natural’ or ‘functorial’ propertiesof these long exact sequences: if

0 L M N 0� � � �

λ μ ν

0 L′ M ′ N ′ 0

f g

f ′ g′� � � �

� � �

is a commutative diagram of R-modules and R-homomorphisms with exactrows, then, for each i ∈ N0, we not only have a commutative diagram

Hia(L)

Hia(f)

Hia(M)

Hia(g)

Hia(N)� �

Hia(λ) Hi

a(μ) Hia(ν)

Hia(L

′)Hi

a(f′)

Hia(M

′)Hi

a(g′)

Hia(N

′)� ��

(simply because Hia is a functor!), but we also have a commutative diagram

Hia(N) Hi+1

a (L)�

Hia(ν) Hi+1

a (λ)

Hia(N

′) Hi+1a (L′)�

� �

in which the horizontal maps are the appropriate connecting homomorphisms.


The following remark will be used frequently in applications. It is an easyconsequence of Exercise 1.1.3 and the definition of local cohomology functorsin 1.2.1.

1.2.3 Remark. Let b be a second ideal of R such that√a =

√b. Then

Hia = Hi

b for all i ∈ N0, so that Hia(M) = Hi

b(M) for each R-module Mand all i ∈ N0.

The next four exercises might help the reader to consolidate the properties oflocal cohomology modules listed in 1.2.2. The first three of these exercises (forwhich non-trivial results from commutative algebra about injective dimensionover the relevant rings are very helpful) give a tiny foretaste of results aboutthe vanishing of local cohomology modules which are central to the subject,and which will feature prominently later in the book.

1.2.4 Exercise. Show that, for every Abelian group (that is, Z-module) Gand for every a ∈ Z, we have Hi

Za(G) = 0 for all i ≥ 2.

1.2.5 Exercise. Suppose that (R,m) is a regular local ring of dimension d.Show that, for each R-module M , we have Hi

a(M) = 0 for all i > d.

1.2.6 Exercise. Suppose that (R,m) is a Gorenstein local ring (see, for ex-ample, Matsumura [50, p. 142]) of dimension d. Show that, for each finitelygenerated R-module M of finite projective dimension, we have Hi

a(M) = 0

for all i > d. (Here is a hint: use the fact [50, Theorem 18.1] that the injectivedimension of R as an R-module is d, and then use induction on the projectivedimension of M .)

The next exercise investigates the behaviour of local cohomology modulesunder fraction formation: its results show that, speaking loosely, the local co-homology functors ‘commute’ with fraction formation. This is a fundamentalfact in the subject; however, we shall actually derive it as an immediate conse-quence of a more general result in Chapter 4 concerning the behaviour of localcohomology under flat base change (and we shall not make use of it until afterChapter 4). Nevertheless, even at this early stage, its proof should not presentmuch difficulty for a reader familiar with the fact (proved in 10.1.14) that, if Iis an injective R-module and S is a multiplicatively closed subset of R, thenS−1I is an injective S−1R-module.

1.2.7 Exercise. LetM be anR-module and let S be a multiplicatively closedsubset of R. Show that S−1(Γa(M)) = ΓaS−1R(S

−1M), and that, for alli ∈ N0, there is an isomorphism of S−1R-modules

S−1(Hia(M)) ∼= Hi

aS−1R(S−1M).


It is now time for us to relate the a-torsion functor Γa to a functor defined interms of direct limits of ‘Hom’ modules. Fundamental to the discussion is thenatural isomorphism, for an R-module M and n ∈ N,

φ := φan,M : HomR(R/an,M)

∼=−→ (0 :M an)

for which φ(f) = f(1+ an) for all f ∈ HomR(R/an,M). In fact, we are go-

ing to put the various φan,M (n ∈ N) together to obtain a natural isomorphismlim−→n∈N

HomR(R/an,M)

∼=−→ Γa(M), but before we do this it might be helpful

to the reader if we give some general considerations about functors and directlimits, as the principles involved will be used numerous times in this book.

1.2.8 Remarks. Let (Λ,≤) be a (non-empty) directed partially ordered set,and suppose that we are given an inverse system of R-modules (Wα)α∈Λ overΛ, with constituent R-homomorphisms hαβ : Wα → Wβ (for each (α, β) ∈Λ × Λ with α ≥ β). Let T : C(R) × C(R) → C(R) be an R-linear functorof two variables which is contravariant in the first variable and covariant in thesecond. (A functor U : C(R)×C(R) −→ C(R) is said to beR-linear preciselywhen it is additive and U(rf, g) = rU(f, g) = U(f, rg) for all r ∈ R and allhomomorphisms f, g of R-modules.) We show now how these data give riseto a covariant, R-linear functor

lim−→α∈Λ

T (Wα, • ) : C(R) −→ C(R).

Let M,N be R-modules and let f : M −→ N be an R-homomorphism.For α, β ∈ Λ with α ≥ β, the homomorphism hαβ : Wα −→ Wβ induces anR-homomorphism

T (hαβ ,M) : T (Wβ ,M) −→ T (Wα,M),

and the fact that T is a functor ensures that the T (hαβ ,M) turn the family(T (Wα,M))α∈Λ into a direct system of R-modules and R-homomorphismsover Λ. We may therefore form lim

−→α∈Λ

T (Wα,M). Moreover, again for α, β ∈ Λ

with α ≥ β, we have a commutative diagram

T (Wβ ,M)T (hα

β ,M)T (Wα,M)�

T (Wβ ,f) T (Wα,f)

T (Wβ , N)T (hα

β ,N)T (Wα, N) ;�

� �


therefore the T (Wα, f) (α ∈ Λ) constitute a morphism of direct systems andso induce an R-homomorphism

lim−→α∈Λ

T (Wα, f) : lim−→α∈Λ

T (Wα,M) −→ lim−→α∈Λ

T (Wα, N).

It is now straightforward to check that, in this way, lim−→α∈Λ

T (Wα, • ) becomes

a covariant, R-linear functor from C(R) to itself. Observe that, since passageto direct limits preserves exactness, if T is left exact, then so too is this newfunctor.

1.2.9 Examples. Here we present some examples that are central for oursubject.

(i) Probably the most important examples for us of the ideas of 1.2.8 con-cern the case where we take for Λ the set N of positive integers with itsusual ordering and the inverse system (R/an)n∈N of R-modules underthe natural homomorphisms hnm : R/an → R/am (for n,m ∈ N withn ≥ m) (in such circumstances, an ⊆ am, of course). In this way, weobtain covariant, R-linear functors

lim−→n∈N

HomR(R/an, • ) and lim

−→n∈N

ExtiR(R/an, • ) (i ∈ N0)

from C(R) to itself. Of course, the natural equivalence between the leftexact functors HomR and Ext0R leads to a natural equivalence betweenthe left exact functors

lim−→n∈N

HomR(R/an, • ) and lim

−→n∈N

Ext0R(R/an, • )

which we shall use without further comment.(ii) Very similar considerations, this time based on the inclusion maps an →

am (for n,m ∈ N with n ≥ m), lead to functors (which are againcovariant and R-linear)

lim−→n∈N

HomR(an, • ) and lim

−→n∈N

ExtiR(an, • ) (i ∈ N0)

from C(R) to itself, and a natural equivalence between the left exactfunctors

lim−→n∈N


−→n∈N

Ext0R(an, • ).

These functors will be considered in detail in Chapter 2.


It will be convenient for us to consider situations slightly more general thanthat studied in 1.2.9(i) above.

1.2.10 Definition and Example. Let (Λ,≤) be a (non-empty) directed par-tially ordered set. By an inverse family of ideals (ofR) over Λ, we mean a fam-ily (bα)α∈Λ of ideals of R such that, whenever (α, β) ∈ Λ × Λ with α ≥ β,we have bα ⊆ bβ .

For example, if

b1 ⊇ b2 ⊇ · · · ⊇ bn ⊇ bn+1 ⊇ · · ·

is a descending chain of ideals ofR, then (bn)n∈N is an inverse family of idealsover N (with its usual ordering). In particular, the family (an)n∈N is an inversefamily of ideals over N.

Let (bα)α∈Λ be an inverse family of ideals of R over Λ. Then the naturalR-homomorphisms hαβ : R/bα → R/bβ (for α, β ∈ Λ with α ≥ β) turn(R/bα)α∈Λ into an inverse system over Λ, and so we can apply the ideas of1.2.8 to produce covariant, R-linear functors

lim−→α∈Λ

HomR(R/bα, • ) and lim−→α∈Λ

ExtiR(R/bα, • ) (i ∈ N0)

(from C(R) to itself), the first two of which are left exact and naturally equiv-alent.

1.2.11 Theorem. Let B = (bα)α∈Λ be an inverse family of ideals of R overΛ, as in 1.2.10.

(i) There is a covariant,R-linear functor ΓB : C(R)→ C(R) which is suchthat, for an R-module M ,

ΓB(M) =⋃α∈Λ

(0 :M bα),

and, for a homomorphism f : M −→ N of R-modules, ΓB(f) :

ΓB(M) −→ ΓB(N) is just the restriction of f to ΓB(M).(ii) There is a natural equivalence

φ′ (= φ′B) : lim−→α∈Λ

HomR(R/bα, • )∼=−→ ΓB

(of functors from C(R) to itself) which is such that, for an R-moduleM and α ∈ Λ, the image under φ′M of the natural image of an h ∈HomR(R/bα,M) is h(1 + bα). Consequently, ΓB is left exact.


(iii) In particular, there is a natural equivalence

φ0 (= φ0a) : lim−→n∈N

HomR(R/an, • )

∼=−→ Γa

which is such that, for an R-module M and n ∈ N, the image under φ0Mof the natural image of an h ∈ HomR(R/a

n,M) is h(1 + an).

Proof. (i) This can be proved by straightforward modification of the ideas of1.1.1, and so will be left to the reader.

(ii) Let f : M −→ N be a homomorphism of R-modules. For each α ∈ Λ,let φbα,M : HomR(R/bα,M) −→ (0 :M bα) be the R-isomorphism forwhich φbα,M (h) = h(1 + bα) for all h ∈ HomR(R/bα,M). Let α, β ∈ Λ

with α ≥ β, and let hαβ : R/bα → R/bβ be as in 1.2.10. Since the diagram

HomR(R/bβ ,M)φbβ,M

∼= (0 :M bβ)�

HomR(hαβ ,M)

HomR(R/bα,M)φbα,M

∼= (0 :M bα)��

(in which the right-hand vertical map is inclusion) commutes, it follows thatthere is indeed an R-isomorphism

φ′M : lim−→α∈Λ

HomR(R/bα,M)∼=−→ ΓB(M) =

⋃α∈Λ

(0 :M bα)

as described in the statement of the theorem. It is easy to check that the diagram

lim HomR(R/bα,M)−→α∈Λ

φ′M

∼= ΓB(M)�

lim HomR(R/bα,f)−→α∈Λ

ΓB(f)

lim HomR(R/bα, N)−→α∈Λ

φ′N

∼= ΓB(N)��

commutes, and the final claim is then immediate from 1.2.10.(iii) This is immediate from (ii), since when we apply (ii) to the family of

ideals B := (an)n∈N, the functor ΓB of (i) is just the a-torsion functor Γa.

We commented earlier that it would in time become transparent that Γa isleft exact: we had 1.2.11 in mind when we made that comment.

1.2.12 �Exercise. Provide a proof for part (i) of 1.2.11.


1.3 Connected sequences of functors

In this section, we are going to use the concepts of ‘connected sequence offunctors’ and ‘strongly connected sequence of functors’. These are explainedon p. 212 of Rotman’s book [71]. For the reader’s convenience, we recall hererelevant definitions in the case of negative connected sequences, as we shall beparticularly concerned with this case.

1.3.1 Definition. Let R′ be a commutative ring.A sequence (T i)i∈N0 of covariant functors from C(R) to C(R′) is said to

be a negative connected sequence (respectively, a negative strongly connectedsequence) if the following conditions are satisfied.

(i) Whenever 0 −→ Lf−→M

g−→ N −→ 0 is an exact sequence in C(R),there are defined connecting R′-homomorphisms

T i(N) −→ T i+1(L) for all i ∈ N0

such that the long sequence

0 T 0(L)T 0(f)

T 0(M)T 0(g)

T 0(N)

T 1(L)T 1(f)

T 1(M)T 1(g)

T 1(N)

· · · · · ·

T i(L)T i(f)

T i(M)T i(g)

T i(N)

T i+1(L) · · ·

� � �

� � �

�

� � �

� �

is a complex (respectively, is exact).(ii) Whenever

0 L M N 0� � � �

λ μ ν

0 L′ M ′ N ′ 0� � � ��

is a commutative diagram of R-modules and R-homomorphisms withexact rows, then there is induced, by λ, μ and ν, a chain map of the longcomplex of (i) for the top row into the corresponding long complex forthe bottom row.

It might help if we remind the reader of the convention regarding the rais-ing and lowering of indices in a situation such as that of 1.3.1, under which

1.3 Connected sequences of functors 11

T i would be written as T−i: with this convention, (T i)i≥0 can be written as(Tj)j≤0.

We also point out that, if T : C(R)→ C(R′) is an additive covariant functor,such as Γa, then its sequence of right derived functors (RiT )i∈N0 is a nega-tive strongly connected sequence of covariant functors from C(R) to C(R′);furthermore, if T is left exact, thenR0T is naturally equivalent to T . We shallbe concerned so often with left exact, additive, covariant functors that it willconsiderably simplify the exposition if we adopt now the following conventionwhich will be in force for the rest of the book.

1.3.2 Convention. Whenever R′ is a commutative ring and T : C(R) −→C(R′) is a covariant, additive functor which is left exact, then we shall identifyT with its 0-th right derived functor R0T in the natural way. Likewise, weshall identify Ext0R with HomR in the natural way.

1.3.3 Definition. Let R′ be a commutative ring, and let (T i)i∈N0 , (U i)i∈N0

be negative connected sequences of covariant functors from C(R) to C(R′). Ahomomorphism Ψ : (T i)i∈N0 −→ (U i)i∈N0 of connected sequences is a family(ψi)i∈N0 where, for each i ∈ N0, ψi : T i → U i is a natural transformation offunctors, and which is such that the following condition is satisfied: whenever0 −→ L −→ M −→ N −→ 0 is an exact sequence of R-modules andR-homomorphisms, then, for each i ∈ N0, the diagram

T i(N) T i+1(L)�

ψiN ψi+1

L

U i(N) U i+1(L)��

(in which the horizontal maps are the appropriate connecting homomorphismsarising from the connected sequences) commutes.

A homomorphism Ψ = (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 of connectedsequences is said to be an isomorphism (of connected sequences) preciselywhen ψi : T i → U i is a natural equivalence of functors for each i ∈ N0.

We hope the reader is sufficiently adept at techniques similar to those onpp. 212–214 of [71] to find the following exercise straightforward; if not, heor she might like to study Theorem 10 (and its Corollary) of Section 6.5 ofNorthcott [60], which together provide a solution.

1.3.4 �Exercise. LetR′ be a commutative ring, and let (T i)i∈N0 , (U i)i∈N0 betwo negative connected sequences of covariant functors from C(R) to C(R′).


(i) Let ψ0 : T 0 → U0 be a natural transformation of functors. Assume that

(a) the sequence (T i)i∈N0 is strongly connected, and(b) T i(I) = 0 for all i ∈ N and all injective R-modules I .

Show that there exist uniquely determined natural transformations

ψi : T i → U i (i ∈ N)

such that (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 is a homomorphism ofconnected sequences.

(ii) Let ψ : T 0 → U0 be a natural equivalence of functors. Assume that

(a) the sequence (T i)i∈N0 is strongly connected,(b) the sequence (U i)i∈N0 is strongly connected, and(c) T i(I) = U i(I) = 0 for all i ∈ N and all injective R-modules I .

By part (i), there is a unique homomorphism of connected sequencesΨ := (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 for which ψ0 = ψ. Show thatΨ is actually an isomorphism of connected sequences.

We shall not state explicitly the analogues of 1.3.1, 1.3.3 and 1.3.4 for pos-itive connected sequences, but we warn the reader now that we shall use suchanalogues in Chapters 11 and 12.

The following consequence of 1.3.4(ii) essentially provides a characteriza-tion of the right derived functors of a left exact, additive, covariant functorfrom C(R) to C(R′), where R′ is a commutative ring.

1.3.5 Theorem. Let R′ be a commutative ring, and let T be a left exact,additive, covariant functor from C(R) to C(R′). Let (T i)i∈N0 be a negativestrongly connected sequence of covariant functors from C(R) to C(R′) suchthat there exists a natural equivalence ψ : T 0

∼=−→ T and such that T i(I) = 0

for all i ∈ N and all injective R-modules I .Then there is a unique isomorphism of connected sequences

Ψ = (ψi)i∈N0 : (T i)i∈N0

∼=−→ (RiT )i∈N0

(of functors from C(R) to C(R′)) such that ψ0 = ψ. (Of course, we are em-ploying Convention 1.3.2.)

The next exercise strengthens Exercise 1.2.7.

1.3.6 Exercise. Let S be a multiplicatively closed subset of R. Show that(S−1(Hi

a( • )))i∈N0

and(Hi

aS−1R(S−1( • ))

)i∈N0

are isomorphic connected sequences of functors (from C(R) to C(S−1R)).


1.3.7 Remarks. Let B = (bα)α∈Λ be an inverse family of ideals of R overΛ, as in 1.2.10.

Let us temporarily write U i := lim−→α∈Λ

ExtiR(R/bα, • ) for i ∈ N0. These

functors were introduced in 1.2.10. We are going to show now how they fittogether into a negative strongly connected sequence of functors (from C(R)to itself).

First of all, whenever 0 −→ L −→ M −→ N −→ 0 is an exact sequenceof R-modules and R-homomorphisms, there are induced, for each α ∈ Λ,connecting homomorphisms

ExtiR(R/bα, N) −→ Exti+1R (R/bα, L) (i ∈ N0)

which make the induced long sequence

0 HomR(R/bα, L) HomR(R/bα,M) HomR(R/bα, N)

Ext1R(R/bα, L) Ext1R(R/bα,M) Ext1R(R/bα, N)

· · · · · ·

ExtiR(R/bα, L) ExtiR(R/bα,M) ExtiR(R/bα, N)

Exti+1R (R/bα, L) · · ·

� � �

� � �

�

� � �

� �

exact. Moreover, these connecting homomorphisms are such that, for α, β ∈ Λ

with α ≥ β, the diagram

ExtiR(R/bβ , N) Exti+1R (R/bβ , L)�

ExtiR(hαβ ,N) Exti+1

R (hαβ ,L)

ExtiR(R/bα, N) Exti+1R (R/bα, L)�

� �

(in which the horizontal maps are the appropriate connecting homomorphismsand hαβ : R/bα → R/bβ is the natural homomorphism) commutes for eachi ∈ N0. It follows that these diagrams induce ‘connecting’ R-homomorphisms

U i(N) = lim−→α∈Λ

ExtiR(R/bα, N) −→ U i+1(L) = lim−→α∈Λ

Exti+1R (R/bα, L)

(for i ∈ N0); moreover, the fact that passage to direct limits preserves exactness


ensures that the resulting long sequence

0 U0(L) U0(M) U0(N)

U1(L) U1(M) U1(N)

· · · · · ·

U i(L) U i(M) U i(N)

U i+1(L) · · ·

� � �

� � �

�

� � �

� �

is exact. Next, standard properties of the extension functors ensure that, when-ever

0 L M N 0� � � �

λ μ ν

0 L′ M ′ N ′ 0� � � ��

is a commutative diagram of R-modules and R-homomorphisms with exactrows, then, for all α ∈ Λ, the diagram

ExtiR(R/bα, N) Exti+1R (R/bα, L)�

ExtiR(R/bα,ν) Exti+1R (R/bα,λ)

ExtiR(R/bα, N′) Exti+1

R (R/bα, L′)�

� �

(in which the horizontal maps are the appropriate connecting homomorphisms)commutes for each i ∈ N0. It therefore follows that the diagram

lim ExtiR(R/bα, N)−→α∈Λ

lim Exti+1R (R/bα, L)−→

α∈Λ

�

lim ExtiR(R/bα,ν)−→α∈Λ

lim Exti+1R (R/bα,λ)−→

α∈Λ

lim ExtiR(R/bα, N′)−→

α∈Λlim Exti+1

R (R/bα, L′)

−→α∈Λ

��

(in which the horizontal maps are again the appropriate connecting homomor-phisms) commutes for all i ∈ N0.


We have thus made

(lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

into a negative strongly

connected sequence of covariant functors from C(R) to C(R). Since we havelim−→α∈Λ

ExtiR(R/bα, I) = 0 for all i ∈ N whenever I is an injective R-module,

it now follows from 1.3.5 that there is a unique isomorphism of connectedsequences

Ψ =(ψi)i∈N0

:

(lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

∼=−→(RiΓB

)i∈N0

for which ψ0 is the natural equivalence φ′B of 1.2.11(ii); furthermore, boththese connected sequences are isomorphic to the negative (strongly) connectedsequence of functors formed by the right derived functors of

lim−→α∈Λ

HomR(R/bα, • ).

A special case of 1.3.7 describes local cohomology modules as direct limitsof Ext modules. As this description is of crucial importance for our subject,we state it separately.

1.3.8 Theorem. There is a unique isomorphism of connected sequences (offunctors from C(R) to C(R))

Φa =(φia

)i∈N0

:

(lim−→n∈N

ExtiR(R/an, • )

)i∈N0

∼=−→(Hi

a

)i∈N0

which extends the natural equivalence φ0a : lim−→n∈N

HomR(R/an, • )

∼=−→ Γa of

1.2.11(iii). Consequently, for each R-module M and each i ∈ N0,

Hia(M) ∼= lim

−→n∈N

ExtiR(R/an,M).

1.3.9 �Exercise. Let M be an R-module, not necessarily finitely generated.Let a1, . . . , an be an M -sequence (of elements of R) (see [50, p. 123]). Leta′1 ∈ R. Show that

(i) if a′1, a2, . . . , an is also an M -sequence, then so too is a1a′1, a2, . . . , an;(ii) if h1, . . . , hn ∈ N, then ah1

1 , . . . , ahnn is also an M -sequence; and

(iii) if a1, . . . , an all belong to a, then ExtiR(R/a,M) = Hia(M) = 0 for all

i = 0, . . . , n− 1. (This theme will be pursued in Chapter 6.)

2

Torsion modules and ideal transforms

The first section of this chapter contains the essence of a useful reduction tech-nique in the study of local cohomology modules of finitely generated modules.The main points are these: if M is a finitely generated R-module, then it turnsout that the R-module M := M/Γa(M) is a-torsion-free, and that a containsa non-zerodivisor r onM ; moreover, for i > 0, the local cohomology modulesHi

a(M) and Hia(M) are isomorphic, so that the study of these higher local co-

homology modules of M with respect to a can be reduced to the study of thecorresponding local cohomology modules of M ; the advantage of this is thatthe exact sequence 0 −→ M

r−→ M −→ M/rM −→ 0 provides a route tofurther progress. There are several places later in the book where this strategyis used.

In the second section of this chapter, we develop the basic theory of thefunctorDa := lim

−→n∈N

HomR(an, • ) which was mentioned in 1.2.9(ii). For anR-

module M , the module Da(M) is called the a-transform of M , and we plan toshow that such transforms provide a powerful algebraic tool. The use of theseideal transforms is an important part of our approach to local cohomology; weshow that they have a certain universal property, and that universal propertywill help us with many technical details later in the book. In §2.2, we shallactually develop the theory of the generalized ideal transform functor with re-spect to what we call a system of ideals of R; one example of such a systemof ideals is the family (an)n∈N of powers of a, and the generalized ideal trans-form functor with respect to this system is just the a-transform functor. Ourmotivation for working in this generality is our wish to apply generalized idealtransforms to the theory of S2-ifications in Chapter 12.

Towards the end of the chapter, we show that, in certain cases, ideal trans-forms have geometrical significance: we describe the ring of regular functionson a non-empty open subset of an affine variety V over an algebraically closed

2.1 Torsion modules 17

field as an ideal transform of O(V ), the ring of regular functions on V . Actu-ally, this is only a brief foretaste of what is to come at the end of the book, inChapter 20, where we explore the relationship between ideal transforms andsheaf cohomology.

2.1 Torsion modules

We begin with results concerning a-torsion-free modules and a-torsion mod-ules. Part (i) of our first lemma is related to Exercise 1.3.9(iv).

2.1.1 Lemma. Let M be an R-module.

(i) If a contains a non-zerodivisor on M , then M is a-torsion-free, that is,Γa(M) = 0.

(ii) Assume now thatM is finitely generated. ThenM is a-torsion-free if andonly if a contains a non-zerodivisor on M .

Proof. (i) Let r ∈ a be a non-zerodivisor on M , and let m ∈ Γa(M). Thismeans that there exists n ∈ N with anm = 0. Thus rnm = 0, from which wededuce that m = 0.

(ii) One implication follows from (i), and so we assume that a consists en-tirely of zerodivisors on M . Then a ⊆

⋃p∈AssM p by [81, Corollary 9.36],

and, sinceM is finitely generated, AssM is finite. Hence, by the Prime Avoid-ance Theorem [81, 3.61], a ⊆ p for some p ∈ AssM . Since M has a sub-module whose annihilator is exactly p, it follows that (0 :M a) = 0, so thatΓa(M) = 0. This completes the proof.

2.1.2 Lemma. For anR-moduleM , the moduleM/Γa(M) is a-torsion-free.

Proof. Let m ∈ M be such that the element m + Γa(M) of M/Γa(M) isannihilated by an, where n ∈ N. Our aim is to show that m + Γa(M) = 0,that is, that m ∈ Γa(M).

Now anm ⊆ Γa(M). Since anm is a finitely generated submodule ofΓa(M), and each element of anm is annihilated by some power of a, it followsthat there exists t ∈ N such that atanm = 0. Therefore m ∈ (0 :M an+t) ⊆Γa(M).

2.1.3 Remarks. The following points should be noted.

(i) If M is an a-torsion R-module, that is, if M = Γa(M), then all sub-modules of M and all R-homomorphic images of M are also a-torsion.

18 Torsion modules and ideal transforms

(ii) Consequently, for each R-module L and each i ∈ N0, the i-th localcohomology module Hi

a(L) is an a-torsion R-module. To see this, let

I• : 0 −→ I0 −→ I1 −→ · · · −→ Ii −→ Ii+1 −→ · · ·

be an injective resolution of L; use this in conjunction with 1.2.2(i) tosee that Hi

a(L) is a homomorphic image of a submodule of the a-torsionmodule Γa(I

i); and then appeal to (i).

Our next aim is to show that, ifM is an a-torsionR-module, thenHia(M) =

0 for all i ∈ N. We approach this by first considering the effect of Γa on aninjective R-module.

2.1.4 Proposition. Let I be an injective R-module. Then Γa(I) is also aninjective R-module.

Proof. Let b be an ideal of R, and let h : b −→ Γa(I) be a homomorphismof R-modules. By the Baer Criterion (see [71, Theorem 3.20]), it is enough forus to show that there exists m ∈ Γa(I) such that h(r) = rm for all r ∈ b.

Since I is injective, there exists w ∈ I such that h(r) = rw for all r ∈ b.Since R is Noetherian, h(b) is a finitely generated submodule of Γa(I), andso there exists t ∈ N such that ath(b) = 0. Now h(b) is a submodule ofthe finitely generated R-module Rw, and so, by the Artin–Rees Lemma [50,Theorem 8.5], there exists c ∈ N such that, for all integers n ≥ c,

an(Rw) ∩ h(b) = an−c(ac(Rw) ∩ h(b)).

Hence at+c(Rw) ∩ h(b) ⊆ ath(b) = 0. Consequently, we can define an R-homomorphism h : at+c + b −→ Γa(I) for which h(s + r) = rw for all s ∈at+c and r ∈ b: this follows because, if r1, r2 ∈ b are such that r1−r2 ∈ at+c,then r1w − r2w = (r1 − r2)w ∈ at+c(Rw) ∩ h(b) = 0.

Now use again the fact that I is R-injective to see that there exists m ∈ I

such that h(r) = rm for all r ∈ at+c + b. Since h extends h, the proof will becomplete if we show that m ∈ Γa(I).

To achieve this, just note that, for all s ∈ at+c,

sm = h(s) = h(s+ 0) = 0w = 0,

by definition of h. This completes the proof.

2.1.5 Corollary. Let I be an injective R-module. Then the canonical exactsequence 0 −→ Γa(I) −→ I −→ I/Γa(I) −→ 0 splits.

Proof. This is immediate from the fact, established in 2.1.4, that Γa(I) isinjective.

2.1 Torsion modules 19

2.1.6 Corollary. Let M be an a-torsion R-module. Then there exists an in-jective resolution of M in which each term is an a-torsion R-module.

Proof. First note that, if N is an arbitrary R-module, then there exists aninjective R-module I and an R-monomorphism h : N → I . Application ofthe left exact functor Γa yields a monomorphism Γa(h) : Γa(N) → Γa(I),and Γa(I) is injective by 2.1.4.

If we apply the above paragraph to the a-torsion R-module M , we see thatM can be embedded in an a-torsion injective R-module I0. Suppose, induc-tively, that n ∈ N0 and we have constructed an exact sequence

� � � � �0 M I0 · · · In−1dn−1

In

of R-modules and R-homomorphisms in which I0, . . . , In−1, In are all a-torsion injective R-modules. Let C := Coker dn−1, and note that, by 2.1.3(i),C is an a-torsion module because In is. Apply the first paragraph of this proofto C to deduce that there is an a-torsion injective R-module In+1 and an R-monomorphism g : C → In+1. Let dn : In → In+1 be the composition ofthe natural epimorphism from In to C and g.

This completes the inductive step, and the proof.

2.1.7 Corollary. Let M and N be R-modules such that M is a-torsion. Then

(i) Hia(M) = 0 for all i > 0;

(ii) Hia(Γa(N)) = 0 for all i > 0; and

(iii) the natural map π : N → N/Γa(N) induces isomorphisms Hia(π) :

Hia(N)

∼=−→ Hia(N/Γa(N)) for all i > 0.

Proof. (i) It was remarked in 1.2.2(i) that we can use any injective resolutionof M to calculate (up to isomorphism) the Hi

a(M): by 2.1.6, there is an injec-tive resolution of M in which each term is an a-torsion R-module, and use ofthis shows that Hi

a(M) = 0 for all i > 0.(ii) This is immediate from (i) because Γa(N) is an a-torsion R-module.(iii) This is immediate from (ii) on use of the long exact sequence of local

cohomology modules induced by the short exact sequence

0 −→ Γa(N) −→ Nπ−→ N/Γa(N) −→ 0.

2.1.8 Exercise. Let M be an a-torsion R-module, and let

I• : 0d−1

−→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

be an injective resolution of M . Show that Γa(I•), that is, the complex

0 � � � � �Γa(d−1)

Γa(I0) · · · Γa(I

i)Γa(d

i)Γa(I

i+1) · · · ,


is also an injective resolution of M .

2.1.9 �Exercise. Let b be a second ideal of R, and let M be a b-torsion R-module. Prove that Hi

a+b(M) ∼= Hia(M) for all i ∈ N0.

In Chapter 1, we indicated that we shall, at times, find it convenient to con-sider an inverse family B = (bα)α∈Λ of ideals of R over a (non-empty) di-rected partially ordered set Λ, as in 1.2.10. We were able to produce, for sucha B, a functor ΓB in 1.2.11, and the right derived functors RiΓB (i ∈ N0) ofΓB are generalizations of the local cohomology functors Hi

a (i ∈ N0).However, we cannot expect the theory of ΓB to imitate local cohomology

theory completely unless we impose additional conditions on B. For example,the analogue for ΓB of 2.1.4 is not true in general. However, this difficultydoes not occur for the concept introduced in the next definition.

2.1.10 Definition. Let (Λ,≤) be a (non-empty) directed partially ordered set.A system of ideals (of R) over Λ is an inverse family B = (bα)α∈Λ of idealsof R over Λ in the sense of 1.2.10 with the additional property that, for allα, γ ∈ Λ, there exists δ ∈ Λ such that bδ ⊆ bαbγ . (It is clear that the δ in thiscondition can be chosen so that δ ≥ α and δ ≥ γ, since (Λ,≤) is a directedset and, by 1.2.10, whenever (μ, ν) ∈ Λ× Λ with μ ≥ ν, we have bμ ⊆ bν .)

For such a system of ideals B, we shall denote RiΓB by HiB (for all i ∈

N0). The reader should keep in mind that (an)n∈N is a fundamental example ofa system A of ideals (over N); however, we shall continue to use the notationHi

a of 1.2.1 (rather than HiA).

2.1.11 Examples. Here are some further examples of systems of ideals.

(i) Let A be a non-empty set of ideals of R. Then the set of all productsof finite families of ideals taken from A forms a system of ideals in anobvious way.

In particular, if A is a non-empty multiplicatively closed set of idealsof R, then A itself forms a system of ideals.

(ii) Let n ∈ N0. The height, ht b, of a proper ideal b of R is defined in [50,p. 31] and [81, 15.6]. Interpret the height of the improper idealR ofR as∞. Then {b : b is an ideal of R and ht b ≥ n} forms a system of idealsof R.

2.1.12 �Exercise. Let B be a system of ideals over Λ in the sense of 2.1.10.Let α, β ∈ Λ with α ≥ β. Show that, for each R-module M ,

Γbβ(M) ⊆ Γbα(M) ⊆ ΓB(M) =

⋃δ∈Λ

Γbδ(M).

2.2 Ideal transforms and generalized ideal transforms 21

2.1.13 �Exercise. Let B be a system of ideals over Λ in the sense of 2.1.10.Let M be an R-module. We shall say that M is B-torsion-free precisely whenΓB(M) = 0, and that M is B-torsion precisely when M = ΓB(M).

(i) Show that M/ΓB(M) is B-torsion-free.(ii) Show that, if I is an injective R-module, then ΓB(I) is also an injective

R-module. Deduce that, if N is a B-torsion R-module, then there existsan injective resolution ofN in which each term is a B-torsionR-module,and conclude that Hi

B(N) = 0 for all i > 0.(iii) Prove that the natural epimorphism π : M → M/ΓB(M) induces iso-

morphisms HiB(π) : Hi

B(M)∼=−→ Hi

B(M/ΓB(M)) for all i > 0.

2.1.14 �Exercise. Let M be an R-module. Show that the sets Ass(Γa(M))

and Ass(M/Γa(M)) are disjoint, and that

AssM = Ass(Γa(M)) ∪Ass(M/Γa(M)).

2.2 Ideal transforms and generalized ideal transforms

The principal object of study in this section will be the ideal transform of anR-module M with respect to a. This is defined as follows.

2.2.1 Definitions. In 1.2.9(ii), we constructed covariant, R-linear functors

Da := lim−→n∈N


−→n∈N

ExtiR(an, • ) (i ∈ N0)

from C(R) to itself. We shall refer to Da as the a-transform functor; note that,by 1.2.8, this functor is left exact.

For anR-moduleM , we callDa(M) = lim−→n∈N

HomR(an,M) the ideal trans-

form of M with respect to a, or the a-transform of M .

However, instead of working with the powers of a single ideal a, we aregoing to work in this section in the more general framework of a system ofideals. This generality is motivated by an application that will be presented inChapter 12. The reader should keep in mind, throughout this section, that abasic example of a system of ideals is (an)n∈N.

2.2.2 Notation. Throughout this section, (Λ,≤) will denote a (non-empty)directed partially ordered set, and B = (bα)α∈Λ will denote a system of idealsof R over Λ in the sense of 2.1.10.

The functors ΓB and lim−→α∈Λ

HomR(R/bα, •) were introduced in 1.2.11(i) and


1.2.10; for i ∈ N0, the functor lim−→α∈Λ

ExtiR(R/bα, • ) was discussed in 1.3.7,

where it was shown to be naturally equivalent to RiΓB, the i-th right derivedfunctor of ΓB; in 2.1.10, we agreed to denote the R-linear functor RiΓB byHi

B; we shall refer to this as the i-th generalized local cohomology functorwith respect to B (and we shall use natural extensions of this terminology).We denote by

ΦB =(φiB)i∈N0

:

(lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

∼=−→(Hi

B

)i∈N0

the unique isomorphism of connected sequences for which φ0B is the naturalequivalence φ′B of 1.2.11(ii): see 1.3.7.

2.2.3 Definitions. The functors

DB := lim−→α∈Λ

HomR(bα, • ) and lim−→α∈Λ

ExtiR(bα, • ) (i ∈ N0)

(from C(R) to itself) can be defined using the ideas of 1.2.8 in conjunctionwith the inclusion maps bα → bβ (for α, β ∈ Λ with α ≥ β). We shall refer toDB as the B-transform functor; note that, by 1.2.8, this functor is left exact.

For an R-module M , we call DB(M) = lim−→α∈Λ

HomR(bα,M) the general-

ized ideal transform of M with respect to B, or, alternatively, the B-transformof M .

2.2.4 �Exercise. For i ∈ N0, we use RiDB to denote the i-th right derivedfunctor of DB.

Modify the ideas of 1.3.7 to show that

(lim−→α∈Λ

ExtiR(bα, • )

)i∈N0

is a nega-

tive strongly connected sequence of functors from C(R) to itself. Use 1.3.5 toshow that there is a unique isomorphism of connected sequences (of functorsfrom C(R) to itself)

ΨB =(ψiB

)i∈N0

:(RiDB

)i∈N0

∼=−→(lim−→α∈Λ

ExtiR(bα, • )

)i∈N0

which extends the identity natural equivalence from DB to itself.

2.2.5 �Exercise. Let M be an R-module. For each α ∈ Λ, let

πMα : HomR(bα,M)→ DB(M)

be the natural homomorphism.


(i) Let α, β ∈ Λ, and let f ∈ HomR(bα, R) and g ∈ HomR(bβ , R). SinceB is a system of ideals of R, there exists δ ∈ Λ such that bδ ⊆ bαbβ .Observe that f� bδ

, the restriction of f to bδ , maps bδ into bβ . Show thatg ◦ (f� bδ

) = f ◦ (g� bδ).

(ii) Show that there is a binary operation ∗ onDB(R) which is such that, forf ∈ HomR(bα, R) and g ∈ HomR(bβ , R),

πRα (f) ∗ πR

β (g) = πRδ (g ◦ (f� bδ

))

for any choice of δ ∈ Λ with bδ ⊆ bαbβ ; show further that DB(R) isa commutative ring with identity with respect to its R-module additionand ∗ as multiplication.

(iii) Show that DB(M) has the structure of a DB(R)-module such that, forα, β ∈ Λ and for f ∈ HomR(bα, R) and h ∈ HomR(bβ ,M),

πRα (f)(πMβ (h))= πM

δ (h ◦ (f� bδ))

for any choice of δ ∈ Λ with bδ ⊆ bαbβ .(iv) Show that DB is an additive, left exact, covariant functor from C(R) to

C(DB(R)). Thus all the RiDB (i ∈ N0) can be considered as additivefunctors from C(R) to C(DB(R)).

2.2.6 Theorem. Denote the identity functor on the category C(R) by Id.

(i) There are natural transformations of functors (from C(R) to itself)

ξ (= ξB) : ΓB −→ Id, η (= ηB) : Id −→ DB

ζ0 (= ζ0B) : DB −→ H1B

such that, for each R-module M ,

(a) ξM : ΓB(M) −→M is the inclusion map,(b) for each g ∈ M , ηM (g) is the natural image in DB(M) of the

homomorphism fα,g ∈ HomR(bα,M) given by fα,g(r) = rg forall r ∈ bα (for any α ∈ Λ), and

(c) the sequence

0 −→ ΓB(M)ξM−→M

ηM−→ DB(M)ζ0M−→ H1

B(M) −→ 0

is exact.

(ii) Let i ∈ N, and M be an R-module. For each α ∈ Λ, the connecting


homomorphism βiα,M : ExtiR(bα,M) −→ Exti+1

R (R/bα,M) is an iso-morphism, and passage to the direct limit yields an R-isomorphism

βiM : lim

−→α∈Λ

ExtiR(bα,M)∼=−→ lim

−→α∈Λ

Exti+1R (R/bα,M).

Define γiM : RiDB(M)∼=−→ Hi+1

B (M) by

γiM := φi+1B,M ◦ βi

M ◦ ψiB,M ,

where φi+1B and ψi

B are the natural equivalences of 2.2.2 and 2.2.4 re-spectively. Then, as M varies through the category C(R), the γiM con-stitute a natural equivalence of functors γi : RiDB

∼=−→ Hi+1B .

(iii) For each i ∈ N, set ζi (= ζiB) := (−1)iγi. Then(ζj)j∈N0

:(RjDB

)j∈N0

−→(Hj+1

B

)j∈N0

is the unique homomorphism of connected sequences which extends thenatural transformation ζ0 : DB −→ H1

B of part (i).

Note. When B = (an)n∈N, we shall write ξa, ηa, ζ0a instead of ξB, ηB, ζ0B.

Proof. (i), (ii) Let α, δ ∈ Λ with α ≥ δ; let jαδ : bα −→ bδ be the inclusionmap; and let hαδ : R/bα −→ R/bδ be the natural epimorphism. Also, letM,N

be R-modules and let f :M −→ N be an R-homomorphism.The commutative diagram

0 bα R R/bα 0� � � �

jαδ hαδ

0 bδ R R/bδ 0� � � ��

(in which the rows are the canonical exact sequences) induces a chain map ofthe long exact sequence of Ext•R( • ,M) modules induced by the top row tothat induced by the bottom row. Since R is a projective R-module, and sinceHomR(R,M) is naturally isomorphic to M , we therefore obtain a commuta-tive diagram

0 HomR(R/bδ,M) M HomR(bδ,M) Ext1R(R/bδ,M) 0� � � � �

0 HomR(R/bα,M) M HomR(bα,M) Ext1R(R/bα,M) 0� � � � ��


(in which the rows are exact), and, for each i ∈ N, a commutative diagram

ExtiR(bδ,M) Exti+1R (R/bδ,M)

βiδ,M

∼=�

ExtiR(jαδ ,M) Exti+1R (hα

δ ,M)

ExtiR(bα,M) Exti+1R (R/bα,M) .

βiα,M

∼=�

� �

Now pass to the direct limits, bearing in mind the exactness-preserving proper-ties of this process, and use the natural equivalences φ0B, φ1B of 2.2.2 to obtainan exact sequence of R-modules and R-homomorphisms



B(M) −→ 0

(where ξM and ηM are as described in (a) and (b) of the statement of thetheorem) and, for each i ∈ N, an isomorphism

βiM : lim

−→α∈Λ

ExtiR(bα,M)∼=−→ lim

−→α∈Λ

Exti+1R (R/bα,M).

Moreover, since the diagram

0 HomR(R/bα,M) M HomR(bα,M) Ext1R(R/bα,M) 0� � � � �

0 HomR(R/bα, N) N HomR(bα, N) Ext1R(R/bα, N) 0� � � � ��

f

� �

and, for each i ∈ N, the diagram

ExtiR(bα,M) Exti+1R (R/bα,M)

βiα,M

∼=�

ExtiR(bα, N) Exti+1R (R/bα, N)

βiα,N

∼=�

� �

(in all of which, all the unmarked vertical maps are induced by f : M −→N ), are all commutative, it follows that, as M varies through C(R), the ξM ,ηM , ζ0M and βi

M constitute natural transformations of functors ξ, η, ζ0 and βi

respectively. This completes the proof of parts (i) and (ii).(iii) Of course

(RjDB

)j∈N0

is a negative strongly connected sequence of


covariant functors from C(R) to itself; also,(Hj+1

B

)j∈N0

is a negative con-

nected sequence of covariant functors from C(R) to itself. It is immediatefrom 1.3.4(i) that there is a unique homomorphism of the first of these con-nected sequences to the second which extends the natural transformation ζ0 :

DB −→ H1B of part (i). The fact that this unique homomorphism is actually(

(−1)jγj)j∈N0

follows from Rotman [71, Theorem 11.24], which shows that,for each j ∈ N0, each exact sequence 0 −→ L −→ M −→ N −→ 0 ofR-modules and R-homomorphisms and each α ∈ Λ, the diagram

ExtjR(bα, N) Extj+1R (bα, L)�

Extj+1R (R/bα, N) Extj+2

R (R/bα, L) ,��

in which all the homomorphisms are the obvious connecting homomorphisms,is anticommutative.

2.2.7 Remark. Let the situation be as in 2.2.6. It is immediate from the exactsequence



B(M) −→ 0

of 2.2.6(i)(c) that ηM : M −→ DB(M) is an isomorphism if and only ifΓB(M) = H1

B(M) = 0.

2.2.8 Corollary. Let M be an R-module, not necessarily finitely generated.Assume that a contains an M -sequence of length 2. Then

ηM :M −→ Da(M)

is an isomorphism.

Proof. By Exercise 1.3.9(iv), we have Γa(M) = H1a(M) = 0, and so the

claim is immediate from 2.2.7.

2.2.9 Remark. It follows from 2.2.6 that, for each R-module M , there is anR-monomorphism θM : M/ΓB(M) −→ DB(M), induced by ηM , such thatthe sequence

0 −→M/ΓB(M)θM−→ DB(M)

ζ0M−→ H1

B(M) −→ 0

is exact. As θM is induced by ηM , a precise formula for it can be extractedfrom the statement of 2.2.6. Note also that, as M varies through C(R), the θM


constitute a natural transformation of functors since, whenever f : M −→ N

is a homomorphism of R-modules, the diagram

M/ΓB(M)θM

DB(M)�

f∗ DB(f)

N/ΓB(N)θN

DB(N)��

(in which f∗ denotes the homomorphism induced by f ) commutes.This remark can be particularly helpful when used in conjunction with the

fact (see 2.1.13(iii)) that the canonical epimorphism π : M → M/ΓB(M)

induces isomorphisms HiB(π) : Hi

B(M)∼=−→ Hi

B(M/ΓB(M)) for all i > 0.We use this idea in the proof of 2.2.10 below.

The exact sequence



B(M) −→ 0

of Theorem 2.2.6(i)(c), and the particular case

0 −→ Γa(M)ξM−→M

ηM−→ Da(M)ζ0M−→ H1

a(M) −→ 0

of it, are fundamental. In the next corollary, we present some important appli-cations of such sequences.

2.2.10 Corollary. Let M be an R-module; we use the notation of 2.2.6 and2.2.9. Let π : M −→ M/ΓB(M) be the canonical epimorphism. Then thefollowing hold:

(i) DB(ΓB(M)) = 0;(ii) DB(π) : DB(M) −→ DB(M/ΓB(M)) is an isomorphism;

(iii) DB(ηM ) = ηDB(M) : DB(M) −→ DB(DB(M)) is an isomorphism;(iv) ΓB(DB(M)) = 0 = H1

B(DB(M));(v) Hi

B(ηM ) : HiB(M) −→ Hi

B(DB(M)) is an isomorphism for everyi > 1.

Proof. (i) Since ΓB(M) is a B-torsion R-module, it is enough, in order toprove this part, to show that, if N is a B-torsion R-module, then DB(N) = 0.Now, for such an N , we have H1

B(N) = 0 by 2.1.13(ii), and ξN : ΓB(N) →N is the identity map. Hence the exact sequence of 2.2.6(i)(c) for N reducesto

0 −→ ΓB(N)∼=−→ N −→ DB(N) −→ 0,


and so DB(N) = 0, as required.(ii) By 2.2.3, the functor DB is left exact. Therefore the canonical exact

sequence 0 −→ ΓB(M) −→ Mπ−→ M/ΓB(M) −→ 0 induces an exact

sequence

0 DB(ΓB(M)) DB(M)DB(π)

DB(M/ΓB(M)) R1DB(ΓB(M)).� � � �

Now DB(ΓB(M)) = 0 by (i), while R1DB(ΓB(M)) ∼= H2B(ΓB(M)) by

2.2.6(ii). Since H2B(ΓB(M)) = 0 by 2.1.13(ii), it follows that DB(π) is an

isomorphism.(iii) It is left as an interesting exercise on direct limits for the reader to show

that DB(ηM ) = ηDB(M). We show that DB(ηM ) is an isomorphism.Since ηM = θM ◦ π (where θM is as defined in 2.2.9) and we have al-

ready shown in (ii) that DB(π) is an isomorphism, it is enough for us to showthat DB(θM ) is an isomorphism. Since DB is left exact (by 2.2.3), the exactsequence


ζ0M−→ H1

B(M) −→ 0

of 2.2.9 yields a further exact sequence

0 DB(M/ΓB(M))DB(θM )

DB(DB(M))DB(ζ0

M )DB(H1

B(M)).� � �

But H1B(M) is B-torsion, and so DB(H1

B(M)) = 0 by (i). Hence DB(θM )

is an isomorphism, as required.(iv) This is now immediate from (iii) and 2.2.7.(v) We again use the fact that ηM = θM ◦ π. We already know, from

2.1.13(iii), that HiB(π) is an isomorphism for all i ∈ N. It is therefore enough,

in order to complete the proof, to show thatHiB(θM ) is an isomorphism for all

i > 1.However, for each i > 1, the exact sequence


ζ0M−→ H1

B(M) −→ 0

of 2.2.9 induces an exact sequence

· · · Hi−1B (H1

B(M))

HiB(M/ΓB(M))

HiB(θM )

HiB(DB(M))

HiB(ζ0

M )Hi

B(H1B(M))

· · · ,

�

� ��

�


and since H1B(M) is a B-torsion R-module, it follows from 2.1.13(ii) that

Hi−1B (H1

B(M)) = HiB(H1

B(M)) = 0; hence HiB(θM ) is an isomorphism, as

required.

2.2.11 �Exercise. Complete the proof of 2.2.10(iii). In other words, showthat, in the notation of 2.2.10, DB(ηM ) = ηDB(M).

2.2.12 �Exercise. Recall that, by Exercise 2.2.5, the B-transform DB(R) ofR has the structure of a commutative ring. Show that ηR : R −→ DB(R) is aring homomorphism.

In order to exploit our results on the generalized ideal transform DB(M),we are going to obtain a description of DB(M) in terms of objects which areperhaps more familiar. This work will, of course, apply to the ideal transformDa(M). Towards the end of the section, we shall obtain a particularly simpledescription of Da(M) in the case in which a is principal. One approach tothese results uses the fact that, for an R-module M , the homomorphism ηM :

M −→ DB(M) can be viewed as the solution to a universal problem. Ournext proposition, which uses ideas similar to ones used by K. Suominen in[85, §1], provides the key to this approach. The influence of the ideas of P.Gabriel [21] should also be acknowledged.

2.2.13 Proposition. (See R. Y. Sharp and M. Tousi [82, Lemma 1.3].) Lete : M −→ M ′ be a homomorphism of R-modules such that Ker e andCoker e are both B-torsion. Let ψ :M −→ K be a further homomorphism ofR-modules.

(i) The map DB(e) : DB(M) −→ DB(M ′) is an isomorphism.(ii) There is a unique R-homomorphism ψ′ :M ′ −→ DB(K) such that the

diagram

Me

M ′�

ψ ψ′

KηK

DB(K)��

commutes. In fact, ψ′ = DB(ψ) ◦DB(e)−1 ◦ ηM ′ .(iii) If ψ and ηM ′ : M ′ −→ DB(M ′) are both isomorphisms, then the ho-

momorphism ψ′ of part (ii) is also an isomorphism.

Proof. (i) We shall use the exact sequences

0 −→ Ker eτ−→M

λ−→ Im e −→ 0


and

0 −→ Im eρ−→M ′ σ−→ Coker e −→ 0,

in which the maps are the obvious homomorphisms. Now e = ρ◦λ; it is there-fore enough for us to show that DB(ρ) and DB(λ) are both isomorphisms.

The first of the above exact sequences induces an exact sequence

0 DB(Ker e) DB(M)DB(τ) DB(λ)

DB(Im e) R1DB(Ker e).� � � �

However, R1DB(Ker e) ∼= H2B(Ker e) by 2.2.6(ii). By hypothesis, Ker e is

B-torsion. Hence DB(Ker e) = H2B(Ker e) = 0, by 2.2.10(i) and 2.1.13(ii).

Therefore DB(λ) is an isomorphism.Next, from the exact sequence 0 −→ Im e

ρ−→ M ′ σ−→ Coker e −→ 0 weobtain an induced exact sequence

0 DB(Im e) DB(M ′)DB(ρ) DB(σ)

DB(Coker e).� � �

However, by hypothesis, Coker e is B-torsion, and so DB(Coker e) = 0, by2.2.10(i). Hence DB(ρ) is an isomorphism.

(ii) For this part of the proof, it will be convenient for us to write K ′ :=DB(K) and h := ηK : K −→ DB(K) = K ′. Application of the naturaltransformation η : Id −→ DB to the modules and homomorphisms in thediagram

Me

M ′�

ψ

Kh

K ′��

yields a commutative diagram

M

e

M ′ψ

K

h

K ′

DB(M)

DB(e)∼=

DB(M ′)

DB(ψ)

DB(K)

DB(h)∼=

DB(K ′) .

�

�

�

�

ηM

ηM′ηK

ηK′∼=

� �

��

��

��

��


It should be noted that DB(e) is an isomorphism, by part (i), that DB(K) =

K ′, and that, by 2.2.10(iii),

DB(h) = DB(ηK) = ηDB(K) = ηK′

is also an isomorphism.Thus, if there were anR-homomorphism ψ′ :M ′ −→ K ′ such that ψ′ ◦e =

h◦ψ, then it would satisfyDB(ψ′)◦DB(e) = DB(h)◦DB(ψ) and we wouldhave to have (since η : Id −→ DB is a natural transformation)

ψ′ = DB(h)−1 ◦ ηK′ ◦ ψ′ = DB(h)−1 ◦DB(ψ′) ◦ ηM ′

= DB(ψ) ◦DB(e)−1 ◦ ηM ′ .

On the other hand, one can easily verify by means of an elementary diagramchase that DB(ψ) ◦DB(e)−1 ◦ ηM ′ ◦ e = h ◦ ψ.

(iii) This is immediate from part (ii), since, if ψ is an isomorphism, then sotoo is DB(ψ).

2.2.14 Remark. Let h : M −→ N be a homomorphism of R-modules.Recall from 2.2.6(i) that ηB : Id −→ DB is a natural transformation offunctors and that Ker ηM and Coker ηM are both B-torsion. It therefore fol-lows from 2.2.13 that DB(h) : DB(M) −→ DB(N) must be the uniqueR-homomorphism from DB(M) to DB(N) which makes the diagram

MηM

DB(M)�

h

NηN

DB(N)��

commute.

2.2.15 Corollary. Let e : M −→ M ′ be a homomorphism of R-modulessuch that Ker e and Coker e are both B-torsion.

(i) The map DB(e) : DB(M) −→ DB(M ′) is an isomorphism.(ii) There is a unique R-homomorphism ψ′ : M ′ → DB(M) such that the

diagram

Me

M ′�

ηMψ′

DB(M)

��


commutes. In fact, ψ′ = DB(e)−1 ◦ ηM ′ .(iii) The map ψ′ of (ii) is an isomorphism if and only if ηM ′ is an iso-

morphism, and, by 2.2.7, this is the case if and only if ΓB(M ′) =

H1B(M ′) = 0.

Proof. Use 2.2.13 with IdM :M −→M in the role of ψ :M −→ K.

We are now going to examine the special case of 2.2.15 in which M = R.Recall from 2.2.5 that DB(R) is a commutative ring, and from 2.2.12 thatηR : R −→ DB(R) is a ring homomorphism. We shall meet several exam-ples of the situation of 2.2.15 in which e : R −→ R′ is a homomorphism ofcommutative rings such that, when R′ is regarded as an R-module by meansof e, both Ker e and Coker e are B-torsion: 2.2.17 below shows that, in thesecircumstances, the R-homomorphism ψ′ : R′ −→ DB(R) given by 2.2.15 isactually a ring homomorphism too.

2.2.16 Proposition. Let R′ be a ring (with identity, but not necessarily com-mutative), and let e : R −→ R′ be a ring homomorphism such that Im e is con-tained in the centre ofR′ and, whenR′ is regarded as a leftR-module by meansof e, both Ker e and Coker e are B-torsion. Assume also that ΓB(R′) = 0.Then the ring R′ is commutative.

Proof. By 2.2.15, there is a unique R-homomorphism ψ′ : R′ −→ DB(R)

such that the diagram

Re

R′�

ηRψ′

DB(R)

��

commutes. Let r′1, r′2 ∈ R′. Since Coker e is B-torsion, there exist α, δ ∈ Λ

such that bα (respectively bδ) annihilates the natural image in Coker e of r′1(respectively r′2). Let b1 ∈ bα and b2 ∈ bδ; then there exist r1, r2 ∈ R suchthat bir′i = e(ri), that is e(bi)r′i = e(ri), for i = 1, 2. Therefore, since Im e iscontained in the centre of R′, we have

e(b1)e(b2)r′1r′2 = e(b1)r

′1e(b2)r

′2 = e(r1)e(r2)

= e(r2)e(r1) = e(b2)r′2e(b1)r

′1 = e(b1)e(b2)r

′2r′1.

Therefore b1b2(r′1r′2−r′2r′1) = 0. Hence the element r′1r

′2−r′2r′1 is annihilated

by bαbδ . But B is a system of ideals, so that there exists μ ∈ Λ such that


bμ ⊆ bαbδ . Since ΓB(R′) = 0, we deduce that r′1r′2 − r′2r

′1 = 0 and R′ is

commutative.

2.2.17 Proposition. Let R′ be a commutative ring (with identity), and lete : R −→ R′ be a ring homomorphism for which the R-modules Ker e andCoker e are B-torsion.

Then the unique R-homomorphism ψ′ : R′ −→ DB(R) such that the dia-gram

Re

R′�

ηRψ′

DB(R)

��

commutes (the existence of which follows from 2.2.15) is a ring homomor-phism, and therefore an R-algebra homomorphism.

Proof. Let r′1, r′2 ∈ R′. Since Coker e is B-torsion, there exist α, δ ∈ Λ

such that bα (respectively bδ) annihilates the natural image in Coker e of r′1(respectively r′2). Let b1 ∈ bα and b2 ∈ bδ; then there exist r1, r2 ∈ R suchthat bir′i = e(ri), that is e(bi)r′i = e(ri), for i = 1, 2.

Note also that, in the commutative ring DB(R), we have

b1b2ψ′(r′1)ψ

′(r′2) = ψ′(b1r′1)ψ

′(b2r′2) = ψ′(e(r1))ψ

′(e(r2))

= ηR(r1)ηR(r2) = ηR(r1r2) = ψ′(e(r1r2))

= ψ′(e(r1)e(r2)) = ψ′(b1r′1b2r

′2) = b1b2ψ

′(r′1r′2),

so that b1b2 (ψ′(r′1)ψ′(r′2)− ψ′(r′1r′2)) = 0. Hence the element

ψ′(r′1)ψ′(r′2)− ψ′(r′1r′2) ∈ DB(R)

is annihilated by bαbδ . But there exists μ ∈ Λ such that bμ ⊆ bαbδ . SinceDB(R) is B-torsion-free by 2.2.10(iv), we have ψ′(r′1r

′2) = ψ′(r′1)ψ

′(r′2).Also ψ′(1R′) = ψ′(e(1R)) = ηR(1R) = 1DB(R). Therefore ψ′ is a ringhomomorphism.

2.2.18 Corollary. Let M be an R-module, and let S be a multiplicativelyclosed subset ofR which consists entirely of non-zerodivisors onM , and whichis such that S ∩ bα = ∅ for all α ∈ Λ. Then there is a unique R-isomorphism

ψ′M :⋃α∈Λ

(M :S−1M bα) −→ DB(M)


for which the diagram

M⊆ ⋃

α∈Λ(M :S−1M bα)�

ηMψ′

M

DB(M)

��

commutes. In the special case in which M = R (and S consists of non-zerodivisors on R), the map ψ′R is actually a ring isomorphism.

Note. Of course, since S consists entirely of non-zerodivisors on M , thecanonical R-homomorphism M → S−1M is injective; we are using this toidentify M as an R-submodule of S−1M .

Proof. Set M ′ :=⋃

α∈Λ(M :S−1M bα), and let e : M −→ M ′ denote theinclusion homomorphism. Since Coker e = ΓB(S−1M/M) is B-torsion, it isimmediate from 2.2.15 that there is a unique R-homomorphism ψ′M :M ′ −→DB(M) which makes the above diagram commute; it also follows that, inorder to show that ψ′M is an isomorphism, it is sufficient for us to show thatΓB(M ′) = H1

B(M ′) = 0. This we do.We show first that Hi

B(S−1M) = 0 for each i ∈ N0. Let y ∈ HiB(S−1M).

Then there exists α ∈ Λ such that bαy = 0. By hypothesis, there exists s ∈bα ∩ S, so that sy = 0. Since the functor Hi

B is R-linear, multiplication bys on Hi

B(S−1M) must provide an automorphism; therefore y = 0. ThereforeHi

B(S−1M) = 0, as claimed. Hence ΓB(M ′) = 0. It now follows from theexact sequence

0 −→M ′ −→ S−1M −→ S−1M/M ′ −→ 0

that H1B(M ′) ∼= ΓB(S−1M/M ′). However,

S−1M/M ′ ∼= (S−1M/M)/(M ′/M) = (S−1M/M)/ΓB(S−1M/M),

and this is B-torsion-free by 2.1.13(i). Hence H1B(M ′) = 0. The final claim

follows from 2.2.17.

Of course, all our work so far in this section applies to the particular systemof ideals (an)n∈N, and, indeed, that is a very important example of such asystem. As we have now presented, in this section, enough of the theory ofgeneralized ideal transforms for our later needs, we are going to concentrate,for the remainder of this section, on the ordinary a-transform functor Da.

In the case when a is principal, there is a result similar to 2.2.18, but under


weaker hypotheses, which has important consequences for our work in Chapter3. We present this result next. Recall that, for an R-module M and a ∈ R,the notation Ma denotes the module of fractions of M with respect to themultiplicatively closed subset

{ai : i ∈ N0

}.

2.2.19 Theorem. Let a ∈ R. There is a natural equivalence of functors

ω′ : DRa = lim−→n∈N

HomR(Ran, • ) −→ ( • )a

(from C(R) to C(R)) such that, for an R-module M , and an f ∈ DRa(M)

represented by ft ∈ HomR(Rat,M) (for some t ∈ N), we have ω′M (f) =

ft(at)/at.

Proof. Let M be an R-module. It is immediate from 2.2.15 that there is aunique R-isomorphism νM :Ma −→ DRa(M) such that the diagram

M Ma�

ηRa,MνM∼=

DRa(M)

��

(in which the horizontal homomorphism is the natural one) commutes: notethat Hi

Ra(Ma) = 0 for all i ∈ N0 because multiplication by a provides anautomorphism on all these local cohomology modules. Define ω′M := ν−1

M . Itis straightforward to use the commutativity of the above diagram to show thatω′M satisfies the formula given in the statement of the theorem. Furthermore,it is easy to use that formula to show that, as M varies through the categoryC(R), the ω′M constitute a natural equivalence of functors.

2.2.20 Remark. With the notation of 2.2.19, consider, for an R-module M ,the fundamental exact sequence of 2.2.6(i)(c) in the particular case in whichB = (an)n∈N and a = Ra: we have

0 −→ ΓRa(M)ξM−→M

ηM−→ DRa(M)ζ0M−→ H1

Ra(M) −→ 0.

Note that ω′M ◦ ηM : M −→ Ma is just the canonical R-homomorphism τM .Set σM := ζ0M ◦ (ω′M )−1; then there is an exact sequence of R-modules andR-homomorphisms

0 −→ ΓRa(M)ξM−→M

τM−→MaσM−→ H1

Ra(M) −→ 0


such that, as M varies through C(R), the τM and σM constitute natural trans-formations of functors

τ : Id −→ ( • )a, σ : ( • )a −→ H1Ra.

2.2.21 Corollary. Let M be an R-module and let a ∈ R.

(i) The kernel of the natural homomorphism τM : M → Ma is preciselyΓRa(M), and so, in view of 2.2.20, M/ΓRa(M) can be identified as asubmodule of Ma. With this identification,

H1Ra(M) ∼=Ma/(M/ΓRa(M)).

(ii) For all i ∈ N with i > 1, we have HiRa(M) = 0.

Proof. (i) This is immediate from 2.2.20.(ii) Let i ∈ N with i > 1. Now Hi

Ra(M) ∼= Ri−1DRa(M) by 2.2.6(ii).However, by 2.2.19, the functor DRa is naturally equivalent to ( • )a; as thelatter functor is exact and i − 1 > 0, it follows that Ri−1DRa(N) = 0 for allR-modules N . Hence Hi

Ra(M) = 0.

The above corollary will prove useful in Chapter 3, as it provides a basis foran argument which uses induction on the number of generators of a, and whichrelies on the Mayer–Vietoris Sequence for local cohomology for the inductiveargument. This Mayer–Vietoris Sequence forms a major part of the subjectmatter of Chapter 3.

2.2.22 Exercise. Use 2.2.18 to obtain an alternative proof of Corollary 2.2.8.More precisely, let M be an R-module, not necessarily finitely generated,

and assume that a contains an M -sequence x, y of length 2. Use 2.2.18 (withthe choice S := {xi : i ∈ N0}) and 1.3.9(ii) to show that the map ηM :M −→Da(M) of 2.2.6(i) is an isomorphism.

2.2.23 �Exercise. Let b be a second ideal of R.

(i) Suppose that a ⊆ √b. Show that there is a unique natural transformationof functors (from C(R) to itself) αb,a : Db −→ Da such that, for eachR-module M , the diagram

Mηb,M

Db(M)�

ηa,Mαb,a,M

Da(M)

��

commutes.


(ii) Deduce that, if√a =

√b, then Da and Db are naturally equivalent.

It will be convenient, in our discussion of geometric examples in the nextsection, for us to have available a result which shows that, in a certain sense, theideal transform is ‘independent of the base ring’. To be more precise, consider asecond commutative Noetherian ring R′ and a ring homomorphism f : R −→R′; let M ′ be an R′-module. At times when we wish to be absolutely precise,we shall use M ′� R to indicate that we are regarding M ′ as an R-module bymeans of f . Note that �R can be regarded as a functor from C(R′) to C(R). Wecan form the ideal transform DaR′(M ′) of M ′ with respect to the extensionaR′ of a to R′ via f , and then regard this as an R-module by means of f : thisis, then, the R-module DaR′(M ′)� R. Alternatively, we can regard M ′ as theR-module M ′� R, and form Da(M

′� R). Our next result will show, amongother things, that there is an R-isomorphism

DaR′(M ′) = DaR′(M ′)� R

∼=−→ Da(M′� R) =: Da(M

′),

so that, speaking loosely, it does not matter whether we calculate these idealtransforms overR orR′. The advantage of this in practice is that there is some-times an obvious choice of an ‘R′’ over which the calculations are easy.

2.2.24 Theorem. Let R′ be a second commutative Noetherian ring and letf : R −→ R′ be a ring homomorphism. We use the notation introduced above.

There is a natural equivalence of functors

ε : DaR′( • )� R −→ Da( • � R)

(from C(R′) to C(R)) which is such that, for each R′-module M ′, the diagram

M ′ηaR′,M′

DaR′(M ′)�

εM′

M ′ηa,M′

Da(M′)�

�

commutes.

Proof. Let M ′ be an R′-module. By 2.2.6(i)(c), the R′-homomorphism

ηaR′,M ′ :M ′ −→ DaR′(M ′)

has kernel and cokernel which are, respectively, isomorphic to ΓaR′(M ′) andH1

aR′(M ′). Hence ηaR′,M ′� R has kernel and cokernel which are a-torsion. It


therefore follows from 2.2.15 that there is a unique R-homomorphism εM ′ :

DaR′(M ′) −→ Da(M′) such that the diagram

M ′ηaR′,M′

DaR′(M ′)�

εM′

M ′ηa,M′

Da(M′)�

�

commutes. In fact, it is easy to use the uniqueness aspect of Proposition 2.2.13to show that, as M ′ varies through the category C(R′), the εM ′ constitute anatural transformation of functors, and so it remains only to show that eachεM ′ is an isomorphism.

The fact that M ′ is an (R,R′)-bimodule means that Da(M′) inherits a nat-

ural structure as an R′-module (such that Da(r′ IdM ′), for r′ ∈ R′, provides

multiplication by r′). Furthermore, since ε is a natural transformation of func-tors, we have

Da(r′ IdM ′) ◦ εM ′ = εM ′ ◦DaR′(r′ IdM ′) = εM ′ ◦ (r′ IdDaR′ (M ′))

for all r′ ∈ R′, so that εM ′ is an R′-homomorphism. Likewise, ηa,M ′ becomesan R′-homomorphism.

Another use of 2.2.6(i)(c) shows that ηa,M ′ : M ′ −→ Da(M′) has kernel

and cokernel which are a-torsion, so that, when we consider ηa,M ′ as an R′-homomorphism, its kernel and cokernel are aR′-torsion. It therefore followsfrom 2.2.15 that there is a unique R′-homomorphism λM ′ : Da(M

′) −→DaR′(M ′) such that the diagram

M ′ηa,M′

Da(M′)�

λM′

M ′ηaR′,M′

DaR′(M ′)��

commutes. The uniqueness aspect of 2.2.15, together with the facts that εM ′

and λM ′ are both R- and R′-homomorphisms, now yields that

λM ′ ◦ εM ′ = IdDaR′ (M ′) and εM ′ ◦ λM ′ = IdDa(M ′),

so that εM ′ is an isomorphism.

2.2.25 �Exercise. Let the situation be as in 2.2.24, and consider the natural

2.3 Geometrical significance 39

equivalence ε of that theorem. Show that ε−1R′ ◦Da(f) : Da(R) −→ DaR′(R′)

is the unique ring homomorphism which makes the diagram

Rηa,R

Da(R)�

f

R′ηaR′,R′

DaR′(R′)��

commute. (You might find it helpful to adapt the argument used in the proof of2.2.17.)

2.2.26 Exercise. Let the situation be as in 2.2.24. Show that there is a naturalequivalence of functors

H1aR′( • )� R −→ H1

a( • � R)

(from C(R′) to C(R)).

The result of Exercise 2.2.26 is a particular case of a more general (and veryimportant) result concerning ‘independence of the base ring’ which will beestablished in Chapter 4.

2.3 Geometrical significance

We are now going to show that the ideal transform Da(R) has an importantgeometrical significance in the case when R is the ring of regular functions onan (irreducible) affine algebraic variety over an algebraically closed field. Sev-eral examples in this book will be concerned with affine algebraic geometryover the field of complex numbers C, and it is convenient for us to introducesome notation which will be consistently used in connection with such exam-ples.

2.3.1 Notation. Let K be an algebraically closed field. For n ∈ N, we shalluse An(K) to denote affine n-space over K, that is, Kn endowed with theZariski topology; we shall use An to denote complex affine n-space An(C).All unexplained mentions of topological notions, including ‘open’ and ‘closed’subsets, in connection with affine spaces will refer to the Zariski topology.

By an affine variety over K we shall mean an irreducible closed subset ofAn(K) (with the induced topology), and by a quasi-affine variety over K we


shall mean a non-empty open subset of an affine variety over K (again withthe induced topology).

For a quasi-affine variety U over K, we shall use O(U) to denote the ringof regular functions on U and K(U) to denote the function field of U . Thus,when U is affine, O(U) is just the coordinate ring of U ; this is actually anintegral domain (since U is irreducible), and its field of fractions is just K(U).

We shall regard the polynomial ring K[X1, . . . , Xn] as the coordinate ringO(An(K)) of An(K) in the obvious way (although we shall tend to use X,Yinstead of X1, X2 in the case when n = 2).

For f1, . . . , ft ∈ K[X1, . . . , Xn], we shall use VAn(K)(f1, . . . , ft) to denotethe affine algebraic set

{p ∈ An(K) : f1(p) = · · · = ft(p) = 0}

corresponding to the ideal (f1, . . . , ft) of K[X1, . . . , Xn]. If V denotes thisaffine algebraic set, then, for 1 ≤ i ≤ n, the restriction of the coordinatefunction Xi of An(K) to V will be denoted by Xi� V or xi.

For a quasi-affine variety W over K and a function f ∈ O(W ), we shalluse UW (f) to denote the open subset {p ∈W : f(p) = 0} of W .

We are now going to show that the ring of regular functions on a quasi-affinevariety can be expressed in terms of an ideal transform. Before doing so, weremind the reader of the following point. For a non-empty open subset U of anaffine variety V over the algebraically closed field K, the ring O(U) can beidentified in a natural way as a subring of K(V ), and, when this identificationis made, the restriction homomorphism

� U : O(V ) −→ O(U)

is just the inclusion map.

2.3.2 Theorem. Let V be an affine variety over the algebraically closed fieldK. Let b be a non-zero ideal of O(V ), let V (b) denote the closed subset of Vdetermined by b, and let U be the open subset V \ V (b) of V .

There is a unique O(V )-isomorphism νV,b : O(U)∼=−→ Db(O(V )) for

which the diagram

O(V ) U O(U)�

ηO(V )

νV,b∼=

Db(O(V ))

��


commutes. Furthermore, νV,b is a ring isomorphism.

Proof. In view of 2.2.18, it is enough for us to show that the submodulesO(U) and

⋃n∈N(O(V ) :K(V ) b

n) of K(V ) are equal.Since the ring O(V ) is Noetherian, b is finitely generated, by h1, . . . , ht,

say. We can, and do, assume that h1, . . . , ht are all non-zero. Note that U =⋃ti=1 UV (hi), and that O(UV (hi)) = O(V )hi

when these two rings are iden-tified with subrings of K(V ) in the natural ways.

Let f ∈ O(U). Choose i ∈ N with 1 ≤ i ≤ t. Since f� UV (hi) ∈O(UV (hi)) = O(V )hi , it follows that there exists ni ∈ N such that hni

i f ∈O(V ). We deduce that there exists n ∈ N such that bnf ∈ O(V ). Hence

O(U) ⊆⋃n∈N

(O(V ) :K(V ) bn).

Now let f ∈⋃

n∈N(O(V ) :K(V ) bn). Thus there exists n ∈ N such thatgi := hni f ∈ O(V ) for all i = 1, . . . , t. Let p ∈ U , and let m(p) be themaximal ideal of O(V ) corresponding to p. Now p ∈ UV (hi) for some i with1 ≤ i ≤ t, and since hi(p) = 0, we have hi ∈ O(V ) \m(p) and

f =gihni∈ O(V )m(p) = OV, p,

the local ring of V at p. It follows that f ∈⋂

p∈U OV, p = O(U). Hence

O(U) ⊇⋃n∈N

(O(V ) :K(V ) bn),

and the proof is complete.

2.3.3 Remark. It follows from 2.3.2 that, in the notation of that proposition,the map ηO(V ) : O(V ) −→ Db(O(V )) is an epimorphism if and only if therestriction map �U : O(V ) −→ O(U) is surjective, that is, if and only if everyregular function on U can be extended to a regular function on V . However,by 2.2.6(i)(c), ηO(V ) fails to be an epimorphism if and only ifH1

b(O(V )) = 0.Thus we can, in a sense, regard non-zero elements of the local cohomologymodule H1

b(O(V )) as obstructions to the extension of regular functions on Uto regular functions on V . We shall exploit this observation later in the chapterin our discussion of the geometric significance of ideal transforms and localcohomology modules in particular examples.

The next exercise establishes a certain ‘naturality’ property of the isomor-phisms given by Theorem 2.3.2.

2.3.4 �Exercise. Let β :W −→ V be a morphism of affine varieties over thealgebraically closed field K. Let b be a non-zero ideal of O(V ), and assume


that the extension bO(W ) of b under the induced K-algebra homomorphismβ∗ : O(V ) −→ O(W ) is also non-zero.

We use the notation of Theorem 2.3.2. Since β−1(V (b)) = V (bO(W )), therestriction of β provides a morphism β� : W \ V (bO(W )) −→ V \ V (b) ofquasi-affine varieties.

Use Exercise 2.2.25 to show that, with the notation of that exercise, thediagram

O(V \ V (b))β∗

O(W \ V (bO(W )))�

νV,b νW,bO(W )∼= ∼=

Db(O(V ))ε−1O(W )

◦Db(β∗)

DbO(W )(O(W ))��

of ring homomorphisms commutes.

2.3.5 �Exercise. Let ι : A2 \ {(0, 0)} → A2 denote the inclusion morphismof varieties. Prove that the induced C-algebra homomorphism

ι∗ : O(A2)→ O(A2 \ {(0, 0)})

is an isomorphism, and deduce that the quasi-affine variety A2 \{(0, 0)} is notaffine.

2.3.6 �Exercise. Let K be a field, and let q be a proper ideal of height 2 inthe ring of polynomials K[X,Y ] in the indeterminates X,Y . Set R := K + q,a subring of K[X,Y ]. Observe that q is a maximal ideal of R, and that thesimple R-module R/q is isomorphic to K, where the R-module structure onK is such that fa = 0 for all f ∈ q and all a ∈ K.

(i) Show that the vector space dimension t := dimK(K[X,Y ]/q) is finite,and deduce that there is a monic polynomial in K[X] ∩ q of degree notexceeding t: let pX be such a polynomial of smallest possible degree.Similarly, let pY be a monic polynomial inK[Y ]∩q of smallest possibledegree. Show that K[X,Y ] is a finitely generated R-module.

(ii) Let c1, . . . , ch generate q (as an ideal of K[X,Y ]), let u := deg pX andv := deg pY . Set

S := {ciXjY l : 1 ≤ i ≤ h, 0 ≤ j < u, 0 ≤ l < v} ∪ {pX , pY } ⊂ R.

Show that R = K[S], so that R is a finitely generated K-algebra, andtherefore Noetherian.


(iii) Show that there is a unique R-isomorphism ψ′ : K[X,Y ] −→ Dq(R)


R⊆

K[X,Y ]�

ηRψ′

Dq(R)

��

commutes, and that ψ′ is a ring isomorphism. Deduce that there is anexact sequence 0 −→ R −→ K[X,Y ] −→ H1

q(R) −→ 0 in C(R), andthat dimK(H1

q(R)) = t− 1.(iv) In the special case in which q := XK[X,Y ]+Y (Y −1)K[X,Y ], show

that R = K[X,XY, Y (Y − 1), Y 2(Y − 1)], that q is the maximal ideal(X,XY, Y (Y − 1), Y 2(Y − 1)) of R, and that H1

q(R)∼= K.

(v) In the special case in which q := XK[X,Y ] + Y 2K[X,Y ], show thatR = K[X,Y 2, XY, Y 3], that q is the maximal ideal (X,Y 2, XY, Y 3)

of R, and that H1q(R)

∼= K.

We are now in a position to present one of the geometric examples whichwas promised earlier.

2.3.7 Example. (See R. Hartshorne [28, 3.4.2].) With the notation of 2.3.1,let V be the affine algebraic set in A4 given by

V := VA4(X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

It is easy to check that the morphism of varieties α : A2 → A4 for whichα((c, d)) = (c, cd, d(d− 1), d2(d− 1)) for all (c, d) ∈ A2 satisfies Imα ⊆ V .In fact, Imα = V , because the map β : V → A2 defined by

β((c1, c2, c3, c4)) =

⎧⎪⎪⎨⎪⎪⎩(c1, c2/c1) if c1 = 0,

(c1, c4/c3) if c3 = 0,

(0, 0) if c1 = c3 = 0

(for all (c1, c2, c3, c4) ∈ V ) satisfies α ◦ β = IdV , the identity map on V .Since A2 is irreducible, it follows that V is irreducible, too. We propose

to study the coordinate ring, R := O(V ), of this affine variety. Note that theC-algebra homomorphism

α∗ : O(V ) −→ O(A2) = C[X,Y ]

induced by α is injective (because α is surjective). Also, if ι : V → A4 denotes


the inclusion morphism of varieties, then ι∗ : C[X1, X2, X3, X4] → O(V ) isthe natural surjective C-algebra homomorphism given by restriction. Further-more, Imα∗ = C[X,XY, Y (Y − 1), Y 2(Y − 1)], since

α∗(ι∗(X1)) = X, α∗(ι∗(X2)) = XY,

α∗(ι∗(X3)) = Y (Y − 1), α∗(ι∗(X4)) = Y 2(Y − 1).

It follows from 2.3.6(iv) that, if we use m to denote the maximal ideal ofR corresponding to the point (0, 0, 0, 0) of V , then H1

m(R)∼= C. Note that

the open subset V \ V (m) is just V \ {(0, 0, 0, 0)}. By 2.3.2 and 2.3.3, theremust be a regular function on V \ {(0, 0, 0, 0)} that cannot be extended to aregular function on V . The reader might find it interesting for us to find such afunction.

Observe that

V \ {(0, 0, 0, 0)} = UV (x1) ∪ UV (x3)

and that, onUV (x1)∩UV (x3), the functions x2/x1 and x4/x3 are both definedand are equal. Thus β2 : V \ {(0, 0, 0, 0)} → C defined by

β2((c1, c2, c3, c4)) =

{c2/c1 if c1 = 0,

c4/c3 if c3 = 0

(for all (c1, c2, c3, c4) ∈ V \ {(0, 0, 0, 0)}) is a regular function on the setV \{(0, 0, 0, 0)}. (In fact, the map from V \{(0, 0, 0, 0)} to A2\{(0, 0), (0, 1)}given by (c1, c2, c3, c4) �→ (c1, β2((c1, c2, c3, c4))) is actually an isomorphismof (quasi-affine) varieties, because it is inverse to α� : A2 \ {(0, 0), (0, 1)} −→V \ {(0, 0, 0, 0)}.)

Suppose that β2 can be extended to a regular function β′2 : V → C, andlook for a contradiction. Now for v ∈ UV (x3), we have β′2(v) = x4(v)/x3(v).Since x4(v)2 = x3(v)

3 + x3(v)x4(v), it follows that

β′2(v)2 − β′2(v) =

x4(v)2

x3(v)2− x4(v)

x3(v)

=x3(v)

3 + x3(v)x4(v)

x3(v)2− x4(v)

x3(v)= x3(v)

for all v ∈ UV (x3). Since UV (x3) is a dense open subset of V , we deduce thatβ′22 −β′2 = x3. Hence β′2((0, 0, 0, 0)) = ε, where ε = 0 or 1. Now the map β′ :V −→ A2 \ {(0, 1− ε)} given by (c1, c2, c3, c4) �−→ (c1, β

′2((c1, c2, c3, c4)))

is a morphism of varieties. In fact, it is an isomorphism of varieties, becauseα� : A2 \ {(0, 1− ε)} −→ V is an inverse for it. This shows that the quasi-affine variety A2 \ {(0, 1− ε)} is affine.


However, Exercise 2.3.5 shows that the quasi-affine variety A2 \ {(0, 0)} isnot affine, and a similar argument will show that A2 \{(0, 1)} is not affine. Wehave therefore arrived at a contradiction, and this shows that

β2 : V \ {(0, 0, 0, 0)} → C

is a regular function which cannot be extended to a regular function on V .

2.3.8 Exercise. With the notation of 2.3.1, let V be the affine algebraic set inA4 given by

V := VA4(X21X2 −X2

3 , X32 −X2

4 , X2X3 −X1X4, X1X22 −X3X4).

(i) Show that the morphism of varieties α : A2 → A4 for which

α((c, d)) = (c, d2, cd, d3) for all (c, d) ∈ A2

is injective and that its image is equal to V . Deduce that V is irreducible,and so is an affine variety.

(ii) Show that

α� : A2 \ {(0, 0)} −→ V \ {(0, 0, 0, 0)}

is an isomorphism of (quasi-affine) varieties, with inverse

β : V \ {(0, 0, 0, 0)} −→ A2 \ {(0, 0)}

given by

β((c1, c2, c3, c4)) =

{(c1, c3/c1) if c1 = 0,

(c1, c4/c2) if c2 = 0

(for all (c1, c2, c3, c4) ∈ V \ {(0, 0, 0, 0)}).(iii) Let m denote the maximal ideal of O(V ) corresponding to the point

(0, 0, 0, 0) of V . Show that H1m(O(V )) ∼= C.

(iv) Show that the function β2 : V \ {(0, 0, 0, 0)} → C defined by

β2((c1, c2, c3, c4)) =

{c3/c1 if c1 = 0,

c4/c2 if c2 = 0

(for all (c1, c2, c3, c4) ∈ V \ {(0, 0, 0, 0)}) is a regular function on theset V \ {(0, 0, 0, 0)} that cannot be extended to a regular function on V .

Although Example 2.3.7 and Exercise 2.3.8 look very similar from an alge-braic point of view, there are substantial geometric differences between them.This is illustrated by the next two exercises. By the metric topology on anaffine variety V ⊆ An we mean the (subspace) topology induced on V by thetopology defined on Cn by the standard distance metric of analysis.


2.3.9 Exercise. Let the situation and notation be as in 2.3.7.Prove that there does not exist a mapping β′2 : V −→ C which extends

β2 and which is continuous for the metric topology. (Here is a hint: considerpoints on the path s : {t ∈ R : 0 ≤ t ≤ 1} → V given by s(t) = α((0, t)) =

(0, 0, t(t− 1), t2(t− 1)) (for 0 ≤ t ≤ 1) to show that, in the metric topology,limv→(0,0,0,0) β2(v) does not exist.)

2.3.10 Exercise. Let the situation and notation be as in 2.3.8.Let β′ : V −→ A2 be the map which extends

β : V \ {(0, 0, 0, 0)} −→ A2 \ {(0, 0)}

and is such that β′((0, 0, 0, 0)) = (0, 0). Show that β′ is actually continuousfor the metric topology, and deduce that α : A2 −→ V is a homeomorphismwith respect to the metric topology. (Again we offer a hint: show that β2(v)2 =

x2(v) for all v ∈ UV (x1) ∪ UV (x2) = V \ {(0, 0, 0, 0)}.)

2.3.11 �Exercise. Use the notation of 2.3.1.

(i) Let 0 = g ∈ C[X,Y ] = O(A2). Let U := UA2(g), a quasi-affine varietyover C, and let p ∈ U . Show that each regular function f : U \{p} → Ccan be extended to a regular function on U . (Do not forget that U isaffine!)

(ii) Let f ∈ K(A2). There is a maximum open subset U ′ of A2 on which fis defined: the poles of f are precisely the points of A2 \ U ′. Use (i) toshow that there does not exist an isolated pole of f , that is, a pole q of ffor which there exists an open subset U ′′ of A2 such that q ∈ U ′′ but qis the only pole of f in U ′′.

3

The Mayer–Vietoris sequence

Any reader with a basic grounding in algebraic topology will recall the impor-tant role that the Mayer–Vietoris sequence can play in that subject. There is ananalogue of the Mayer–Vietoris sequence in local cohomology theory, and itcan play a foundational role in this subject. It is our intention in this chapter topresent the basic theory of the Mayer–Vietoris sequence in local cohomology,and to prepare for several uses of the idea during the subsequent development.

The Mayer–Vietoris sequence involves two ideals, and so throughout thischapter, b will denote a second ideal of R (in addition to a). Let M be an R-module. The Mayer–Vietoris sequence provides, among other things, a longexact sequence

0 H0a+b(M) H0

a(M)⊕H0b(M) H0

a∩b(M)

H1a+b(M) H1

a(M)⊕H1b(M) H1

a∩b(M)

· · · · · ·

Hia+b(M) Hi

a(M)⊕Hib(M) Hi

a∩b(M)

Hi+1a+b(M) · · ·

� � �

� � �

�

� � �

� �

of local cohomology modules. Its potential for use in arguments that employinduction on the number of elements in a generating set for an ideal c of R canbe explained as follows. Suppose that c is generated by n elements c1, . . . , cn,where n > 1. Set a = Rc1+· · ·+Rcn−1 and b = Rcn, so that c = a+b. Eachof a and b can be generated by fewer than n elements, but at first sight it seemsthat the ideal a ∩ b, which also appears in the Mayer–Vietoris sequence, couldpresent difficulties. However,

√(a∩b) = √(ab), and so Γa∩b = Γab by 1.1.3;

hence Hia∩b = Hi

ab for all i ∈ N0 (see 1.2.3). Moreover, in our situation,

ab = (Rc1 + · · ·+Rcn−1)Rcn = Rc1cn + · · ·+Rcn−1cn

48 The Mayer–Vietoris sequence

can be generated by n− 1 elements. Thus a, b and ab can all be generated byfewer than n elements, and an appropriate inductive hypothesis would apply toall of them. In addition, we have already obtained (in 2.2.21) a certain amountof information about the local cohomology functorsHi

Ra (i ∈ N0) with respectto a principal idealRa ofR: this can provide a basis for an inductive argument.

3.1 Comparison of systems of ideals

The result that Hia∩b = Hi

ab for all i ∈ N0 can be viewed as a particular ex-ample of a more general phenomenon which concerns a situation where twosystems of ideals are ‘comparable’ in a sense made precise in the proposi-tion below. This comparison result will not only be used to obtain the Mayer–Vietoris sequence; it will also provide an important ingredient, which is thesubject of Exercise 3.1.4, in our proof of the local Lichtenbaum–HartshorneVanishing Theorem in Chapter 8.

3.1.1 Proposition. Let (Λ,≤) and (Π,≤) be (non-empty) directed partiallyordered sets, and let B = (bα)α∈Λ be an inverse family of ideals of R over Λ,as in 1.2.10; let C = (cβ)β∈Π be an inverse family of ideals of R over Π.

Assume that, for all α ∈ Λ, there exists β ∈ Π such that cβ ⊆ bα, and, forall β′ ∈ Π, there exists α′ ∈ Λ such that bα′ ⊆ cβ′ . Then

(i) ΓB = ΓC;(ii) the negative strongly connected sequences of covariant functors(

lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

and

(lim−→β∈Π

ExtiR(R/cβ , • )

)i∈N0

are isomorphic; and(iii) B is a system of ideals of R over Λ (in the sense of 2.1.10) if and only if

C is a system of ideals of R over Π.

Note. The functor ΓB was defined in 1.2.11, while in 1.3.7 it was explainedthat the negative strongly connected sequence (RiΓB)i∈N0 of its right derivedfunctors is isomorphic to(

lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

;

of course, similar comments apply to ΓC.

3.1 Comparison of systems of ideals 49

Proof. Part (i) is clear; henceRiΓB = RiΓC for all i ∈ N0, and the claim in(ii) is an obvious consequence of the above note.

(iii) Suppose that B is a system of ideals over Λ, and let β1, β2 ∈ Π. Byassumption, there exist α1, α2 ∈ Λ such that bα1 ⊆ cβ1 and bα2 ⊆ cβ2 . SinceB is a system of ideals, there exists α3 ∈ Λ such that bα3 ⊆ bα1bα2 . Byassumption, there exists β3 ∈ Π such that cβ3 ⊆ bα3 . Hence

cβ3 ⊆ bα3 ⊆ bα1bα2 ⊆ cβ1cβ2 .

Hence C is a system of ideals over Π.In view of the symmetry of our hypotheses on B and C, the proof is com-

plete.

3.1.2 Example. Consider the descending chain B = (an + bn)n∈N of idealsof R. Since (a+ b)2n−1 ⊆ an + bn ⊆ (a+ b)n for all n ∈ N, it follows from3.1.1 that B is actually a system of ideals of R over N; also, it follows from1.3.7 and 3.1.1 that the (negative strongly) connected sequences of covariantfunctors (from C(R) to C(R))(

Hia+b

)i∈N0

,

(lim−→n∈N

ExtiR(R/(an + bn), • )

)i∈N0

and(Hi

B

)i∈N0

are all isomorphic. In particular, for each i ∈ N0, the functors Hia+b and

lim−→n∈N

ExtiR(R/(an + bn), • ) are naturally equivalent.

3.1.3 Exercise. Let (R,m) be a complete local ring, and let B = (bn)n∈Nbe a descending chain of m-primary ideals of R such that

⋂n∈N bn = 0.

Show that B is a system of ideals of R over N, and that the (negativestrongly) connected sequences of covariant functors (from C(R) to C(R))(

Him

)i∈N0

,

(lim−→n∈N

ExtiR(R/bn, • )

)i∈N0

and(Hi

B

)i∈N0

are all isomorphic. (If you have no idea where to start with this, we suggestthat you look up Chevalley’s Theorem in, for example, [59, §5.2, Theorem 1]or [89, Chapter VIII, §5, Theorem 13].)

3.1.4 �Exercise. Let (R,m) be a complete local domain, and let p be a primeideal of R of dimension 1, that is, such that dimR/p = 1.

(i) Use Chevalley’s Theorem ([59, §5.2, Theorem 1] or [89, Chapter VIII,§5, Theorem 13]) to prove that, for each n ∈ N, there exists t ∈ N suchthat the t-th symbolic power p(t) of p is contained in mn.


(ii) Deduce that (p(n))n∈N is a system of ideals of R over N, and that the(negative strongly) connected sequences of covariant functors(

Hip

)i∈N0

and

(lim−→n∈N

ExtiR(R/p(n), • )

)i∈N0

(from C(R) to C(R)) are isomorphic.

3.1.5 Corollary. The descending chain (an ∩ bn)n∈N is a system of idealsof R (over N), and the (negative strongly) connected sequences of covariantfunctors (from C(R) to C(R))(

lim−→n∈N

ExtiR(R/(an ∩ bn), • )

)i∈N0

and(Hi

a∩b

)i∈N0

are isomorphic. In particular, for each i ∈ N0, the functors Hia∩b = Hi

ab andlim−→n∈N

ExtiR(R/(an ∩ bn), • ) are naturally equivalent.

Proof. As we mentioned at the beginning of this chapter, it follows from 1.2.3that Hi

a∩b = Hiab for all i ∈ N0. Since (a ∩ b)n ⊆ an ∩ bn for all n ∈ N, it

is enough, in view of 3.1.1, for us to show that, for each n ∈ N, there existsq(n) ∈ N such that aq(n) ∩ bq(n) ⊆ (a ∩ b)n. We use the Artin–Rees Lemma[50, Theorem 8.5] to achieve this.

Fix n ∈ N. By the Artin–Rees Lemma, there is c ∈ N such that am ∩ bn =

am−c(ac ∩ bn) for all integers m > c. Hence

an+c ∩ bn+c ⊆ an+c ∩ bn = an(ac ∩ bn) ⊆ anbn ⊆ (a ∩ b)n.

The proof is therefore complete.

3.1.6 Exercise. For each h ∈ N, let a[h] denote the ideal of R generated byall the h-th powers of elements of a. Show that, for each i ∈ N0, the functorHi

a is naturally equivalent to lim−→h∈N

ExtiR(R/a[h], • ).

(A variation on this is the result that, if R contains a subfield of character-istic p > 0, then, for each i ∈ N0, the functor Hi

a is naturally equivalent tolim−→e∈N

ExtiR(R/a[pe], • ). There are situations where this observation can be very

useful: see, for example, C. Peskine and L. Szpiro [66, Chapitre III, Proposi-tion 1.8(3)], and C. L. Huneke and R. Y. Sharp [43, p. 770].)

3.1.7 Exercise. Suppose that the t elements a1, . . . , at generate a. Now Nt

is a directed partially ordered set with respect to the ordering ≤ defined by, for

3.2 Construction of the sequence 51

(u1, . . . , ut), (v1, . . . , vt) ∈ Nt,

(u1, . . . , ut) ≤ (v1, . . . , vt) if and only if uj ≤ vj for all j = 1, . . . , t.

Show that, for each i ∈ N0, the functor Hia is naturally equivalent to

lim−→

(u1,...,ut)∈Nt

ExtiR(R/(au11 , . . . , aut

t ), • ).

3.2 Construction of the sequence

Our first lemma in this section provides a fundamental tool for the constructionof the Mayer–Vietoris sequence.

3.2.1 Lemma. Let N1, N2 be submodules of the R-module M . The sequenceof R-modules and R-homomorphisms

0 −→M/(N1 ∩N2)α−→M/N1 ⊕M/N2

β−→M/(N1 +N2) −→ 0,

in which α(m+N1 ∩N2) = (m+N1,m+N2) for all m ∈M and

β((x+N1, y +N2)) = x− y + (N1 +N2) for all x, y ∈M,

is exact.

Proof. It is clear that α is injective, that β is surjective and that β ◦ α = 0.Let x, y ∈M be such that (x+N1, y +N2) ∈ Kerβ. Then x− y = n1 + n2

for some n1 ∈ N1, n2 ∈ N2, so that x− n1 = y + n2 and

(x+N1, y +N2) = (x− n1 +N1, y + n2 +N2) ∈ Imα.

Briefly, the general strategy for our construction of the Mayer–Vietorissequence is to write down, for each n ∈ N, the long exact sequence of ‘Ext’modules which results from application of the functor HomR( • ,M) to theexact sequence

0 −→ R/(an ∩ bn) −→ R/an ⊕R/bn −→ R/(an + bn) −→ 0

resulting from 3.2.1, pass to direct limits, and then appeal to 3.1.2 and 3.1.5to convert the result into information about local cohomology modules. Thereis, however, one minor technical point that needs attention first, and that is theidentification of Hi

a( • )⊕Hib( • ) with the functor

lim−→n∈N

ExtiR(R/an ⊕R/bn, • )

(obtained using the ideas of 1.2.8 in an obvious way). Although perhaps a littletedious, this is not particularly difficult.


3.2.2 �Exercise. In this exercise, we use, for R-modules L and M , the sym-bolism

0 L L⊕M M 0� � � ��

to denote simultaneously the two split exact sequences associated with the di-rect sum: thus we consider just the arrows pointing to the right to obtain oneof these split exact sequences, while the arrows pointing to the left provide theother.

Let T : C(R)×C(R)→ C(R) be anR-linear functor of two variables whichis contravariant in the first variable and covariant in the second. (For example,T could be HomR or ExtiR for i ∈ N0.)

For n,m ∈ N with n ≥ m, let

hnm : R/an −→ R/am and knm : R/bn −→ R/bm

denote the natural homomorphisms, and consider the commutative diagrams

0 R/an R/an ⊕R/bn R/bn 0� � � ��

0 R/am R/am ⊕R/bm R/bm 0 ,� � � ��

hnm hn

m⊕knm kn

m

� � �

in which the rows represent the canonical split exact sequences. Apply thecontravariant, additive functor T ( • ,M), and pass to direct limits to obtainsplit exact sequences

0�� lim T (R/bn,M)

−→n∈N

�� lim T (R/an ⊕R/bn,M)−→n∈N

�� lim T (R/an,M)−→n∈N

�� 0.

Deduce that there is a natural equivalence between the functors

lim−→n∈N

T (R/an ⊕R/bn, • ) and lim−→n∈N

T (R/an, • )⊕ lim−→n∈N

T (R/bn, • )

(from C(R) to itself).Deduce from 1.3.8 that the functors

lim−→n∈N

ExtiR(R/an ⊕R/bn, • ) and Hi

a( • )⊕Hib( • )

are naturally equivalent for each i ∈ N0.

3.2 Construction of the sequence 53

3.2.3 Theorem: the Mayer–Vietoris sequence. For each R-module M ,there is a long exact sequence (called the Mayer–Vietoris sequence for Mwith respect to a and b)

0 H0a+b(M) H0

a(M)⊕H0b(M) H0

a∩b(M)

H1a+b(M) H1

a(M)⊕H1b(M) H1

a∩b(M)

· · · · · ·

Hia+b(M) Hi

a(M)⊕Hib(M) Hi

a∩b(M)

Hi+1a+b(M) · · ·

� � �

� � �

�

� � �

� �

such that, whenever f : M −→ N is a homomorphism of R-modules, thediagram

Hia+b(M) Hi

a(M)⊕Hib(M) Hi

a∩b(M) Hi+1a+b(M)� � �

Hia+b(N) Hi

a(N)⊕Hib(N) Hi

a∩b(N) Hi+1a+b(N)� � �

Hia+b(f) Hi

a(f)⊕Hib(f) Hi

a∩b(f) Hi+1a+b(f)

� � � �

commutes for all i ∈ N0.

Proof. Let n,m ∈ N with n ≥ m, and let

hnm : R/an −→ R/am and knm : R/bn −→ R/bm

denote the natural homomorphisms. The diagram

0 R/(an ∩ bn) R/an ⊕R/bn R/(an + bn) 0� � � �

0 R/(am ∩ bm) R/am ⊕R/bm R/(am + bm) 0,� � � �

hnm⊕kn

m

� � �

in which the upper row is the exact sequence resulting from application of 3.2.1to the submodules an and bn of R, the lower row is the corresponding exactsequence form instead of n, and the two outer vertical homomorphisms are thenatural ones, commutes. Therefore, application of the functor HomR( • ,M)


to this yields a long exact sequence

0 HomR(R/(am + bm),M) HomR(R/a

m ⊕R/bm,M)� �

HomR(R/(am ∩ bm),M) Ext1R(R/(a

m + bm),M)� �

· · ·�

ExtiR(R/(am + bm),M) ExtiR(R/a

m ⊕R/bm,M)� �

ExtiR(R/(am ∩ bm),M) Exti+1

R (R/(am + bm),M)� �

· · ·�

and a chain map (induced by the vertical homomorphisms in the last commu-tative diagram) of this long exact sequence into the corresponding one with mreplaced by n. Also, a homomorphism f : M → N of R-modules induces achain map of the above displayed long exact sequence into the correspondingsequence withM replaced byN . Now pass to direct limits: it follows that thereis a long exact sequence

0 lim HomR(R/(an + bn),M)

−→n∈N

lim HomR(R/an ⊕R/bn,M)

−→n∈N

� �

lim HomR(R/(an ∩ bn),M)

−→n∈N

lim Ext1R(R/(an + bn),M)−→

n∈N

� �

· · ·�

lim ExtiR(R/(an + bn),M)−→

n∈Nlim ExtiR(R/a

n ⊕R/bn,M)−→n∈N

� �

lim ExtiR(R/(an ∩ bn),M)

−→n∈N

lim Exti+1R (R/(an + bn),M)

−→n∈N

� �

· · ·�

and that f :M → N induces a chain map of this long exact sequence into thecorresponding sequence with M replaced by N . The result is now an immedi-ate consequence of the natural equivalences of functors between

lim−→n∈N

ExtiR(R/(an + bn), • ) and Hi

a+b,

lim−→n∈N

ExtiR(R/an ⊕R/bn, • ) and Hi

a( • )⊕Hib( • )

and

lim−→n∈N

ExtiR(R/(an ∩ bn), • ) and Hi

a∩b

3.3 Arithmetic rank 55

(for i ∈ N0) established in 3.1.2, 3.2.2 and 3.1.5 respectively.

3.2.4 Exercise. Let M be an R-module. Show that the sequence of R-modu-les and R-homomorphisms

0 −→ HomR(a+ b,M)α−→ HomR(a,M)⊕HomR(b,M)

β−→ HomR(a ∩ b,M)

in which α(f) = (f� a, f� b) for all f ∈ HomR(a + b,M), and β((g, h)) =g� a∩b − h� a∩b for all (g, h) ∈ HomR(a,M)⊕HomR(b,M), is exact.

Show also that, if M is an injective R-module, then β is an epimorphism.

3.2.5 Exercise. Prove, perhaps with the aid of Exercise 3.2.4, that, for eachR-module M , there is a long exact sequence

0 Da+b(M) Da(M)⊕Db(M) Da∩b(M)

H2a+b(M) H2

a(M)⊕H2b(M) H2

a∩b(M)

· · · · · ·

Hia+b(M) Hi

a(M)⊕Hib(M) Hi

a∩b(M)

Hi+1a+b(M) · · ·

� � �

� � �

�

� � �

� �

such that each homomorphism f :M → N ofR-modules induces a chain mapof the above long exact sequence into the corresponding long exact sequencefor N .

As we have already mentioned, we shall make substantial use of the Mayer–Vietoris sequence in this book. We present in the next two sections two illus-trations of its use in order to give some idea of its potential to any reader who,wondering why so much effort has been expended in setting up the Mayer–Vietoris sequence, is impatient to see some applications.

3.3 Arithmetic rank

Our first application of the Mayer–Vietoris sequence will be in the proof ofa result which relates the number of elements required to generate the ideala (and, more precisely, its so-called ‘arithmetic rank’) to the vanishing of thelocal cohomology functors Hi

a. This vanishing result provides a powerful toolfor applications of local cohomology to algebraic geometry, and, in particular,will play a crucial role in our presentation of applications to connectivity inChapter 19.


3.3.1 Theorem. Suppose that a can be generated by t elements. Then, forevery R-module M , we have Hi

a(M) = 0 for all i > t.

Proof. This proof is an example of the use of induction outlined in the intro-duction to this chapter.

When t = 0, we have a = 0 and Γa = Γ0R is the identity functor, so thatHi

0R(M) = 0 for all i > 0. The result is therefore true when t = 0, and it wasproved in 2.2.21(ii) in the special case in which t = 1.

Now suppose, inductively, that t > 1 and the result has been proved forideals that can be generated by fewer than t elements. Suppose that a is gener-ated by t elements a1, . . . , at. Set b = Ra1 + · · · + Rat−1 and c = Rat, sothat a = b+ c. By the inductive assumption, Hi

b(M) = 0 for all i > t− 1 andHi

c(M) = 0 for all i > 1. By the Mayer–Vietoris sequence 3.2.3, we have, foran arbitrary i > t, an exact sequence

Hi−1b∩c (M) −→ Hi

a(M) −→ Hib(M)⊕Hi

c(M).

Now Hi−1b∩c (M) = Hi−1

bc (M) by 3.1.5, and, since

bc = (Ra1 + · · ·+Rat−1)Rat = Ra1at + · · ·+Rat−1at

can be generated by t − 1 elements, it follows from the inductive assumptionthat Hi−1

bc (M) = 0. Since Hib(M)⊕Hi

c(M) = 0 also, we have Hia(M) = 0.

This completes the inductive step.

We now combine 3.3.1 with 1.2.3 to obtain a result which is often usedin geometric applications. To formulate this result, let us recall the followingdefinition from basic commutative algebra.

3.3.2 Definition. The arithmetic rank of a, denoted by ara(a), is the leastnumber of elements of R required to generate an ideal which has the sameradical as a. Thus ara(a) is equal to the integer

min {n ∈ N0 : ∃ b1, . . . , bn ∈ R with√(Rb1 + · · ·+Rbn) =

√a} .

Note that ara(0R) = 0.The next corollary is now immediate from 3.3.1 and 1.2.3.

3.3.3 Corollary. For every R-module M , we have Hia(M) = 0 for all i >

ara(a). �

Corollary 3.3.3 leads naturally to the following definition.

3.3.4 Definition. The cohomological dimension of a, denoted cohd(a), isdefined as the greatest integer i for which there exists an R-module M withHi

a(M) = 0 if any such integers exist, and −∞ otherwise (for example

3.3 Arithmetic rank 57

when a = R). It follows from 3.3.3 that this definition makes sense, and thatcohd(a) ≤ ara a.

Theorem 3.3.1 can be used to obtain information about the number of el-ements required to generate a specified ideal; also, in geometric situations,Corollary 3.3.3 can be used to obtain information about the number of ‘equa-tions’ needed to define an algebraic variety. The following example illustratesthis.

3.3.5 Example. This example concerns the affine variety V in A4 studied in2.3.7 and given by

V := VA4(X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

By 2.3.7, V ={(c, cd, d(d− 1), d2(d− 1)) ∈ A4 : c, d ∈ C

}. Thus the lines

L,L′ in A4 given by L := VA4(X2, X3, X4) ={(c, 0, 0, 0) ∈ A4 : c ∈ C

}and L′ := VA4(X1 − X2, X3, X4) =

{(c, c, 0, 0) ∈ A4 : c ∈ C

}are both

contained in V : our aim here is to use 3.3.3 to show that the subvariety L of Vcannot be ‘defined by one equation’, that is, there does not exist a polynomialf ∈ C[X1, X2, X3, X4] such that

L = {p ∈ V : f(x1, x2, x3, x4)(p) = 0} .

Our argument uses the morphism of varieties α : A2 → V of 2.3.7 for whichα((c, d)) = (c, cd, d(d − 1), d2(d − 1)) for all (c, d) ∈ A2; recall that α� :A2 \ {(0, 0), (0, 1)} −→ V \ {(0, 0, 0, 0)} is an isomorphism of (quasi-affine)varieties. Set L :=

{(c, 0) ∈ A2 : c ∈ C

}and L′ :=

{(c, 1) ∈ A2 : c ∈ C

}.

Note that α(L) = L and α(L′) = L′.Write O(A2) = C[X,Y ]; note that L = VA2(Y ) and that (0, 1) ∈ L. It

therefore follows from 2.3.11 that the restriction map

O(A2 \ L) −→ O(A2 \ ({(0, 1)} ∪ L))

is surjective. Now L′ \ {(0, 1)} ⊆ A2 \ ({(0, 1)} ∪ L): our immediate aim isto show that the restriction map O(A2 \ ({(0, 1)} ∪ L)) −→ O(L′ \ {(0, 1)})is not surjective.

Set x := X� L′ and consider the regular function γ : L′ \ {(0, 1)} → Cdefined by γ(v) = x(v)−1 for all v ∈ L′ \ {(0, 1)} = UL′(x). If γ could beextended to a regular function on A2 \ ({(0, 1)} ∪ L), then it would followfrom the consequence of 2.3.11 noted in the previous paragraph that γ couldbe extended to a regular function γ′ : A2 \ L → C. Let γ′′ be the restrictionof γ′ to L′, a subset of A2 \ L. It then follows that, for all v ∈ UL′(x), wehave γ′′(v)x(v) = 1. However, UL′(x) is a dense open subset of L′, and so


γ′′(v)x(v) = 1 for all v ∈ L′. As this implies that 0 = 1, we have obtained acontradiction! Thus the restriction map

O(A2 \ ({(0, 1)} ∪ L)) −→ O(L′ \ {(0, 1)})

is not surjective, as claimed.Next, α(A2 \ ({(0, 1)} ∪ L)) = V \ L and

α(L′ \ {(0, 1)}) = L′ \ {(0, 0, 0, 0)} .

There is therefore a commutative diagram

O(V \ L) O(L′ \ {(0, 0, 0, 0)})�

∼= ∼=

O(A2 \ ({(0, 1)} ∪ L)) O(L′ \ {(0, 1)})��

in which the vertical isomorphisms are induced by (the appropriate restrictionsof) α. Therefore the map σ : O(V \ L) −→ O(L′ \ {(0, 0, 0, 0)}) given byrestriction is not surjective.

Let ι : L′ −→ V be the inclusion morphism of varieties, and let c denotethe vanishing ideal {f ∈ O(V ) : f(v) = 0 for all v ∈ L} of L in O(V ). Withthe notation of 2.3.2 and 2.3.4, we have

V (cO(L′)) = ι−1(V (c)) = ι−1(L) = L ∩ L′ = {(0, 0, 0, 0)} ⊂ L′,

and so cO(L′) = 0. Note also that the restriction map σ of the precedingparagraph is just ι�∗: O(V \ V (c)) −→ O(L′ \ V (cO(L′))). Therefore ι�∗is not surjective, and so it follows from 2.3.4 that Dc(ι

∗) : Dc(O(V )) −→Dc(O(L′)) is not surjective. Hence, if b := Ker(ι∗), we have R1Dc(b) = 0,so that H2

c (b) = 0 by 2.2.6(ii). It therefore follows from 3.3.3 that ara(c) ≥ 2.Thus the subvariety L of V cannot be ‘defined by one equation’.

3.3.6 Exercise. Show that, over the ring R[X1, X2, X3, X4, X5, X6] of pol-ynomials in six indeterminates with coefficients in R,

H4(X1,X2,X3)(X4,X5,X6)

(R[X1, X2, X3, X4, X5, X6]) = 0.

3.3.7 Exercise. Show that, over the polynomial ring R[X1, X2, X3],

H3(X2

1 ,X1X2+X32 ,X

42 )(R[X1, X2, X3]) = 0.

3.4 Direct limits 59

3.4 Direct limits

Our second illustration of the use of the Mayer–Vietoris sequence is in ourproof that the local cohomology functors ‘commute with the formation of di-rect limits’. This is an important property of local cohomology, which playsa significant role in Grothendieck’s development in [25]. It should be notedthat our approach below avoids the use of an injective resolution of a directsystem of R-modules and R-homomorphisms in the category formed by suchdirect systems. It is possible to approach this theory in a different way, basedon the interchange of the order of direct limits; however, we have chosen theapproach below because we consider it to be more illuminating.

3.4.1 Terminology. Let (Λ,≤) be a (non-empty) directed partially orderedset, and let (Wα)α∈Λ be a direct system ofR-modules over Λ, with constituentR-homomorphisms hαβ : Wβ → Wα (for each (α, β) ∈ Λ × Λ with α ≥ β).Set W∞ := lim

−→α∈Λ

Wα, and let hα : Wα → W∞ be the canonical map (for each

α ∈ Λ).Let T : C(R) → C(R) be a covariant functor. It is immediate from the

definition of functor that the T (hαβ ) turn the family (T (Wα))α∈Λ into a directsystem of R-modules and R-homomorphisms over Λ. Also, whenever α, β ∈Λ with α ≥ β, the commutative diagram

Wβ

hαβ

Wα�

hαhβ

W∞

��

induces the commutative diagram

T (Wβ)T (hα

β )T (Wα)�

T (hα)T (hβ)

T (W∞) ,

��

and so there is induced an R-homomorphism

ωT : lim−→α∈Λ

T (Wα) −→ T

(lim−→α∈Λ

Wα

)= T (W∞).


If, for all choices of such a directed set Λ and such a direct system (Wα)α∈Λover Λ, the map ωT is an isomorphism, then we shall say that T commutes withthe formation of direct limits or, more loosely, T commutes with direct limits.

In the next exercise, we again use the notation Ma, for an R-module Mand a ∈ R, to denote the module of fractions of M with respect to the multi-plicatively closed subset

{ai : i ∈ N0

}. In fact, we consider the functor ( • )a :

C(R) −→ C(R).

3.4.2 �Exercise. Let a ∈ R. Show that the functor ( • )a : C(R) −→ C(R)commutes with direct limits.

The next exercise gives another example, again needed later in the book, ofa functor which commutes with direct limits.

3.4.3 �Exercise. Let (Λ,≤) be a (non-empty) directed partially ordered set,and let (Wα)α∈Λ be a direct system of R-modules over Λ, with constituentR-homomorphisms hαβ : Wβ → Wα (for each α, β ∈ Λ with α ≥ β). SetW∞ := lim

−→α∈Λ

Wα.

(i) Show that

(lim−→α∈Λ

ExtiR( • ,Wα)

)i∈N0

can be made into a negative stron-

gly connected sequence of contravariant functors from C(R) to C(R) insuch a way that the natural homomorphisms (of 3.4.1) give rise to ahomomorphism

Ψ = (ψi)i∈N0 :

(lim−→α∈Λ

ExtiR( • ,Wα)

)i∈N0

−→(ExtiR( • ,W∞)

)i∈N0

of connected sequences.(ii) Prove that ψ0

R is an isomorphism, and deduce that ψ0F is an isomorphism

whenever F is a finitely generated free R-module.(iii) Deduce that ψ0

M is an isomorphism whenever M is a finitely generatedR-module.

(iv) Prove, by induction on i, that, for each i ∈ N0, the homomorphism ψiM

is an isomorphism whenever M is a finitely generated R-module.(v) Deduce that, whenever M is a finitely generated R-module, the functor

ExtiR(M, • ) commutes with direct limits, for each i ∈ N0.

Our main aim during the remainder of this chapter is to show that the localcohomology functors Hi

a (i ∈ N0) commute with direct limits; we show firstthat the a-torsion functor commutes with direct limits.


3.4.4 Proposition. The a-torsion functor Γa commutes with direct limits.

Proof. We use the notation of 3.4.1, so that (Λ,≤) is a (non-empty) directedpartially ordered set, and (Wα)α∈Λ is a direct system of R-modules over Λ,with constituentR-homomorphisms hαβ :Wβ →Wα (for each (α, β) ∈ Λ×Λ

with α ≥ β). We must show that the R-homomorphism

ωΓa: lim−→α∈Λ

Γa(Wα) −→ Γa

(lim−→α∈Λ

Wα

)is an isomorphism. We write F∞ := lim

−→α∈Λ

Γa(Wα) and W∞ := lim−→α∈Λ

Wα, and,

for α ∈ Λ, we use fα : Γa(Wα)→ F∞ to denote the canonical map.A typical element of KerωΓa

can be expressed as fα(wα) for some α ∈ Λ

and wα ∈ Γa(Wα), where Γa(hα)(wα) = 0 in Γa(W∞). Thus hα(wα) =

0 in W∞, and so there exists γ ∈ Λ with γ ≥ α such that hγα(wα) = 0.Since wα ∈ Γa(Wα), this means that Γa(h

γα)(wα) = 0 in Γa(Wγ), and so

fα(wα) = fγ (Γa(hγα)(wα)) = 0. Hence ωΓa

is injective.We now show that ωΓa

is surjective. Let y ∈ Γa(W∞). There exists α ∈ Λ

and wα ∈ Wα such that y = hα(wα). We know that y is annihilated by apower of a: let j ∈ N be such that ajy = 0. Let aj be generated by r1, . . . , rn.For each i = 1, . . . , n, we have hα(riwα) = 0, so that there exists βi ∈ Λ withβi ≥ α such that hβi

α (riwα) = 0. Since Λ is directed, there exists γ ∈ Λ suchthat γ ≥ βi for all i = 1, . . . , n. We now have

rihγα(wα) = rih

γβi(hβi

α (wα)) = hγβi(hβi

α (riwα)) = 0

for all i = 1, . . . , n, so that ajhγα(wα) = 0 and hγα(wα) ∈ Γa(Wγ). But

y = hα(wα) = hγ(hγα(wα)) = Γa(hγ)(h

γα(wα)) ∈ Im(Γa(hγ)).

Set zγ := hγα(wα), an element of Γa(Wγ). Now fγ(zγ) ∈ F∞, and

ωΓa(fγ(zγ)) = Γa(hγ)(zγ) = y.

Hence ωΓais surjective and the proof is complete.

3.4.5 �Exercise. Let T : C(R) → C(R) be a covariant, additive functorwhich commutes with direct limits. Let (Lθ)θ∈Ω be a non-empty family ofR-modules; for each φ ∈ Ω, let qφ : Lφ −→

⊕θ∈Ω Lθ be the canonical

injection.Prove that T

(⊕θ∈Ω Lθ

)is the direct sum of its family of submodules

(ImT (qθ))θ∈Ω, so that

T

(⊕θ∈Ω

Lθ

)=⊕θ∈Ω

ImT (qθ) ∼=⊕θ∈Ω

T (Lθ).


We shall describe this result by the statement that T commutes with directsums.

To show that the i-th right derived functors of Γa (for i ∈ N) also commutewith direct limits, we plan to argue by induction on the number of elementsneeded to generate a, and to use the Mayer–Vietoris sequence to complete theinductive step. The following lemma will be helpful.

3.4.6 Lemma. Let Q,S, T, U, V be covariant functors (from C(R) to C(R))and let σ : Q −→ S, τ : S −→ T, μ : T −→ U, ν : U −→ V be naturaltransformations of functors such that, for each R-module M , the sequence

Q(M)σM−→ S(M)

τM−→ T (M)μM−→ U(M)

νM−→ V (M)

is exact. Suppose that Q,S, U, V all commute with direct limits. Then T alsocommutes with direct limits.

Proof. We again use the notation of 3.4.1. For each α, β ∈ Λ with α ≥ β,there is a commutative diagram

Q(Wβ) S(Wβ) T (Wβ) U(Wβ) V (Wβ)� � � �σWβ

τWβμWβ

νWβ

Q(hαβ ) S(hα

β ) T (hαβ ) U(hα

β ) V (hαβ )

Q(Wα) S(Wα) T (Wα) U(Wα) V (Wα)� � � �σWα τWα μWα νWα

� � � � �

with exact rows, and so there is induced an exact sequence

Q∞σ∞−→ S∞

τ∞−→ T∞μ∞−→ U∞

ν∞−→ V∞,

where Q∞ := lim−→α∈Λ

Q(Wα), etc., and σ∞ := lim−→α∈Λ

σWα , etc. It is easy to see

from the definition of ωQ, . . . , ωV in 3.4.1 that the diagram

Q∞ S∞ T∞ U∞ V∞� � � �σ∞ τ∞ μ∞ ν∞

ωQ ωS ωT ωU ωV

Q(W∞) S(W∞) T (W∞) U(W∞) V (W∞)� � � �σW∞ τW∞ μW∞ νW∞

� � � � �

commutes. Since this diagram has exact rows and ωQ, ωS , ωU and ωV are iso-morphisms by hypothesis, it follows from the Five Lemma that ωT is an iso-morphism too.


3.4.7 Remark. It is clear that the identity functor Id : C(R)→ C(R) and thezero functor 0 : C(R) → C(R) (for which 0(M) = 0 for all R-modules M )commute with direct limits.

3.4.8 Corollary. Suppose that, for some i ∈ N0, the functors Hia and Hi

b

commute with direct limits. Then Hia( • )⊕Hi

b( • ) commutes with direct limits.

Proof. This is an easy consequence of 3.4.6 and 3.4.7: for each R-moduleM , consider the canonical split exact sequences

0 Hia(M) Hi

a(M)⊕Hib(M) Hi

b(M) 0.� � � ��

3.4.9 Proposition. Let a ∈ R. Then the local cohomology functor H1Ra com-

mutes with direct limits.

Proof. By 2.2.20, there are natural transformations of functors

τ : Id −→ ( • )a and σ : ( • )a −→ H1Ra

such that, for all R-modules M , the sequence

MτM−→Ma

σM−→ H1Ra(M) −→ 0 −→ 0

is exact. We propose to use 3.4.6, and with this in mind we observe that, by3.4.2, the functor ( • )a : C(R) −→ C(R) commutes with direct limits. (Thiswill be obvious to any reader familiar with the facts that ( • )a is naturallyequivalent to ( • )⊗RRa : C(R) −→ C(R) and that, for an arbitrary R-moduleN , the functor ( • ) ⊗R N : C(R) −→ C(R) commutes with direct limits.Alternatively, it is easy to prove directly.)

The claim now follows immediately from 3.4.6 and 3.4.7.

We are now able to prove that the local cohomology functors Hia (i ∈ N0)

all commute with direct limits.

3.4.10 Theorem. For all i ∈ N0, the local cohomology functorHia commutes

with direct limits.

Proof. This proof is another example of the use of induction outlined in theintroduction to this chapter.

We suppose that a can be generated by t elements and proceed by inductionon t. When t = 0, we have a = 0 and Γa is the identity functor, so thatHi

a = 0

for all i ∈ N. The result is therefore immediate from 3.4.7 in this case.We proved in 3.4.4 that Γa commutes with direct limits. When t = 1, it

follows from 3.3.1 thatHia = 0 for all i > 1, and from 3.4.9 thatH1

a commuteswith direct limits. The result is therefore also proved in the case when t = 1.


Now suppose, inductively, that t > 1 and the result has been proved for ide-als that can be generated by fewer than t elements. Suppose that a is generatedby t elements a1, . . . , at. Set b = Ra1 + · · · + Rat−1 and c = Rat, so thata = b+ c. Note that

bc = (Ra1 + · · ·+Rat−1)Rat = Ra1at + · · ·+Rat−1at

can be generated by t − 1 elements, and that Hib∩c = Hi

bc for all i ∈ N0 by3.1.5.

It therefore follows from the inductive hypothesis and what we have alreadyproved that the local cohomology functors Hi

b (i ∈ N0), Hic (i ∈ N0) and

Hib∩c (i ∈ N0) all commute with direct limits. Hence, by 3.4.8, all the functors

Hia( • )⊕Hi

b( • ) (i ∈ N0) commute with direct limits.It now follows from 3.4.6 and the Mayer–Vietoris sequence (we use here

the full statement of 3.2.3) that the functors Hib+c (i ∈ N0) all commute with

direct limits. Since a = b+ c, the inductive step is complete.

3.4.11 Corollary. The a-transform functor Da commutes with direct limits.Consequently, by 3.4.5, the functor Da commutes with direct sums (in the

sense of Exercise 3.4.5).

Proof. By 2.2.6(i), there are natural transformations of functors

ξ : Γa −→ Id, η : Id −→ Da, ζ0 : Da −→ H1a

such that, for each R-module M , the sequence

Γa(M)ξM−→M

ηM−→ Da(M)ζ0M−→ H1

a(M) −→ 0

is exact. Since Γa and H1a commute with direct limits (by 3.4.4 and 3.4.10

respectively), the claim follows from 3.4.6.

3.4.12 Exercise. Suppose that, for some i ∈ N0, we have Hia(R) = 0. Show

that Hia(P ) = 0 for every projective R-module P .

3.4.13 Exercise. Let K ′ denote the full ring of fractions of R; that is, K ′ :=S−1R, where S is the set of all non-zerodivisors of R. Show that, if, for someinteger i ∈ N0, we have Hi

a(R) = 0, then Hia(K

′) = 0 too.Thus, in particular, if R is an integral domain with field of fractions K and

Hia(R) = 0 for some i ∈ N0, then Hi

a(K) = 0.

4

Change of rings

The main results of this chapter concern a homomorphism of commutativeNoetherian rings f : R −→ R′. More precisely, we shall prove two funda-mental comparison results for local cohomology modules in this context. Thefirst of these, which we shall call the ‘Independence Theorem’, compares, foran R′-module M ′ and an i ∈ N0, the local cohomology modules Hi

a(M′)

and HiaR′(M ′): to form the first of these, we consider M ′ as an R-module

by restriction of scalars using f ; also, aR′ denotes the extension of a to R′

under f . Our second main result, which we shall refer to as the ‘Flat BaseChange Theorem’, compares the local cohomology modules Hi

a(M) ⊗R R′

and HiaR′(M ⊗R R′) for i ∈ N0 and an arbitrary R-module M under the

additional assumption that the ring homomorphism f is flat.Our main results rely on the fact that certain modules are acyclic with respect

to torsion functors. We say that an R-module A is Γa-acyclic precisely whenHi

a(A) = 0 for all i > 0. As was explained in 1.2.2, the most basic methodfor calculation, for an R-module M and an i ∈ N0, of Hi

a(M) is to take aninjective resolution I• of M , apply Γa to I• to obtain the complex Γa(I

•),and take the i-th cohomology module of this complex: we have Hi

a(M) =

Hi(Γa(I•)). However, it is an easy exercise to show that a resolution of M by

Γa-acyclic R-modules will serve this purpose just as well.Of course, injective R-modules are Γa-acyclic, but in general the class of

Γa-acyclic R-modules is larger than the class of injective R-modules. Let f :

R −→ R′ be a homomorphism of commutative Noetherian rings. We shallshow that, if I ′ is an injective R′-module, then, when viewed as an R-moduleby restriction of scalars, I ′ is Γa-acyclic. We shall also show that, if the ringhomomorphism f is flat, then, for each injective R-module I , the R′-moduleI ⊗R R′ is ΓaR′ -acyclic. These results pave the way for the proofs later inthe chapter of the Independence Theorem (4.2.1) and the Flat Base ChangeTheorem (4.3.2).

66 Change of rings

4.1 Some acyclic modules

For completeness, we begin with the formal definition.

4.1.1 Definition. We say that an R-module A is Γa-acyclic precisely whenHi

a(A) = 0 for all i > 0. If B is a system of ideals (of R) over Λ as in 2.1.10,then we say that A is ΓB-acyclic if and only if Hi

B(A) = 0 for all i > 0.

Our presentation does not depend on the following exercise, but the readershould be aware of the result it contains.

4.1.2 Exercise. Show that, given an R-module M , its local cohomology mo-dules with respect to a can be calculated by means of a resolution of M byΓa-acyclic modules as follows. Let

A• : 0d−1

−→ A0 d0

−→ A1 −→ · · · −→ Ai di

−→ Ai+1 −→ · · ·

be a Γa-acyclic resolution of M , so that A0, A1, . . . , Ai, . . . are all Γa-acyclicR-modules and there is an R-homomorphism α : M → A0 such that thesequence

0 −→Mα−→ A0 d0

−→ A1 −→ · · · −→ Ai di

−→ Ai+1 −→ · · ·

is exact. Show that

Hia(M) ∼= Ker(Γa(d

i))/ Im(Γa(di−1)) for all i ∈ N0.

We shall have two applications of our first proposition in this section.

4.1.3 Proposition. Let M be an R-module, and let C be a set of ideals of Rsuch that

(a) C is closed under the formation of finite sums and products,(b) 0R ∈ C, and(c) each ideal in C is the sum of finitely many principal ideals which belong

to C.

Assume that H1c (M) = 0 for all c ∈ C. Then M is Γc-acyclic for all c ∈ C.

Remark. Of course, if M is Γc-acyclic for all c ∈ C, then it is automatic thatH1

c (M) = 0 for all c ∈ C. Thus this theorem could be phrased as an ‘if andonly if’ result.

Proof. Since Γ0R is the identity functor on C(R), it is clear that M is Γ0R-acyclic. Also, it was proved in 2.2.21(ii) that, for a ∈ R, we haveHi

Ra(M) = 0

for all i ∈ N with i > 1; hence M is Γc-acyclic whenever c is a principal idealin C.

4.1 Some acyclic modules 67

Now suppose, inductively, that t > 1 and we have proved that, wheneverc′ ∈ C can be expressed as a sum of fewer than t principal ideals which allbelong to C, then M is Γc′ -acyclic. This is certainly the case when t = 2. Letc = Rc1 + · · · + Rct, where c1, . . . , ct ∈ R and Rc1, . . . , Rct ∈ C. We shalluse the Mayer–Vietoris sequence to show that M is also Γc-acyclic.

Set a = Rc1+ · · ·+Rct−1 and b = Rct, so that c = a+b. Note that b ∈ C,and, by hypothesis (a) on C, each of a, ab, Rc1ct, . . . , Rct−1ct belongs to C.Since b is principal, and a and

ab = (Rc1 + · · ·+Rct−1)Rct = Rc1ct + · · ·+Rct−1ct

are both expressible as sums of t− 1 principal ideals belonging to C, it followsfrom 2.2.21(ii) and the inductive hypothesis that M is Γa-acyclic, Γb-acyclicand Γab-acyclic.

By the Mayer–Vietoris sequence 3.2.3, we have, for an arbitrary i > 1,an exact sequence Hi−1

a∩b(M) −→ Hic(M) −→ Hi

a(M) ⊕ Hib(M). Now

Hi−1a∩b(M) = Hi−1

ab (M) by 3.1.5, and so, by what we have already established,Hi−1

a∩b(M) = Hia(M) = Hi

b(M) = 0. Thus Hic(M) = 0 for all i > 1. Since

H1c (M) = 0 by hypothesis (because c ∈ C), this completes the inductive step.Since each ideal in C can be expressed as the sum of finitely many principal

ideals which belong to C, the proof is complete.

The two important applications of Proposition 4.1.3 which we have in mindconcern a homomorphism of commutative Noetherian rings, and it is appro-priate for us to clarify notation at this point.

4.1.4 Notation. Throughout this chapter, R′ will denote a second commuta-tive Noetherian ring and f : R −→ R′ will denote a ring homomorphism. Asin Chapter 2, for an ideal b of R, we shall use bR′ to denote the extension of bto R′ under f .

For an R-module M , we shall use ZdvR(M) (or Zdv(M) if there is no am-biguity about the underlying ring involved) to denote the set of elements of Rwhich are zerodivisors on M . Thus, for example, if M ′ is an R′-module, thenR \ ZdvR(M ′) denotes the set of elements of R which are non-zerodivisorson M ′ when the latter is regarded as an R-module by means of f .

4.1.5 Lemma. Let M ′ be a finitely generated R′-module that is aR′-torsion-free. Then a ⊆ ZdvR(M

′); in other words, a contains a non-zerodivisor onM ′ when the latter is regarded as an R-module by means of f .

Proof. Suppose that a ∩ (R \ ZdvR(M ′)) = ∅ and look for a contradiction.Then f(a) ⊆ ZdvR′(M ′) =

⋃P∈AssR′ M ′ P. Since M ′ is a finitely generated

R′-module, it follows from the Prime Avoidance Theorem in the form given in

68 Change of rings

[81, 3.61] that f(a) ⊆ P for some P ∈ AssR′ M ′. Since there existsm′ ∈M ′

with (0 :R′ m′) = P, we deduce that 0 = m′ ∈ (0 :M ′ aR′). This is acontradiction.

We are now ready for the proof of one of the main results of this section.

4.1.6 Theorem. Let I ′ be an injective R′-module. Then I ′, when viewed asan R-module by means of f , is Γa-acyclic.

Proof. The strategy of this proof is to show first that H1a(I

′) = 0, and thenappeal to Proposition 4.1.3.

By 2.2.24, there is an R-isomorphism εI′ : DaR′(I ′) −→ Da(I′) such that

the diagram

I ′ηaR′,I′

DaR′(I ′)�

εI′∼=

I ′ηa,I′

Da(I′)�

�

commutes. Since H1aR′(I ′) = 0, it follows from 2.2.6(i)(c) that ηaR′,I′ is sur-

jective. Hence ηa,I′ is surjective, and so H1a(I

′) = 0, again by 2.2.6(i)(c).It follows that H1

a(I′) = 0 for an arbitrary ideal a of R. We can now apply

4.1.3 (with the set of all ideals ofR as the set C) to deduce that I ′ is Γb-acyclicfor every ideal b of R.

We now move on to consider the situation in which the ring homomorphismf : R→ R′ is flat, that is, such that R′, when viewed as an R-module via f , isflat. We need two preliminary results for this situation; for the first, we appealto [50] for a proof.

4.1.7 Lemma. (See [50, Theorem 7.11].) Assume that the ring homomor-phism f : R → R′ is flat. Let M be an R-module. Then there is a naturaltransformation of functors

μ : HomR( • ,M)⊗R R′ −→ HomR′(( • )⊗R R

′,M ⊗R R′)

(from C(R) to C(R′)) which is such that

(i) μN (g ⊗ r′) = r′(g ⊗ IdR′) for each R-module N , each r′ ∈ R′ andeach g ∈ HomR(N,M), and

(ii) μN is an isomorphism if N is a finitely generated R-module. �

4.1 Some acyclic modules 69

4.1.8 Proposition. Assume that the ring homomorphism f : R → R′ is flat;let I be an injective R-module. Then the natural R′-homomorphism

β : I ⊗R R′ −→ HomR′(aR′, I ⊗R R

′)

(for which β(m′)(r′) = r′m′ for all r′ ∈ aR′ and all m′ ∈ I ⊗R R′) issurjective.

Proof. Since f is flat, the natural R′-epimorphism γ : a⊗R R′ −→ aR′ (for

which γ(∑t

i=1 ai ⊗ r′i) =∑t

i=1 f(ai)r′i for all t ∈ N, a1, . . . , at ∈ a and

r′1, . . . , r′t ∈ R′) is an isomorphism (see [50, Theorem 7.7]).

Since a is a finitely generated R-module, the R′-homomorphism

μa : HomR(a, I)⊗R R′ −→ HomR′(a⊗R R

′, I ⊗R R′)

of 4.1.7 is an isomorphism. Since I is R-injective, the R-homomorphism σ :

I → HomR(a, I) (for which σ(m)(a) = am for all a ∈ a and all m ∈ I) issurjective, so that, since tensor product is right exact,

σ ⊗ IdR′ : I ⊗R R′ −→ HomR(a, I)⊗R R

′

is an R′-epimorphism. It follows that the composition

HomR′(γ−1, IdI⊗RR′)◦μa ◦(σ⊗IdR′) : I⊗RR′ −→ HomR′(aR′, I⊗RR

′)

is surjective. Since this composition is just β, the proof is complete.

We are now in a position to prove a theorem which has some similarities toTheorem 4.1.6.

4.1.9 Theorem. Assume that the ring homomorphism f : R → R′ is flat,and let I be an injective R-module. Then I ⊗R R

′ is ΓaR′ -acyclic.

Proof. We shall employ a strategy similar to that which we used in the proofof 4.1.6: we shall show first that H1

aR′(I⊗RR′) = 0, and then we shall appeal

to Proposition 4.1.3.For each n ∈ N, the natural R′-homomorphism

I ⊗R R′ −→ HomR′(anR′, I ⊗R R

′)

of 4.1.8 is surjective, so that, since anR′ = (aR′)n, the R′-homomorphism

ηI⊗RR′ : I ⊗R R′ −→ DaR′(I ⊗R R

′) = lim−→n∈N

HomR′((aR′)n, I ⊗R R′)

of 2.2.6(i) is surjective. Hence H1aR′(I ⊗R R

′) = 0, by 2.2.6(i)(c).We can now apply 4.1.3 (with the set C taken as the set of all ideals of R′

which are extended from ideals of R) to deduce that I ⊗R R′ is ΓbR′ -acyclic

for every ideal b of R.

70 Change of rings

Theorems 4.1.6 and 4.1.9 pave the way for us to present, in the next twosections, the two main results of this chapter.

4.2 The Independence Theorem

In this section, we prove the Independence Theorem, which shows, looselyspeaking, that local cohomology is ‘independent of the base ring’. We havealready seen results of a similar type for ideal transforms in 2.2.24, and for firstlocal cohomology modules in Exercise 2.2.26. Let us again use � R : C(R′)→C(R) to denote the functor obtained from restriction of scalars (using f ): thus,if M ′ is an R′-module and i ∈ N0, we can form the R-modules Hi

aR′(M ′)� R

andHia(M

′�R). For the first, we form the i-th local cohomology module ofM ′

with respect to aR′ and then regard the resulting R′-module as an R-moduleby means of f ; for the second, we first consider M ′ as an R-module via f , andthen take the i-th local cohomology module with respect to a. Theorem 4.2.1below shows, among other things, that there is an R-isomorphism

HiaR′(M ′) = Hi

aR′(M ′)� R

∼=−→ Hia(M

′� R) =: Hia(M

′),

so that, speaking loosely, it does not matter whether we calculate these localcohomology modules over R or R′.

4.2.1 Independence Theorem. The functors ΓaR′(•)�R and Γa(•�R) (fromC(R′) to C(R)) are the same, and there is a unique isomorphism

Λ = (λi)i∈N0 :(Hi

aR′( • )� R

)i∈N0

∼=−→(Hi

a( • � R))i∈N0

of negative (strongly) connected sequences of covariant functors (from C(R′)to C(R)) such that λ0 is the identity natural equivalence. In particular, for eachi ∈ N0, the functors Hi

aR′( • )� R and Hia( • � R) are naturally equivalent.

Proof. Since � R : C(R′)→ C(R) is covariant and exact, it is clear that

(HiaR′( • )� R)i∈N0 and (Hi

a( • � R))i∈N0

are negative strongly connected sequences of covariant functors from C(R′) toC(R).

Now anR′ = (aR′)n for all n ∈ N, and so ΓaR′( • )� R and Γa( • � R) arethe same functor. Furthermore, whenever I ′ is an injective R′-module, it is,of course, automatic that Hi

aR′(I ′) = 0 for all i ∈ N, while it follows from4.1.6 that Hi

a(I′) = 0 for all i ∈ N. The result now follows immediately on

application of 1.3.4(ii).

4.2 The Independence Theorem 71

Remark. Some readers might prefer to approach the final claim of 4.2.1 inthe following way. Observe first that ΓaR′( • )� R and Γa( • � R) are the samefunctor. Let M ′ be an R′-module, and let

I ′• : 0 −→ I ′0 −→ I ′1 −→ · · · −→ I ′i −→ I ′i+1 −→ · · ·

be an injective resolution ofM ′ overR′. By 1.2.2, the i-th cohomology moduleof the complex ΓaR′(I ′•) is isomorphic to Hi

aR′(M ′) (for i ∈ N0).By 4.1.6, the complex

I ′•� R : 0 −→ I ′0� R −→ I ′1� R −→ · · · −→ I ′i� R −→ I ′i+1� R −→ · · ·

is a resolution of M ′� R by Γa-acyclic R-modules, and so, by 4.1.2, the i-thcohomology module of the complex Γa(I

′•� R) is isomorphic to Hia(M

′� R)

(for i ∈ N0). Since ΓaR′( • )� R = Γa( • � R), this gives a quick and relativelytransparent proof that Hi

aR′(M ′)�R ∼= Hia(M

′�R) for i ∈ N0. Note, however,that our use of 1.3.4 in the above proof of 4.2.1 led rather rapidly to additionalinformation.

4.2.2 Example. Suppose that the R-module M is annihilated by the ideal bof R. Then, since M is b-torsion, it is immediate from Exercise 2.1.9 thatHi

a(M) ∼= Hia+b(M) for all i ∈ N0. However, the reader might find the

following alternative approach, which uses the Independence Theorem 4.2.1,illuminating.

Since M is annihilated by b, we can regard M as a module over R/b in anatural way. Let i ∈ N0. By the Independence Theorem 4.2.1, there is an R-isomorphism Hi

a(M) ∼= Hia(R/b)(M). But a(R/b) = (a + b)/b, and another

use of the Independence Theorem 4.2.1 provides us with an R-isomorphismHi

(a+b)/b(M) ∼= Hia+b(M), so that Hi

a(M) ∼= Hia+b(M), as claimed.

4.2.3 Remark. In 3.3.4 we defined the cohomological dimension cohd(a)

of a. It is immediate from the Independence Theorem 4.2.1 that cohd(aR′) ≤cohd(a).

4.2.4 Exercise. Let n ∈ N and consider the ring R[X1, . . . , Xn] of polyno-mials over R.

(i) Show that the R[X1]-module H1(X1)

(R[X1]) is not finitely generated,and therefore non-zero. (Here is a hint: you might find 2.2.21 helpful.)

(ii) Use 1.3.9(iv) to show that, when n > 1,

Hn−1(X1,...,Xn)

(R[X1, . . . , Xn]) = 0.

72 Change of rings

(iii) The ring homomorphism ν : R[X1, . . . , Xn] → R[X1, . . . , Xn−1]

(again for n > 1) obtained by evaluation at X1, . . . , Xn−1, 0 allowsus to regardR[X1, . . . , Xn−1] as anR[X1, . . . , Xn]-module. Show that,when this is done,

Hn(X1,...,Xn)

(R[X1, . . . , Xn−1]) = 0.

(iv) Assume again that n > 1. Deduce from the exact sequence

0 −→ R[X1, . . . , Xn]Xn−→ R[X1, . . . , Xn]ν−→ R[X1, . . . , Xn−1] −→ 0

ofR[X1, . . . , Xn]-modules thatHn(X1,...,Xn)

(R[X1, . . . , Xn]) has a sub-module isomorphic to Hn−1

(X1,...,Xn−1)(R[X1, . . . , Xn−1]) (the latter be-

ing regarded as an R[X1, . . . , Xn]-module by means of ν), and that

Hn(X1,...,Xn)

(R[X1, . . . , Xn]) = XnHn(X1,...,Xn)

(R[X1, . . . , Xn]).

(v) Prove by induction that, for all n ∈ N, the R[X1, . . . , Xn]-moduleHn

(X1,...,Xn)(R[X1, . . . , Xn]) is not finitely generated, and so is non-

zero.

4.2.5 Exercise. Use 4.2.4 to show that, over the ring R[X1, X2, X3, X4] ofpolynomials in four indeterminates with coefficients in R,

H3(X1,X2)∩(X3,X4)

(R[X1, X2, X3, X4]) = 0.

Deduce that the affine algebraic set W in A4 defined by

W := VA4(X1, X2) ∪ VA4(X3, X4)

cannot be ‘defined by two equations’, that is, there do not exist two polynomi-als f, g ∈ C[X1, X2, X3, X4] such that W =

{p ∈ A4 : f(p) = g(p) = 0

}.

We are now going to use the Independence Theorem to prove a proposi-tion which presents a non-vanishing result for certain local cohomology mod-ules. We shall use this proposition in Chapter 6 to prove a more general non-vanishing result. The strategy of the proof of the proposition has some similar-ities with the strategy used in Exercise 4.2.4.

4.2.6 Proposition. Suppose that (R,m) is a regular local ring of dimensiond > 0. Then the R-module Hd

m(R) is not finitely generated, and thereforenon-zero.

4.2 The Independence Theorem 73

Proof. We argue by induction on d. When d = 1, the maximal ideal m isgenerated by one element, π say. Let K denote the field of fractions of R.Apply 2.2.18 (in the case where B = (Rπn)n∈N, M = R and S is the set ofnon-zero elements of R), and note that

⋃n∈N(R :K πn) = K: it follows from

this and 2.2.6(i)(c) that there is an exact sequence

0 −→ R −→ K −→ H1Rπ(R) −→ 0.

Since R is finitely generated as an R-module but K is not, we deduce thatH1

Rπ(R) is not finitely generated.Now suppose, inductively, that d > 1 and that the result has been proved

for regular local rings of smaller (positive) dimensions. Let u1, . . . , ud be delements of m which generate this maximal ideal. Let R := R/udR, a regularlocal ring of dimension d− 1 with maximal ideal m := m/udR. By the induc-tive hypothesis, the R-module Hd−1

m (R) is not finitely generated. Hence, bythe Independence Theorem 4.2.1, the R-module Hd−1

m (R/udR) is not finitelygenerated. The exact sequence

0 −→ Rud−→ R −→ R/udR −→ 0

induces an exact sequence Hd−1m (R) −→ Hd−1

m (R/udR) −→ Hdm(R). But

Hd−1m (R) = 0 by 1.3.9(iv) since u1, . . . , ud is an R-sequence. Hence Hd

m(R)

has a submodule isomorphic to Hd−1m (R/udR); since the latter is not finitely

generated, neither is Hdm(R).

4.2.7 Exercise. Assume that R contains a field K as a subring. Supposethat a can be generated by t elements a1, . . . , at. The ring homomorphismf : K[X1, . . . , Xt] −→ R obtained by evaluation at a1, . . . , at allows us toregard each R-module as a module over K[X1, . . . , Xt].

Show that, for an R-module M , there is a K[X1, . . . , Xt]-isomorphism

Hi(X1,...,Xt)

(M) ∼= Hia(M) for each i ∈ N0.

Use the fact that the ring K[X1, . . . , Xt] has global dimension t to show thatHi

a(M) = 0 for all i > t. Compare this with Theorem 3.3.1.

4.2.8 Exercise. (In this exercise, there is no assumption about R beyondthe standard ones.) Suppose that a can be generated by t elements. Use anappropriate ring homomorphism Z[X1, . . . , Xt] −→ R in conjunction withthe Independence Theorem 4.2.1 and the fact that the global dimension ofZ[X1, . . . , Xt] is t + 1 to show that Hi

a(M) = 0 for all i > t + 1. Com-pare this with Theorem 3.3.1.

The following refinement of the Independence Theorem 4.2.1 is occasion-ally useful.

74 Change of rings

4.2.9 Exercise. Let the situation be as in the Independence Theorem 4.2.1.Let M ′ be an R′-module. Show that, for each i ∈ N0, the R-module Hi

a(M′)

actually has a natural structure as an R′-module under which

r′x = Hia(r

′ IdM ′)(x) for all r′ ∈ R′ and x ∈ Hia(M

′).

Show further that, if f ′ :M ′ −→ N ′ is a homomorphism of R′-modules, thenHi

a(f′) : Hi

a(M′) −→ Hi

a(N′) is an R′-homomorphism.

Deduce that (Hia(•�R))i∈N0 is a negative strongly connected sequence of co-

variant functors from C(R′) to itself, and, as such, is isomorphic to (HiaR′)i∈N0

.

4.3 The Flat Base Change Theorem

The second fundamental result about local cohomology that we propose toderive from the work in §4.1 is the Flat Base Change Theorem. This result isa consequence of Theorem 4.1.9 and concerns the situation in which f : R→R′ is flat: we are going to show that, in this situation, speaking loosely, theformation of local cohomology ‘commutes’ with ‘extension of the base ringfrom R to R′’. More precisely, we shall show that, for each R-module M andeach i ∈ N0, there is anR′-isomorphismHi

aR′(M⊗RR′)

∼=−→ Hia(M)⊗RR

′.Although our proof of this result has some similarities with our proof of 4.2.1above, a preparatory lemma will be helpful in this case.

4.3.1 Lemma. Assume that the ring homomorphism f : R → R′ is flat.There is a natural equivalence of functors (from C(R) to C(R′))

ρ0 : Γa( • )⊗R R′ −→ ΓaR′(( • )⊗R R

′)

which is such that, for each R-module M , we have ρ0M (m⊗ r′) = m⊗ r′ forall m ∈ Γa(M) and all r′ ∈ R′.

Proof. Let M be an R-module. Since f is flat, the inclusion map induces anR′-monomorphism Γa(M) ⊗R R′ → M ⊗R R′ whose image is containedin ΓaR′(M ⊗R R′). It follows easily that there is a natural transformation offunctors ρ0 : Γa( • )⊗R R

′ −→ ΓaR′(( • )⊗R R′) which is such that, for each

R-module M , ρ0M is monomorphic and satisfies

ρ0M

(∑ti=1mi ⊗ r′i

)=∑t

i=1mi ⊗ r′i

for all t ∈ N, m1, . . . ,mt ∈ Γa(M) and r′1, . . . , r′t ∈ R′. It remains to show

that ρ0M is surjective for each R-module M .

4.3 The Flat Base Change Theorem 75

We can deduce this from 4.1.7 as follows. Let n ∈ N. Since R/an is afinitely generated R-module, 4.1.7 provides us with an R′-isomorphism

μR/an : HomR(R/an,M)⊗R R

′ ∼=−→ HomR′((R/an)⊗R R′,M ⊗R R

′)

such that μR/an(h⊗ r′) = r′(h⊗ IdR′) for all h ∈ HomR(R/an,M) and all

r′ ∈ R′. We can incorporate μ := μR/an into the composition

(0 :M an)⊗R R′ ∼=−→ HomR(R/a

n,M)⊗R R′

μ−→ HomR′((R/an)⊗R R′,M ⊗R R

′)∼=−→ HomR′(R′/anR′,M ⊗R R

′)∼=−→ (0 :M⊗RR′ (aR′)n)

of R′-isomorphisms, in which the other three isomorphisms are the obviousnatural ones. The reader can easily check that this composition ε is such thatε(∑t

i=1mi ⊗ r′i) =∑t

i=1mi ⊗ r′i (for t ∈ N, m1, . . . ,mt ∈ (0 :M an) andr′1, . . . , r

′t ∈ R′). It is now clear that ρ0M is surjective.

4.3.2 Flat Base Change Theorem. Assume that the ring homomorphism f :

R −→ R′ is flat. There is a unique isomorphism

(ρi)i∈N0 : (Hia( • )⊗R R

′)i∈N0

∼=−→ (HiaR′(( • )⊗R R

′))i∈N0

of negative (strongly) connected sequences of covariant functors (from C(R)to C(R′)) which extends the natural equivalence ρ0 of 4.3.1. In particular, foreach i ∈ N0, the functors Hi

a( • )⊗R R′ and Hi

aR′(( • )⊗R R′) are naturally

equivalent.

Proof. As in the proof of 4.2.1, we use 1.3.4. Since ( • ) ⊗R R′ : C(R) −→

C(R′) is exact, it is clear that the two sequences given in the statement of4.3.2 are indeed negative strongly connected sequences of covariant functorsfrom C(R) to C(R′). Furthermore, whenever I is an injective R-module, itis, of course, automatic that Hi

a(I) = 0 for all i ∈ N, while it follows from4.1.9 that Hi

aR′(I ⊗R R′) = 0 for all i ∈ N. The result is now an immediateconsequence of 4.3.1 and 1.3.4(ii).

Remark. Again, some readers might prefer the following direct approach tothe final claim of 4.3.2. Let M be an R-module, and let

I• : 0 −→ I0 −→ I1 −→ · · · −→ Ii −→ Ii+1 −→ · · ·

be an injective resolution of M over R. For i ∈ N0, the i-th cohomology

76 Change of rings

module of the complex Γa(I•) is, of course, Hi

a(M). Since ( • ) ⊗R R′ :

C(R) −→ C(R′) is additive and exact, we have an R′-isomorphism

Hia(M)⊗R R

′ = Hi(Γa(I•))⊗R R

′ ∼= Hi(Γa(I•)⊗R R

′).

In view of the natural equivalence of 4.3.1, there is an R′-isomorphism

Hi(Γa(I•)⊗R R

′) ∼= Hi(ΓaR′(I• ⊗R R′)).

But, by 4.1.9, the complex I• ⊗R R′ is a ΓaR′ -acyclic resolution of the R′-module M ⊗R R

′, and so, by 4.1.2, there is an R′-isomorphism

Hi(ΓaR′(I• ⊗R R′)) ∼= Hi

aR′(M ⊗R R′).

This gives us, fairly directly, an R′-isomorphism

Hia(M)⊗R R

′ ∼= HiaR′(M ⊗R R

′).

Note again, however, that our use of 1.3.4 in the above proof of 4.3.2 led veryquickly to additional information.

If S is a multiplicatively closed subset of R, then, as is very well known, thenatural ring homomorphism R → S−1R is flat. Thus we can obtain the resultof Exercise 1.3.6 as a special case of the Flat Base Change Theorem 4.3.2.

4.3.3 Corollary. Let S be a multiplicatively closed subset of R. Then(S−1(Hi

a( • )))i∈N0

and(Hi

aS−1R(S−1( • ))

)i∈N0

are isomorphic connected sequences of functors (from C(R) to C(S−1R)).In particular, for every i ∈ N0 and every R-module M , there is anS−1R-isomorphism S−1(Hi

a(M)) ∼= HiaS−1R(S

−1M). �

4.3.4 Exercise. Suppose that R is regular; that is, Rp is a regular local ringfor every p ∈ Spec(R).

Suppose that a is proper and has height t > 0. Show that the R-moduleHt

a(R) is not finitely generated, and therefore non-zero.

The first part of the next exercise is for those readers who have reached thispoint without needing the fact that, for an arbitrary R-module N , the functor( • )⊗R N : C(R) −→ C(R) commutes with direct limits!

4.3.5 �Exercise. Let N be an R-module.

(i) Show that the functor (•)⊗RN : C(R) −→ C(R) commutes with directlimits (see 3.4.1).


(ii) Assume that the ring homomorphism f : R→ R′ is flat. Show that thereis a natural equivalence of functors

ε′ : Da( • )⊗R R′ −→ DaR′(( • )⊗R R

′)

(from C(R) to C(R′)) which is such that, for each R-module M , thediagram

M ⊗R R′ ηa,M⊗RR′

Da(M)⊗R R′�

ε′M

M ⊗R R′

ηaR′,M⊗RR′DaR′(M ⊗R R

′)��

commutes. (The notations ηa,M and ηaR′,M⊗RR′ are explained in thenote following the statement of 2.2.6.)

(iii) Deduce that, for a multiplicatively closed subset S of R, the functors

S−1(Da( • )) and DaS−1R(S−1( • ))

(from C(R) to C(S−1R)) are naturally equivalent.

We shall end this chapter with a geometric example and exercise; the fol-lowing remark will be useful.

4.3.6 Remark. Let r1, r2 ∈ R. Then

{p ∈ Spec(R) : p ⊇ r1r2R+ a}

={p ∈ Spec(R) : p ⊇

√(r1R+ a) ∩

√(r2R+ a)

},

and so√

(r1r2R+ a) =√(r1R+ a) ∩

√(r2R+ a).

4.3.7 Example. This example concerns the affine variety V in A4 studied in2.3.7 and 3.3.5 and given by

V := VA4(X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

In 3.3.5 we showed that the subvariety

L := VA4(X2, X3, X4) ={(c, 0, 0, 0) ∈ A4 : c ∈ C

}of V (a line) cannot be ‘defined by one equation’; here, we shall show that Vitself cannot be ‘defined by two equations’.

Write, for convenience, R := C[X1, X2, X3, X4], and let c be the ideal ofR given by

c = (X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

78 Change of rings

We wish to show that ara(c) > 2, and we shall achieve this by showing thatH3

c (R) = 0 and then appealing to 3.3.3. Let R := C[[X1, X2, X3, X4]]; sincethe natural ring homomorphism R → R is flat, it follows from the Flat BaseChange Theorem 4.3.2 thatH3

c (R)⊗R R ∼= H3cR(R), and so it will be enough,

in order to achieve our aim, for us to show that H3cR(R) = 0. This is what we

shall do.Let u ∈ R be given by

u := 1+ 2X3 − 2X23 +4X3

3 − 10X43 + · · ·+ (−1)n−12(2n− 2)!

n!(n− 1)!Xn

3 + · · · .

In fact, u2 = 1 + 4X3; note that, since the latter is a unit in R (by [81, 1.43]),so too is u. Write

f1 := X1X4−X2X3, f2 := X21X3+X1X2−X2

2 , f3 := X33+X3X4−X2

4 ,

so that cR = (f1, f2, f3)R. Now

4f2 = 4X21X3 +X2

1 − (4X22 − 4X1X2 +X2

1 )

= X21u

2 − (2X2 −X1)2

= (X1u+ 2X2 −X1)(X1u− 2X2 +X1)

= (X1(u− 1) + 2X2)(X1(u+ 1)− 2X2).

A similar calculation shows that

4f3 = 4X33 +X2

3 − (4X24 − 4X3X4 +X2

3 )

= (X3(u− 1) + 2X4)(X3(u+ 1)− 2X4).

Set

g1 := X1(u− 1) + 2X2, g2 := X1(u+ 1)− 2X2,

h1 := X3(u− 1) + 2X4, h2 := X3(u+ 1)− 2X4.

Note that

h1g2 − h2g1 = X1(u+ 1)2X4 −X3(u− 1)2X2

− 2X2X3(u+ 1) + 2X4X1(u− 1)

= 4u(X1X4 −X2X3) = 4uf1,

so that f1 = u−1(h1g2 − h2g1)/4.


Therefore, on use of 4.3.6, we have√cR =

√(f1, f2, f3)R =

√(f1, g1g2, h1h2)R

=

√(f1, g1, h1h2)R ∩

√(f1, g2, h1h2)R

=√d11 ∩

√d12 ∩

√d21 ∩

√d22

where dij = (f1, gi, hj)R for i, j = 1, 2. We now use the equation f1 =

u−1(h1g2 − h2g1)/4 to see that d11 = (f1, g1, h1)R = (g1, h1)R and d22 =

(f1, g2, h2)R = (g2, h2)R; the same equation and 4.3.6 show that

√d12 =

√(f1, g1, h2)R

=

√(h1g2, g1, h2)R =

√(h1, g1, h2)R ∩

√(g2, g1, h2)R

and√d21 =

√(f1, g2, h1)R =

√(h2, g2, h1)R ∩

√(g1, g2, h1)R.

Now put all this information together to see that√cR =

√(g1, h1)R ∩

√(g2, h2)R.

Next, note that, since u is a unit of R, the four elements g1, g2, h1, h2 gen-erate the maximal ideal (X1, X2, X3, X4)R of the 4-dimensional regular localring R, and so, by [81, 15.38] for example, p := (g1, h1)R and q := (g2, h2)R

are prime ideals of R. Thus we have the interesting situation where, although√c is, by 2.3.7, a prime ideal of R, the ideal

√cR of R is the intersection of

two distinct prime ideals of height 2.Note that p + q = m := (X1, X2, X3, X4)R, the maximal ideal of R.

Since X1, X2, X3, X4 is an R-sequence in m, it follows from 1.3.9(iv) thatH3

p+q(R) = 0. However, Proposition 4.2.6 shows that H4p+q(R) = 0. By

3.3.1, we have Hip(R) ⊕ Hi

q(R) = 0 for all i > 2. It therefore follows from4.3.2, 1.2.3 and the Mayer–Vietoris sequence 3.2.3 that

H3c (R)⊗R R ∼= H3

cR(R) = H3

p∩q(R)∼= H4

m(R) = 0.

Hence H3c (R) = 0, and so ara(c) > 2 by 3.3.3. Therefore V cannot be ‘de-

fined by two equations’.

4.3.8 Exercise. Let K be a field of characteristic 0, and let R denote the ringK[X1, X2, X3, X4]; let c be the ideal of R given by

c = (X21 +X3

1 −X22 , X

23 +X1X

23 −X2

4 , X2X3 −X1X4).

80 Change of rings

Let R := K[[X1, X2, X3, X4]] and let u ∈ R be given by

u := 1+1

2X1−

1

8X2

1+1

16X3

1−5

128X4

1+· · ·+(−1)n−12(2n− 2)!

4nn!(n− 1)!Xn

1 +· · · ,

so that u2 = 1+X1. Let z1 := X1u−X2, z2 := X3u−X4, z3 := X1u+X2

and z4 := X3u+X4.

(i) Show that z1, z2, z3, z4 generate the maximal ideal of R.(ii) Show that, in R,

X2X3 −X1X4 = X1z2 −X3z1 = X3z3 −X1z4

and 2u(X2X3 −X1X4) = z2z3 − z1z4. Deduce that√cR = (z1, z2)R ∩ (z3, z4)R.

(iii) Prove that H3c (R) = 0.

4.3.9 Exercise. With the notation of 2.3.1, let V be the affine algebraic set inA4 given by

V := VA4(X21 +X3

1 −X22 , X

23 +X1X

23 −X2

4 , X2X3 −X1X4).

(i) By considering the morphism of varieties α : A2 → A4 for whichα((c, d)) = (c2 − 1, c3 − c, d, cd) for all (c, d) ∈ A2 and the mapping

β : V \ {(0, 0, 0, 0)} −→ A2 \ {(1, 0), (−1, 0)}

given by

β((c1, c2, c3, c4)) =

{(c2/c1, c3) if c1 = 0,

(c4/c3, c3) if c3 = 0

(for all (c1, c2, c3, c4) ∈ V \ {(0, 0, 0, 0)}), show that V is irreducible,and so is an affine variety.

(ii) Deduce from Exercise 4.3.8 above that V cannot be ‘defined by twoequations’.

5

Other approaches

Although we have now developed enough of the basic algebraic theory of lo-cal cohomology so that we could, if we wished, start right away with seriouscalculations with local cohomology modules, there are two other approachesto the construction of local cohomology modules which are useful, and pop-ular with many workers in the subject. One approach uses cohomology ofCech complexes, and the other uses direct limits of homology modules ofKoszul complexes. Links between local cohomology and Koszul complexesand Cech cohomology are described in A. Grothendieck’s foundational lecturenotes [25, §2]; related ideas are present in J.-P. Serre’s fundamental paper [77,§61]. Among other texts which discuss links between local cohomology andthe Cech complex or Koszul complexes are those by W. Bruns and J. Herzog[7, §3.5], D. Eisenbud [10, Appendix 4], M. Herrmann, S. Ikeda and U. Or-banz [32, §35], P. Roberts [70, Chapter 3, §2], J. R. Strooker [83, §4.3] and J.Stuckrad and W. Vogel [84, Chapter 0, §1.3].

We shall make very little use in this book of the descriptions of local coho-mology modules as direct limits of homology modules of Koszul complexes.However, we will use the approach to local cohomology via cohomology ofCech complexes, and so we present the basic ideas of this approach in thischapter. As this work leads naturally to the connection between local coho-mology and direct limits of homology modules of Koszul complexes, we alsopresent some aspects of that connection.

The Cech complex approach is particularly useful for calculations inHna (R)

when a can be generated by n elements. We shall illustrate this in §5.3 in thecase where R has prime characteristic p. Then each local cohomology moduleHi

a(R) (i ∈ N0) of R itself carries a natural ‘Frobenius action’, that is, anAbelian group homomorphism F : Hi

a(R) −→ Hia(R) such that F (rm) =

rpF (m) for all m ∈ Hia(R) and r ∈ R. The Frobenius action on Hn

a (R),

82 Other approaches

where a can be generated by n elements, can be described very simply usingthe Cech complex.

Later in the book, we shall often study aspects of Hdm(R) where (R,m) is

local and of dimension d. Let a1, . . . , ad be a system of parameters for R, andset q := Ra1 + · · · + Rad. Since

√q = m, we have Hd

m(R) = Hdq (R), and

so, in the case when R has prime characteristic p, the above-mentioned Cechcomplex approach facilitates calculations with the natural Frobenius action onHd

m(R). There is an illustration of this idea in §6.5.

5.1 Use of Cech complexes

Throughout this chapter, a1, . . . , an (where n > 0) will denote n elementswhich generate a, and M will denote an arbitrary R-module. Our first taskis to define the (extended) Cech complex C(M)• of M with respect to thesequence a1, . . . , an. This has the form

0 −→ C(M)0d0

−→ C(M)1 −→ · · · −→ C(M)n−1 dn−1

−→ C(M)n −→ 0.

The following lemma will be helpful. Recall again that, for a ∈ R, the nota-tions Ra and Ma denote, respectively, the ring and module of fractions of Rand M with respect to the multiplicatively closed subset

{ai : i ∈ N0

}of R.

5.1.1 Lemma. Let a, b ∈ R. There is an isomorphism of R-modules

μ :Mab −→ (Ma)b

for which μ(m/(ab)i) = (m/ai)/bi for all m ∈M and all i ∈ N0.

Proof. Suppose that m, y ∈ M and i, j ∈ N0 are such that m/(ab)i =

y/(ab)j in Mab. Then there exists k ∈ N0 such that

(ab)k((ab)jm− (ab)iy

)= 0.

Thus, in Ma, we have bk+jm/ai = bk+iy/aj , so that (m/ai)/bi = (y/aj)/bj

in (Ma)b. Therefore there is indeed a mapping μ : Mab −→ (Ma)b given bythe formula in the statement of the lemma. It is now a very easy exercise tocheck that μ is an R-isomorphism.

5.1.2 �Exercise. Complete the proof of Lemma 5.1.1.

5.1.3 Remark. In situations such as that of 5.1.1, we shall use μ to iden-tify the R-modules Mab and (Ma)b without further comment. Thus, when wespeak of the naturalR-homomorphism ω :Ma −→Mab, we shall be referringto the natural R-homomorphism from Ma to (Ma)b and employing the above

5.1 Use of Cech complexes 83

identification, so that ω(m/ai) = bim/(ab)i for allm ∈M and i ∈ N0. Theseideas are employed in the construction of the Cech complex of M with respectto a1, . . . , an in 5.1.5 below.

5.1.4 Notation. For k ∈ N with 1 ≤ k ≤ n, we shall write

I(k, n) :={(i(1), . . . , i(k)) ∈ Nk : 1 ≤ i(1) < i(2) < . . . < i(k) ≤ n

},

the set of all strictly increasing sequences of length k of positive integers takenfrom the set {1, . . . , n}. For i ∈ I(k, n), we shall, for 1 ≤ j ≤ k, denote thej-th component of i by i(j), so that i = (i(1), . . . , i(k)).

Now suppose that k < n, and s ∈ N with 1 ≤ s ≤ k+1. Let j ∈ I(k+1, n).Then by

j s or (j(1), . . . , j(s), . . . , j(k + 1))

we mean the sequence (j(1), . . . , j(s− 1), j(s+ 1), . . . , j(k + 1)) of I(k, n)obtained by omitting the s-th component of j.

It is perhaps worth pointing out here that, if t ∈ N with 1 ≤ t < s, then(j s)t = (j t)s−1.

Again under the assumption that k < n, let i ∈ I(k, n). By the n-comple-ment of i we mean the sequence j ∈ I(n− k, n) such that

{1, . . . , n} = {i(1), . . . , i(k), j(1), . . . , j(n− k)} .

5.1.5 Proposition and Definition. Define a sequence C(M)• of R-modulesand R-homomorphisms

0 −→ C(M)0d0

−→ C(M)1 −→ · · · −→ C(M)n−1 dn−1

−→ C(M)n −→ 0

as follows:

(a) C(M)0 :=M ;(b) for k = 1, . . . , n, and with the notation of 5.1.4,

C(M)k =⊕

i∈I(k,n)Mai(1)...ai(k)

;

(c) d0 : C(M)0 −→ C(M)1 is to be such that, for each h = 1, . . . , n,the composition of d0 followed by the canonical projection fromC(M)1 to Mah

is just the natural map from M to Mah; and

(d) for k = 1, . . . , n− 1, i ∈ I(k, n) and j ∈ I(k + 1, n), the compo-sition

Mai(1)...ai(k)−→ C(M)k

dk

−→ C(M)k+1 −→Maj(1)...aj(k+1)

(in which the first and third maps are the canonical injection and

84 Other approaches

canonical projection respectively) is to be the natural map fromMai(1)...ai(k)

to Mai(1)...ai(k)aj(s)multiplied by (−1)s−1 if i = j s

for an s with 1 ≤ s ≤ k + 1, and is to be 0 otherwise.

Then C(M)• is a complex, called the (extended) Cech complex of M withrespect to a1, . . . , an. (Henceforth, we shall omit the word ‘extended’.)

We denote C(R)• by

C• : 0 −→ C0 d0

−→ C1 −→ · · · −→ Ci di

−→ Ci+1 −→ · · · −→ Cn −→ 0.

Proof. We have only to prove that dk+1 ◦ dk = 0 for all k = 0, . . . , n− 2.Let m ∈ M . To show that d1 ◦ d0 = 0, it is enough to show that, for each

i ∈ I(2, n), the component of d1 ◦d0(m) in the direct summand Mai(1)ai(2)of

C(M)2 is 0. The only contributions to this component that could conceivablybe non-zero must come ‘through’ the direct summands Mai(1)

and Mai(2)of

C(M)1. It follows that the component of d1 ◦ d0(m) in the direct summandMai(1)ai(2)

of C(M)2 is (−1)(m/1) + (m/1) = 0. Hence d1 ◦ d0 = 0.Now consider the case where 1 ≤ k ≤ n−2. In order to show dk+1◦dk = 0,

it is enough to show that, for each i ∈ I(k, n), the restriction of dk+1 ◦ dk tothe direct summand Mai(1)...ai(k)

of C(M)k is zero, and this is what we shalldo. So let m ∈ M, v ∈ N0 and j ∈ I(k + 2, n): we calculate the componentof

dk+1 ◦ dk(

m

(ai(1) . . . ai(k))v

)in the direct summand Maj(1)...aj(k+2)

of C(M)k+2. This component will be(conceivably) non-zero only if i = (j s)t for some integers s, t with 1 ≤ t <

s ≤ k + 2, and, when this is so, will (in view of the penultimate paragraph of5.1.4) be

(−1)t−1(−1)s−2avj(t)avj(s)m

(aj(1) . . . aj(k+2))v+

(−1)s−1(−1)t−1avj(s)avj(t)m

(aj(1) . . . aj(k+2))v,

which is zero. Hence dk+1 ◦ dk = 0, and the proof is complete.

5.1.6 Example. The reader might find it helpful if we write down explicitlythe Cech complex C• = C(R)• of R with respect to a1, . . . , an when n has afairly small value: when n = 3, the complex is

0→ Rd0

−→ Ra1⊕Ra2

⊕Ra3

d1

−→ Ra2a3⊕Ra1a3

⊕Ra1a2

d2

−→ Ra1a2a3→ 0

where the di (i = 0, 1, 2) are described as follows. For r, r1, r2, r3 ∈ R and


v1, v2, v3 ∈ N0,

d0(r) =(r1,r

1,r

1

),

d1((

r1av11

,r2av22

,r3av33

))=

(av32 r3

(a2a3)v3− av23 r2

(a2a3)v2,

av31 r3(a1a3)v3

− av13 r1(a1a3)v1

,av21 r2

(a1a2)v2− av12 r1

(a1a2)v1

)and

d2((

r1(a2a3)v1

,r2

(a1a3)v2,

r3(a1a2)v3

))=

av11 r1

(a1a2a3)v1− av22 r2

(a1a2a3)v2+

av33 r3

(a1a2a3)v3.

5.1.7 �Exercise. Suppose that n > 1 and M is Ran-torsion. Show that theCech complex ofM with respect to a1, . . . , an is the Cech complex ofM withrespect to a1, . . . , an−1.

5.1.8 �Exercise. Show that a homomorphism f : M → N of R-modulesinduces a chain map of complexes C(f)• : C(M)• −→ C(N)• such thatC(f)0 : C(M)0 → C(N)0 is just f :M → N .

Show further that, with these assignments, C( • )• becomes a functor fromthe category C(R) to the category of complexes ofR-modules (andR-homom-orphisms) and chain maps of such complexes.

We remind the reader that we use the notation

C• : 0 −→ C0 d0

−→ C1 −→ · · · −→ Ci di

−→ Ci+1 −→ · · · −→ Cn −→ 0

for the Cech complex of R itself with respect to a1, . . . , an. We interpret Ci as0 for i ∈ Z \ {0, 1, . . . , n}, of course.

It should be clear to the reader that Hi(( • ) ⊗R C•) is, for each i ∈ N0, acovariant R-linear functor from C(R) to itself.

5.1.9 Lemma. The sequence (Hi(( • ) ⊗R C•))i∈N0 is a negative stronglyconnected sequence of covariant functors from C(R) to itself.

Proof. Let

0 L M N 0� � � �

λ μ ν

0 L′ M ′ N ′ 0

α β

α′ β′� � � �

� � �

86 Other approaches

be a commutative diagram of R-modules and R-homomorphisms with exactrows. Now for each i ∈ N, the R-module Ci is flat, since it is a direct sum offinitely many modules of fractions of the form S−1R for suitable choices ofthe multiplicatively closed subset S ofR. It follows that there is a commutativediagram

0 L⊗R C• M ⊗R C

• N ⊗R C• 0� � � �α⊗C• β⊗C•

α′⊗C• β′⊗C•

0 L′ ⊗R C• M ′ ⊗R C

• N ′ ⊗R C• 0� � � �

μ⊗C•λ⊗C• ν⊗C•

� � �

of complexes of R-modules and chain maps of such complexes such that, foreach i ∈ N0, the sequence

0 L⊗R Ci M ⊗R C

i N ⊗R Ci 0� � � �α⊗Ci β⊗Ci

is exact, and a similar property holds for the lower row. Thus the above commu-tative diagram of complexes gives rise to a long exact sequence of cohomologymodules of the complexes in the top row, a similar long exact sequence for thebottom row, and a chain map of the first long exact sequence into the second.The claim follows from this.

The technique used in the above proof will solve the following exercise.

5.1.10 �Exercise. Show that (Hi(C( • )•))i∈N0 is a negative strongly con-nected sequence of covariant functors from C(R) to itself.

5.1.11 �Exercise. Use the natural isomorphisms

M ⊗R Rai(1)...ai(k)

∼=−→Mai(1)...ai(k)(for i ∈ I(k, n))

(where 1 ≤ k ≤ n) to produce an isomorphism of complexes

ω•M :M ⊗R C

• ∼=−→ C(M)•.

Show further that, as M varies through C(R), the ω•M constitute a natural

equivalence of functors ω• : ( • ) ⊗R C• −→ C( • )• (from C(R) to thecategory of all complexes of R-modules (and R-homomorphisms) and chainmaps of such complexes).

Show also that ω• induces an isomorphism

(Hi(ω•))i∈N0 : (Hi(( • )⊗R C•))i∈N0

∼=−→ (Hi(C( • )•))i∈N0


of negative strongly connected sequences of covariant functors (from C(R) toC(R)).

5.1.12 �Exercise. In this and the next few exercises, we shall be concernedwith relationships between Cech complexes of M with respect to different se-quences of elements of R, and, for such discussions, it will help if we use aslightly more complicated notation. Thus, in this and a few subsequent situa-tions, we shall use C(a1, . . . , an;M)• to denote the Cech complex of M withrespect to a1, . . . , an.

(i) Suppose that n ≥ 2, and let m ∈ N be such that 1 ≤ m < n. Letb1, . . . , bn denote the sequence

a1, . . . , am−1, am+1, am, am+2, . . . , an

obtained from the sequence a1, . . . , an by interchange of the m-th and(m+ 1)-th terms.

Let k ∈ N with 1 ≤ k ≤ n. Let i ∈ I(k, n).If neither m nor m+ 1 appears as a term in i, then

Mai(1)...ai(k)=Mbi(1)...bi(k)

:

define ψ(i) : Mai(1)...ai(k)−→ Mbi(1)...bi(k)

to be the identity mappingin this case.

If m+ 1 does not appear as a term in i but m = i(s), then

Mai(1)...ai(k)=Mbi(1)...bi(s−1)bi(s)+1bi(s+1)...bi(k)

:

define ψ(i) : Mai(1)...ai(k)−→ Mbi(1)...bi(s−1)bi(s)+1bi(s+1)...bi(k)

to bethe identity mapping in this case.

If m does not appear as a term in i but m+ 1 = i(s), then

Mai(1)...ai(k)=Mbi(1)...bi(s−1)bi(s)−1bi(s+1)...bi(k)

:

define ψ(i) : Mai(1)...ai(k)−→ Mbi(1)...bi(s−1)bi(s)−1bi(s+1)...bi(k)

to bethe identity mapping in this case.

If m and m+1 both appear in i, with, say, m = i(s) (so that m+1 =

i(s) + 1 = i(s + 1)), then Mai(1)...ai(k)= Mbi(1)...bi(k)

: in this case,define ψ(i) : Mai(1)...ai(k)

−→ Mbi(1)...bi(k)to be the identity mapping

multiplied by −1.Let

ψk =⊕

i∈I(k,n)ψ(i) : C(a1, . . . , an;M)k −→ C(b1, . . . , bn;M)k.

88 Other approaches

Also, let ψ0 :M −→M be the identity mapping. Show that

Ψ =(ψk)0≤k≤n

: C(a1, . . . , an;M)• −→ C(b1, . . . , bn;M)•

is an isomorphism of complexes.(ii) Let σ be a permutation of the set {1, . . . , n}. Show that there is an iso-

morphism of complexes

C(a1, . . . , an;M)•∼=−→ C(aσ(1), . . . , aσ(n);M)•.

5.1.13 Definition. Let m ∈ Z, and let

Q• : · · · Qk Qk+1dkQ• dk+1

Q•Qk+2 · · ·� � � �

be a complex of R-modules and R-homomorphisms. We define {m}Q•, theresult of shifting Q• by m places, as follows: ({m}Q•)k = Qm+k for all k ∈Z, and the k-th ‘differentiation’ homomorphism dk{m}Q• in {m}Q• is given

by dk{m}Q• = dm+kQ• : Qm+k −→ Qm+k+1. Thus changing Q• into {1}Q•

(respectively {−1}Q•) amounts to shifting it one place to the left (respectivelyright).

5.1.14 �Exercise. In this exercise, we use the extended notation for Cechcomplexes introduced in Exercise 5.1.12.

Let b ∈ R. Show that there is a sequence of complexes (of R-modules andR-homomorphisms) and chain maps

0 −→ {−1}C(a1, . . . , an;Mb)• −→ C(a1, . . . , an, b;M)•

−→ C(a1, . . . , an;M)• −→ 0

which is such that, for each i ∈ N0, the sequence

0 −→ ({−1}C(a1, . . . , an;Mb)•)i −→ C(a1, . . . , an, b;M)i

−→ C(a1, . . . , an;M)i −→ 0

is exact.

5.1.15 Exercise. In this exercise, we use the extended notation for Cech com-plexes introduced in Exercise 5.1.12.

Let a ∈ R. Show that Hi(C(a;M)•) ∼= HiRa(M) for all i ∈ N0.

5.1.16 �Exercise. Show that all the cohomology modules of the Cech com-plex C(M)• are a-torsion. (Here are some hints: try induction on the numberof elements needed to generate a, in conjunction with 5.1.14 and 5.1.12.)


5.1.17 Exercise. In this exercise, we again use the extended notation for Cechcomplexes introduced in Exercise 5.1.12.

Suppose that b1, . . . , bm also generate a. Prove that

Hi(C(a1, . . . , an;M)•) ∼= Hi(C(b1, . . . , bm;M)•) for all i ∈ N0,

that is, the cohomology modules of the Cech complex C(M)• are, up to R-isomorphism, independent of the choice of sequence of generators for a. (Wesuggest that you use 5.1.14 to compare

C(a1, . . . , an, b1;M)• and C(a1, . . . , an;M)•.

Also, you may find 5.1.16 and 5.1.12 helpful.)

Our first major aim in this chapter is to show that, for each i ∈ N0, the localcohomology module Hi

a(M) is isomorphic to Hi(C(M)•), the i-th cohomol-ogy module of the Cech complex C(M)•. The next remark is a first step.

5.1.18 Remark. In the Cech complex

C(M)• : 0 −→Md0

−→Ma1 ⊕Ma2 ⊕ · · · ⊕Man −→ . . . ,

we have

H0(C(M)•) = Ker d0

= {m ∈M : for each i = 1, . . . , n, there exists

hi ∈ N with ahii m = 0

}= Γa(M).

It will probably come as no surprise to the reader to learn that we intendto exploit the above Remark 5.1.18 by use of negative strongly connectedsequences of functors and 1.3.5.

5.1.19 Remark. It is immediate from 5.1.18 and 5.1.11 that there is an R-isomorphism γ0M : Γa(M)

∼=−→ H0(M ⊗R C•) for which γ0M (m) = m ⊗ 1

for all m ∈ Γa(M), and it is then clear that, as M varies through C(R), theγ0M constitute a natural equivalence of functors γ0 : Γa −→ H0(( • )⊗R C

•)

from C(R) to itself.

5.1.20 Theorem. (Recall that a1, . . . , an (where n > 0) denote n elementswhich generate a, and C• denotes the Cech complex of R with respect toa1, . . . , an.) There is a unique isomorphism

(δi)i∈N0 : (Hi(C( • )•))i∈N0

∼=−→ (Hia)i∈N0

90 Other approaches

of negative (strongly) connected sequences of covariant functors (from C(R)to C(R)) which extends the identity natural equivalence on Γa.

Consequently, there is a unique isomorphism

(γi)i∈N0 : (Hia)i∈N0

∼=−→ (Hi(( • )⊗R C•))i∈N0

of negative strongly connected sequences of covariant functors (from C(R) toC(R)) which extends the natural equivalence γ0 of 5.1.19.

Proof. Recall from 5.1.18 that H0(C( • )•) = Γa. We shall be able to useTheorem 1.3.5 to prove the first part provided we can show that Hi(C(I)•) =

0 for all i ∈ N whenever I is an injective R-module. We shall achieve thisby induction on n, and we shall use the extended notation for Cech complexesintroduced in Exercise 5.1.12.

When n = 1, the Cech complex C(a1; I)• is just 0 −→ IτI−→ Ia1 −→ 0,

where τI is the natural map. But, by 2.2.20, Coker τI ∼= H1Ra1

(I), and thelatter is zero because I is injective. Thus Hi(C(I)•) = 0 for all i ∈ N. Wenote also that Ker τI = ΓRa1(I), which is injective by 2.1.4. Hence the exactsequence 0 −→ ΓRa1(I) −→ I

τI−→ Ia1 −→ 0 of 2.2.20 splits, and Ia1 is aninjective R-module.

Now suppose, inductively, that n > 1 and the result has been proved forsmaller values of n. By 5.1.14, there is a sequence of complexes (ofR-modulesand R-homomorphisms) and chain maps

0 −→ {−1}C(a1, . . . , an−1; Ian)• −→ C(a1, . . . , an; I)

•

−→ C(a1, . . . , an−1; I)• −→ 0

which is such that, for each i ∈ N0, the sequence

0 −→ ({−1}C(a1, . . . , an−1; Ian)•)i −→ C(a1, . . . , an; I)

i

−→ C(a1, . . . , an−1; I)i −→ 0

is exact. This sequence of complexes therefore induces a long exact sequenceof cohomology modules. Since Ian is an injective R-module (by the imme-diately preceding paragraph of this proof), we can deduce from our inductivehypothesis that

Hi({−1}C(a1, . . . , an−1; Ian)•) = 0 for all i ≥ 2

and

Hi(C(a1, . . . , an−1; I)•) = 0 for all i ≥ 1.

Furthermore, an easy check shows that the connecting homomorphism

H0(C(a1, . . . , an−1; I)•) −→ H1({−1}C(a1, . . . , an−1; Ian)

•)


induced by the above sequence of complexes is just the map

ΓRa1+···+Ran−1(I) −→ ΓRa1+···+Ran−1(Ian)

induced by the natural homomorphism I → Ian , and this is surjective sincethe canonical exact sequence

0 −→ ΓRan(I) −→ I −→ Ian −→ 0

of 2.2.20 splits. It follows from the long exact sequence of cohomology mod-ules that Hi(C(a1, . . . , an; I)

•) = 0 for all i ≥ 1. This completes the induc-tive step. We can now use Theorem 1.3.5 to complete the proof of the firstpart.

For the second part, we recall the isomorphism

(Hi(ω•))i∈N0 : (Hi(( • )⊗R C•))i∈N0

∼=−→ (Hi(C( • )•))i∈N0

of negative strongly connected sequences of 5.1.11, and deduce that((Hi(ω•))i∈N0

)−1 ◦((δi)i∈N0

)−1: (Hi

a)i∈N0

∼=−→ (Hi(( • )⊗R C•))i∈N0

is an isomorphism of connected sequences; moreover, (H0(ω•))−1 ◦ (δ0)−1 isjust the natural equivalence γ0 of 5.1.19, as is easy to check. The uniquenessin the second part follows from 1.3.4.

5.1.21 Remark. The reader should note that it is immediate from Theorem5.1.20 that, when a can be generated by n elements, Hi

a(N) = 0 for all R-modules N whenever i > n: we proved this result by means of the Mayer–Vietoris sequence in Theorem 3.3.1. This is one example of a situation wherethe use of a different approach to the calculation of local cohomology modulescan provide a simpler proof and additional insight. The next exercise providesanother example.

5.1.22 Exercise. Prove that Hna (R) = 0 if and only if there exists k ∈ N

such that, for every t ∈ N, it is the case that

(a1 . . . an)t ∈ Rat+k

1 + · · ·+Rat+kn .

Deduce that, for h ∈ N, in the ring R[X1, . . . , Xh] of polynomials over R, wehave Hh

(X1,...,Xh)(R[X1, . . . , Xh]) = 0.

Compare this approach with that of Exercise 4.2.4.

5.1.23 Proposition. LetK1(M) := Ker d1, where d1 : C(M)1 −→ C(M)2

is the first ‘differentiation’ map in the Cech complex C(M)• ofM with respectto the sequence a1, . . . , an. (It is clear from 5.1.8 that K1( • ) can easily bemade into a functor from C(R) to itself.)

92 Other approaches

There is a natural equivalence of functors ε : K1( • ) −→ Da which is suchthat, for each R-module M , the diagram

Md0

K1(M)�

εM

Mηa,M

Da(M)��

commutes.

Proof. By 5.1.16, the cohomology modules Hi(C(M)•) are all a-torsion,and so both the kernel and cokernel of d0 : M −→ K1(M) are a-torsion. Ittherefore follows from 2.2.15 that there is a unique R-homomorphism εM :

K1(M) −→ Da(M) such that the diagram

Md0

K1(M)�

εM

Mηa,M

Da(M)��

commutes, and that εM = Da(d0)−1 ◦ ηa,K1(M). This formula and 2.2.6(i)(c)

show that εM is injective, since Γa(K1(M)) ⊆ Γa(C

1(M)) = 0. Further-more, it is easy to use the uniqueness aspect of 2.2.13 to deduce that, as Mvaries through the category C(R), the εM constitute a natural transformationof functors.

Let t ∈ N, and let h ∈ HomR(at,M). It is straightforward to check that the

element (h(at1)

at1,h(at2)

at2, . . . ,

h(atn)

atn

)∈ C(M)1

actually belongs to K1(M). (Note that atih(atj) = h(atia

tj) = atjh(a

ti) for

integers i, j with 1 ≤ i, j ≤ n.) Hence there is an R-homomorphism γt,M :

HomR(at,M)→ K1(M) for which

γt,M (h) =

(h(at1)

at1, . . . ,

h(atn)

atn

)for all h ∈ HomR(a

t,M).


Also, it is straightforward to check that, for t, u ∈ N with u ≥ t, the diagram

HomR(at,M)

��

��

γt,MHomR(jut ,M)

HomR(au,M) K1(M)�

γu,M

�

(in which jut : au → at denotes the inclusion map) commutes, and so theγt,M (t ∈ N) induce an R-homomorphism γM : Da(M) −→ K1(M) forwhich γM ◦ ηa,M = d0. Note that εM ◦ γM = IdDa(M) by the uniquenessaspect of 2.2.15. Hence εM is surjective, and so is an isomorphism.

5.1.24 Exercise. Consider the special case of 5.1.23 in which M = R. Notethat C1 = C(R)1 =

∏ni=1Rai has a natural structure as a (commutative

Noetherian) ring (with identity), and that d0 : R → C1 is a ring homomor-phism. Recall also from 2.2.5 that Da(R) has a structure as a commutativering with identity.

Show that K1(R) is a subring of C1, and deduce from 2.2.17 that εR :

K1(R) −→ Da(R) (where εR is as in 5.1.23) is actually a ring isomorphism.

The next exercise concerns the ring K1(R) in a geometrical situation.

5.1.25 Exercise. Let V be an affine variety over the algebraically closed fieldK. Consider C1 and K1(R) (of 5.1.24) in the special case in which R =

O(V ), and assume that the ideal a of O(V ) is non-zero (and generated bya1, . . . , an, all of which are assumed to be non-zero). Let U be the open subsetV \ {p ∈ V : a1(p) = · · · = an(p) = 0} of V .

For each j = 1, . . . , n, let Uj denote the open subset {p ∈ V : aj(p) = 0}of U , identify the subring O(V )aj of K(V ) with the ring O(Uj) of regularfunctions on Uj in the natural way, and let ιj : O(U) −→ O(Uj) be therestriction homomorphism.

Show that the map λ : O(U) −→∏n

j=1O(Uj) for which

λ(g) = (ι1(g), . . . , ιn(g)) for all g ∈ O(U)

is an injective ring homomorphism with image K1(O(V )). In this way, weobtain a ring isomorphism λ′ : O(U)

∼=−→ K1(O(V )). Show that, with the

94 Other approaches

notation of 5.1.24, the diagram

O(V ) U O(U)�

ηa,O(V )εO(V )◦λ′∼=

Da(O(V ))

��

commutes, and deduce from 2.3.2 that εO(V ) ◦ λ′ is the νV,a of that theorem.

5.2 Use of Koszul complexes

We revert to the general situation and remind the reader that, throughout thischapter, a1, . . . , an (where n > 0) denote n elements which generate a, andM denotes an arbitrary R-module.

There is yet another method of calculation of local cohomology modules:this describes them as direct limits of homology modules of Koszul complexes.We present this description here, because it can be derived quickly from ourwork so far in this chapter on the Cech complex. We need to specify our nota-tion for the Koszul complexes that we shall use.

5.2.1 Notation. For all u ∈ N, we denote by K(au)• (or K(au1 , . . . , aun)•)

the usual Koszul complex of R with respect to au1 , . . . , aun. Thus, if F denotes

the free R-module Rn and, for each i = 1, . . . , n, the element

(0, . . . , 0, 1, 0, . . . , 0) ∈ Rn

which has i-th component 1 and all its other components 0 is denoted by ei,then K(au)• has the form

0 � � � � �K(au)n · · · K(au)kd(au)k

K(au)k−1

· · · �K(au)0 � 0,

where

K(au)k =

k∧F =

k∧(Rn) for k = 0, . . . , n

(so thatK(au)0 = R) and, for k = 1, . . . , n and i ∈ I(k, n) (with the notationof 5.1.4),

d(au)k(ei(1) ∧ . . .∧ ei(k)) =k∑

h=1

(−1)h−1aui(h)ei(1) ∧ . . .∧ ei(h) ∧ . . .∧ ei(k),

5.2 Use of Koszul complexes 95

where the ‘ei(h)’ indicates that ei(h) is omitted.Of course, we set K(au)k = 0 for all k ∈ Z \ {0, 1, . . . , n}.

5.2.2 Lemma. Let u, v ∈ N with u ≤ v. There is a chain map

(ψvu)• = ((ψv

u)k)k∈Z : K(au)• −→ K(av)•

of complexes of R-modules and R-homomorphisms such that (ψvu)n is the

identity mapping of∧n

F , such that (ψvu)0 is the endomorphism of R given

by multiplication by (a1 . . . an)v−u, and such that, for k = 1, . . . , n − 1 and

i ∈ I(k, n),

(ψvu)k(ei(1) ∧ . . . ∧ ei(k)) = (aj(1) . . . aj(n−k))

v−uei(1) ∧ . . . ∧ ei(k),

where j ∈ I(n− k, n) is the n-complement of i (see 5.1.4).

Proof. We must check that d(av)k ◦ (ψvu)k = (ψv

u)k−1 ◦ d(au)k for eachk = 1, . . . , n. We leave this to the reader in the case in which k = n. For1 ≤ k < n and i ∈ I(k, n), j ∈ I(n− k, n) as in the statement of the lemma,we have

(d(av)k ◦ (ψvu)k)(ei(1) ∧ . . . ∧ ei(k))

=k∑

h=1

(−1)h−1avi(h)(aj(1) . . . aj(n−k))v−uei(1) ∧ . . . ∧ ei(h) ∧ . . . ∧ ei(k),

and

((ψvu)k−1 ◦ d(au)k)(ei(1) ∧ . . . ∧ ei(k))

= (ψvu)k−1

(k∑

h=1

(−1)h−1aui(h)ei(1) ∧ . . . ∧ ei(h) ∧ . . . ∧ ei(k)

)

=

k∑h=1

(−1)h−1(aj(1) . . . aj(n−k))v−uavi(h)ei(1) ∧ . . . ∧ ei(h) ∧ . . . ∧ ei(k)

= (d(av)k ◦ (ψvu)k)(ei(1) ∧ . . . ∧ ei(k)).

The result follows.

5.2.3 �Exercise. Complete the proof of 5.2.2. In other words, show that, withthe notation of the lemma, d(av)n ◦ (ψv

u)n = (ψvu)n−1 ◦ d(au)n.

5.2.4 Remark. It is clear that, in the notation of 5.2.2, for u, v, w ∈ N withu ≤ v ≤ w, we have (ψw

v )• ◦ (ψvu)• = (ψw

u )•, and that (ψuu)• is the identity

chain map of the complex K(au)• to itself. Thus the (ψvu)• turn the family

96 Other approaches

(K(au)•)u∈N into a direct system of complexes ofR-modules and chain maps.Our immediate aim is to show that the direct limit complex, which we denoteby K(a∞)•, is isomorphic to a shift of the Cech complex C• of 5.1.5.

5.2.5 Theorem. There is an isomorphism of complexes

K(a∞)• = lim−→u∈N

K(au)•∼=−→ {n}C•

between the direct limit complex of Koszul complexes described in 5.2.4 andthe shift {n}C• of the Cech complex of R with respect to a1, . . . , an.

Proof. Let u ∈ N. We first define a chain map of complexes

(gu)• : K(au)• −→ {n}C•.

Define (gu)n : K(au)n −→ ({n}C•)n = C−n+n = C0 = R by requiringthat

(gu)n(re1 ∧ . . . ∧ en) = (−1)1+2+···+(n−1)r for all r ∈ R.

(We point out now, to help the reader discern a pattern in what follows, that∑n−1i=1 i = −n +

∑ni=1 i.) Now let k ∈ {1, . . . , n− 1} and let i be a typical

element of I(k, n) (with the notation of 5.1.4). Define an R-homomorphism

(gu)k : K(au)k −→ ({n}C•)k = Cn−k

by requiring that, for i ∈ I(k, n) as above,

(gu)k(ei(1) ∧ . . . ∧ ei(k)) =(−1)i(1)+···+i(k)−k

(aj(1) . . . aj(n−k))u

in the direct summand Raj(1)...aj(n−k)of Cn−k, where j ∈ I(n − k, n) is the

n-complement of i. Lastly, define an R-homomorphism (gu)0 : K(au)0 =

R −→ ({n}C•)0 = Cn = Ra1...an by

(gu)0(r) =r

(a1 . . . an)ufor all r ∈ R.

In order to show that ((gu)k)0≤k≤n gives rise to a chain map of complexes(gu)•, we must show that (with the notation of 5.1.5 concerning the Cech com-plex C• of R)

(gu)k−1 ◦ d(au)k = dn−k ◦ (gu)k for all k = 1, . . . , n.

We leave this to the reader in the case when k = n, and deal here with the


case when 1 ≤ k < n. Let i ∈ I(k, n), and let j ∈ I(n − k, n) be then-complement of i. Then

((gu)k−1 ◦ d(au)k)(ei(1) ∧ . . . ∧ ei(k))

= (gu)k−1

(k∑

h=1

(−1)h−1aui(h)ei(1) ∧ . . . ∧ ei(h) ∧ . . . ∧ ei(k)

)

=

k∑h=1

(−1)h−1aui(h)(−1)i(1)+···+i(h)+···+i(k)−(k−1)(−1)i(h)

(aj(1) . . . aj(n−k)ai(h))u

=k∑

h=1

(−1)i(h)+h(−1)i(1)+···+i(k)−kaui(h)

(aj(1) . . . aj(n−k)ai(h))u.

On the other hand,

(dn−k ◦ (gu)k)(ei(1) ∧ . . . ∧ ei(k)) = dn−k

((−1)i(1)+···+i(k)−k

(aj(1) . . . aj(n−k))u

).

We now refer back to 5.1.5: in order to evaluate the right-hand side of the aboveequation, we need to know, for each h = 1, . . . , k, at which point i(h) shouldbe inserted in the sequence (j(1), . . . , j(n−k)) in order to make an increasingsequence: if it should occupy the sh-th position in the new sequence, then

{1, 2, . . . , i(h)− 1} = {i(1), . . . , i(h− 1), j(1), . . . , j(sh − 1)} ,

so that i(h)− 1 = h− 1 + sh − 1. We thus see that

(dn−k ◦(gu)k)(ei(1) ∧ . . . ∧ ei(k))

=k∑

h=1

(−1)i(h)−h(−1)i(1)+···+i(k)−kaui(h)

(aj(1) . . . aj(n−k)ai(h))u

= ((gu)k−1 ◦ d(au)k)(ei(1) ∧ . . . ∧ ei(k)).

It follows that we do indeed obtain a chain map of complexes

(gu)• : K(au)• −→ {n}C•.

Next note that, for u, v ∈ N with u ≤ v and the chain map

(ψvu)• = ((ψv

u)k)k∈Z : K(au)• −→ K(av)•

98 Other approaches

of complexes of 5.2.2, we have a commutative diagram

K(au)•(ψv

u)•K(av)•�

(gv)•(gu)•

{n}C• .

��

It follows that there is induced a chain map of complexes

(g∞)• = ((g∞)k)k∈Z : K(a∞)• = lim−→u∈N

K(au)• −→ {n}C•

which is clearly such that (g∞)k is surjective for all k ∈ Z. In order to showthat (g∞)• is an isomorphism, it is enough to show that, for u ∈ N, k ∈ N0

with 0 ≤ k ≤ n and β ∈ Ker(gu)k, there exists v ∈ N with u ≤ v such thatβ ∈ Ker(ψv

u)k. This is very easy (and left to the reader) in the cases whenk = 0 and k = n, and so we suppose that 1 ≤ k < n. Let i ∈ I(k, n). Now(gu)k maps the direct summand Rei(1) ∧ . . . ∧ ei(k) of K(au)k into the directsummand Raj(1)...aj(n−k)

of Cn−k = ({n}C•)k, where j ∈ I(n− k, n) is then-complement of i. In view of this and the fact that I(k, n) is a finite set, wecan assume that β has the form rei(1) ∧ . . . ∧ ei(k) for some r ∈ R. In thiscase, the fact that β ∈ Ker(gu)k means that there exists w ∈ N0 such that(aj(1) . . . aj(n−k))

wr = 0, and then (ψu+wu )k(rei(1) ∧ . . . ∧ ei(k)) = 0.

It follows that (g∞)• is an isomorphism of complexes.

5.2.6 �Exercise. Complete the proof of Theorem 5.2.5.

Because the Cech complex C• can, by 5.1.20, be used to calculate the localcohomology modules Hi

a(M) (i ∈ N0), it is a consequence of Theorem 5.2.5that these local cohomology modules can also be computed in terms of Koszulcomplexes.

The result of the following exercise will be used in the proof of Theorem5.2.9.

5.2.7 �Exercise. Let N be an R-module. Let (Λ,≤) be a (non-empty) di-rected partially ordered set, and let ((Wα)

•)α∈Λ be a direct system of com-plexes of R-modules and chain maps over Λ, with constituent chain maps(hαβ)

• : (Wβ)• → (Wα)

• (for each (α, β) ∈ Λ× Λ with α ≥ β). Set

(W∞)• := lim−→α∈Λ

(Wα)•.

Show that the chain maps N ⊗R (hαβ )• (for α, β ∈ Λ with α ≥ β) turn the


family (N ⊗R (Wα)•)α∈Λ into a direct system of complexes of R-modules

and R-homomorphisms and chain maps over Λ, and show that there is an iso-morphism of complexes

lim−→α∈Λ

(N ⊗R (Wα)•) ∼= N ⊗R (W∞)• = N ⊗R

(lim−→α∈Λ

(Wα)•

).

5.2.8 �Exercise. For each u ∈ N, let K(au,M)• denote M ⊗R K(au)•,the Koszul complex of M with respect to au1 , . . . , a

un. Let α : M −→ N

be a homomorphism of R-modules. For v, w ∈ N with v ≤ w, there is acommutative diagram

M ⊗R K(av)•α⊗K(av)•

N ⊗R K(av)•�

M⊗(ψwv )• N⊗(ψw

v )•

M ⊗R K(aw)•α⊗K(aw)•

N ⊗R K(aw)•��

of complexes and chain maps; use this to show that, for each j ∈ Z, the family(Hj(K(au,M)•))u∈N is a direct system, and that lim

−→u∈N

Hj(K(au, • )•) is a

covariant R-linear functor from C(R) to itself.Show further that (

lim−→u∈N

Hn−i(K(au, • )•)

)i∈N0

is a negative strongly connected sequence of functors from C(R) to itself.

5.2.9 Theorem. In the situation and with the notation of 5.2.8, there is anatural equivalence of functors δ0 : lim

−→u∈N

Hn(K(au, • )•)∼=−→ Γa from C(R) to

itself; furthermore, there is a unique isomorphism

(δi)i∈N0 :

(lim−→u∈N

Hn−i(K(au, • )•)

)i∈N0

∼=−→(Hi

a

)i∈N0

of negative (strongly) connected sequences of covariant functors (from C(R)to C(R)) which extends δ0.

Consequently, for each i ∈ N0 and each R-module M ,

Hia(M) ∼= lim

−→u∈N

Hn−i(K(au,M)•).

100 Other approaches

Proof. We shall use Theorem 1.3.5 again.Let u ∈ N. Note that (with the notation of 5.2.1) the homomorphism

IdM ⊗d(au)n :M ⊗R K(au)n −→M ⊗R K(au)n−1

has kernel {m⊗ (e1 ∧ . . . ∧ en) : m ∈ (0 :M (au1 , . . . , aun))}, and that

Γa(M) =⋃u∈N

(0 :M (au1 , . . . , aun)).

Hence, for each u ∈ N, there is a monomorphism

δu,M : Ker(IdM ⊗d(au)n) −→ Γa(M)

with image (0 :M (au1 , . . . , aun)); furthermore, for u, v ∈ N with u ≤ v, the

diagram

Ker(IdM ⊗d(au)n)��

��

δu,M⊆

Ker(IdM ⊗d(av)n) Γa(M)�δv,M

�

commutes. It follows that there is induced an isomorphism

δ0M : lim−→u∈N

Hn(K(au,M)•)∼=−→ Γa(M),

and it is easy to check that, as M varies through C(R), the δ0M constitute anatural equivalence of functors δ0 : lim

−→u∈N

Hn(K(au, • )•)∼=−→ Γa from C(R) to

itself.Let I be an injective R-module. The fact that passage to direct limits pre-

serves exactness ensures that

lim−→u∈N

Hj(K(au, I)•) ∼= Hj

(lim−→u∈N

(K(au, I)•)

)for all j ∈ Z.

It now follows from Exercise 5.2.7 that there is an isomorphism of complexes

lim−→u∈N

(K(au, I)•) = lim−→u∈N

(I ⊗R K(au)•) ∼= I ⊗R K(a∞)•.

But 5.2.5 shows that K(a∞)• ∼= {n}C•, and 5.1.20 shows that

Hi(I ⊗R C•) ∼= Hi

a(I) for all i ∈ N0.

5.3 Local cohomology in prime characteristic 101

Hence lim−→u∈N

Hn−i(K(au, I)•) = 0 for all i ∈ N.

We can now use Theorem 1.3.5 to complete the proof.

5.3 Local cohomology in prime characteristic

Since a can be generated by n elements, we must have Hn+ja (M) = 0 for all

j ∈ N (by 3.3.3), so that interest focusses on Hna (M). In this section, we are

going to illustrate how the Cech complex approach of this chapter facilitatescalculation in Hn

a (M).

5.3.1 Remark. (Recall that a1, . . . , an (where n > 0) denote n elementswhich generate a.) By 5.1.20, the local cohomology module Hn

a (M) is iso-morphic to Coker dn−1, where

dn−1 : C(M)n−1 =n⊕

i=1

Ma1...ai−1ai+1...an −→ C(M)n =Ma1...an

is the (n − 1)-th homomorphism in the Cech complex of M with respect toa1, . . . , an. We use ‘[ ]’ to denote natural images of elements of Ma1...an

in this cokernel. Thus a typical element of Hna (M) can be represented as[

m/(a1 . . . an)i]

for some m ∈M and i ∈ N0. Note that, for u ∈ {1, . . . , n},we have [

aium

(a1 . . . an)i

]= 0 for all i ∈ N0 and m ∈M.

Thus [m′/(a1 . . . an)i] = 0 wheneverm′ ∈ (ai1, . . . , ain)M . It is important for

us to know exactly when an element of Coker dn−1 is zero, and this is coveredin the next lemma.

5.3.2 Lemma. Denote the product a1 . . . an by a. Let m, g ∈ M and i, j ∈N0. Then, with the notation of 5.3.1,

(i)[m/ai]=[g/aj]

if and only if there exists k ∈ N0 such that k ≥max{i, j} and ak−im− ak−jg ∈

∑nu=1 a

kuM ;

(ii) in particular,[m/ai]= 0 if and only if there exists k ∈ N0 such that

k ≥ i and ak−im ∈∑n

u=1 akuM .

Proof. (i) Since[mai

]−[ gaj

]=

[ajm

ai+j

]−[aig

ai+j

]=

[ajm− aigai+j

],

it is enough for us to prove (ii).


(ii) (⇐) There exist m1, . . . ,mn ∈ M such that ak−im =∑n

u=1 akumu.

Therefore [mai

]=

[ak−im

ak

]=

[∑nu=1 a

kumu

ak

]= 0

because(∑n

u=1 akumu

)/ak ∈ Im dn−1.

(⇒) Since[m/ai]= 0, we have m/ai ∈ Im dn−1, so that there exist

j1, . . . , jn ∈ N0 and m1, . . . ,mn ∈M such that

m

ai=aj11 m1

aj1+ · · ·+ ajnn mn

ajn.

Let j := max{j1, . . . , jn}; then there exist m′1, . . . ,m′n ∈ M such that

m/ai = (aj1m′1 + · · ·+ ajnm

′n)/a

j , so that there is an h ∈ N0 such that

ah(ajm− ai(aj1m′1 + · · ·+ ajnm′n)) = 0.

Take k := i+ j + h to complete the proof.

5.3.3 Notation for the section. In addition to the standard notation for thischapter, we are going to assume, for the remainder of this section, that R hasprime characteristic p. In these circumstances, the map f : R −→ R for whichf(r) = rp for all r ∈ R is a ring homomorphism (simply because the binomialcoefficient

(pi

)is an integer divisible by p for all i ∈ {1, . . . , p − 1}). We call

f the Frobenius homomorphism.In this section, we shall use � f to denote the functor obtained from restric-

tion of scalars using f , rather than the � R of Chapter 4. We are making thischange in the interests of clarity: when the two rings concerned are the same,the notation � R could be confusing.

Thus R� f denotes R considered as an R-module via f .By a Frobenius action on the R-module M , we mean an Abelian group

homomorphism F : M −→ M such that F (rm) = rpF (m) for all m ∈ M

and r ∈ R. For example, the Frobenius homomorphism f : R −→ R is aFrobenius action on R.

For an ideal b of R and n ∈ N0, we shall denote by b[pn] the ideal of R

generated by all the pn-th powers of elements of b. This ideal is called thepn-th Frobenius power of b. Observe that, if b can be generated by b1, . . . , bt,then bp

n

1 , . . . , bpn

t generate b[pn], and that (b[p

n])[p] = b[pn+1].

Among other things, we aim in this section to show that there is a naturalFrobenius action on each local cohomology module Hi

a(R) of R itself withrespect to a, and to give a detailed description of this action in the case wherei = n. (Recall our assumption that a can be generated by n elements.)


5.3.4 Theorem. (Recall that R has prime characteristic p, and that R� f

denotes R considered as an R-module via f .) The Frobenius homomorphismf : R −→ R� f is a homomorphism of R-modules, and thus inducesR-homomorphisms Hi

a(f) : Hia(R) −→ Hi

a(R� f ) for all i ∈ N0.By the Independence Theorem 4.2.1, there is a unique isomorphism

Λ = (λi)i∈N0 :(Hi

f(a)R( • )� f

)i∈N0

∼=−→(Hi

a( • � f ))i∈N0

of negative strongly connected sequences of covariant functors (from C(R) toC(R)) such that λ0 is the identity natural equivalence.

For each i ∈ N0, the map (λiR)−1 ◦Hi

a(f) is a Frobenius action on Hia(R).

Proof. Note that f(a)R = a[p]. Since√a =

√a[p], the local cohomol-

ogy functor with respect to a coincides with the local cohomology functorwith respect to a[p]. Thus (λiR)

−1 is an R-isomorphism from Hia(R� f ) to

Hif(a)R(R)� f = Hi

a[p](R)� f = Hia(R)� f . Thus F := (λiR)

−1 ◦ Hia(f) :

Hia(R) −→ Hi

a(R)� f is an R-homomorphism (and so certainly an Abeliangroup homomorphism). Since F (rh) = f(r)F (h) = rpF (h) for all h ∈Hi

a(R) and r ∈ R, we see that F is a Frobenius action on Hia(R).

5.3.5 Remark. It is important to note that the Frobenius actions defined in5.3.4 on the Hi

a(R) (i ∈ N0) do not depend on any choice of generators for a.However, our next task is to describe the Frobenius action on Hn

a (R) givenby 5.3.4 in terms of our generators a1, . . . , an for a.

5.3.6 Theorem. (Recall that R has prime characteristic p.) The naturalFrobenius action F on Hn

a (R) of 5.3.4 is such that, with the notation of 5.3.1and when we identify Hn(C•) with Hn

a (R) by means of the isomorphism δnRof 5.1.20,

F

([r

(a1 . . . an)k

])=

[rp

(a1 . . . an)kp

]for all r ∈ R and k ∈ N0.

Proof. We have F = (λnR)−1 ◦ Hn

a (f), and so the precise formula that wemust establish is that

(δnR)−1 ◦ (λnR)−1 ◦Hn

a (f) ◦ δnR([

r

(a1 . . . an)k

])=

[rp

(a1 . . . an)kp

].

As in 5.3.4, let � f : C(R) −→ C(R) denote the functor obtained from re-striction of scalars using f . Since δn is a natural equivalence of functors,Hn

a (f) ◦ δnR = δnR f◦Hn(C(f)•); also

Hn(C(f)•)

([r

(a1 . . . an)k

])=

[rp

(a1 . . . an)k

].


It is therefore sufficient for us to show that

(δnM )−1 ◦ (λnM )−1 ◦ δnM f

([m

(a1 . . . an)k

])=

[m

(a1 . . . an)kp

]for all m ∈M and k ∈ N0.

Let b ∈ R. It is straightforward to check that there is an R-isomorphismνM,b : (M� f )b

∼=−→ (Mb)� f for which νM,b(m/bj) = m/bpj for all m ∈M

and j ∈ N0. Therefore, for each k ∈ N with 1 ≤ k ≤ n, we have an R-isomorphism

τkM :=⊕

i∈I(k,n)νM,ai(1)...ai(k)

: C(M� f )k ∼=−→ C(M)k� f

(the notation I(k, n) was defined in 5.1.4). It is straightforward to check thatthe τkM (k ∈ {1, . . . , n}), together with the identity map on M� f , constitutean isomorphism τ •

M : C(M� f )• ∼=−→ C(M)•� f of complexes of R-modules

and R-homomorphisms. Note that

τnM (m/(a1 . . . an)k) = m/(a1 . . . an)

kp for all m ∈M and k ∈ N0.

As M varies through C(R), the τ •M constitute a natural equivalence of func-

tors τ • : C( • � f )•∼=−→ C( • )•� f (from C(R) to the category of all complexes

of R-modules (and chain maps of such complexes)), and τ • induces an iso-morphism

(Hi(τ •))i∈N0 : (Hi(C( • � f )•))i∈N0

∼=−→ (Hi(C( • )•)� f )i∈N0

of negative strongly connected sequences of covariant functors which extendsthe identity natural equivalence on Γa( • � f ) = Γa( • )� f .

We now use the isomorphism of connected sequences

(δi)i∈N0 : (Hi(C( • )•))i∈N0

∼=−→ (Hia)i∈N0

of 5.1.20 to produce further isomorphisms of connected sequences

(δi• f)i∈N0 : (Hi(C( • � f )

•))i∈N0

∼=−→ (Hia( • � f ))i∈N0

and

(δi• � f )i∈N0 : (Hi(C( • )•)� f )i∈N0

∼=−→ (Hia( • )� f )i∈N0

which extend the identity natural equivalence on Γa( • � f ) = Γa( • )� f . Butthen

(δi• f)i∈N0 ◦

((Hi(τ •))i∈N0

)−1 ◦((δi• � f )i∈N0

)−1:(

Hia( • )� f

)i∈N0

∼=−→(Hi

a( • � f ))i∈N0


is an isomorphism of connected sequences which extends the identity naturalequivalence on Γa( • � f ) = Γa( • )� f . By the uniqueness aspect of the Inde-pendence Theorem 4.2.1, this isomorphism must be the Λ = (λi)i∈N0 of 5.3.4.Hence (λnM )−1 = (δnM� f ) ◦Hn(τ •

M ) ◦ (δnM f)−1. Therefore

(δnM )−1 ◦ (λnM )−1 ◦ δnM f

([m

(a1 . . . an)k

])= Hn(τ •

M )

([m

(a1 . . . an)k

])=

[m

(a1 . . . an)kp

],

as required.

5.3.7 Exercise. For u, v ∈ N with u ≤ v, let hvu : R/(au1 , . . . , aun) −→

R/(av1, . . . , avn) be the R-homomorphism induced by multiplication by av−u,

where a := a1 . . . an. These homomorphisms turn the family

(R/(au1 , . . . , aun))u∈N

into a direct system. For each u ∈ N, let

hu : R/(au1 , . . . , aun) −→ lim

−→w∈N

R/(aw1 , . . . , awn ) =: H

be the natural homomorphism. Show that there is an isomorphism α : H∼=−→

Hna (R).Use the isomorphism α and the Frobenius action of 5.3.4 on Hn

a (R) to puta Frobenius action F ′ on H , and show that F ′ is given by the following rule:

F ′(hu(r + (au1 , . . . , aun))) = hpu(r

p + (apu1 , . . . , apun )) for all u ∈ N, r ∈ R.

We plan to exploit the Frobenius action described in 5.3.4 and 5.3.6, but theapplications we have in mind will have to be postponed until the final section ofthe next chapter, by which point we shall have covered the Non-vanishing The-orem and some interactions between local cohomology and regular sequences.

6

Fundamental vanishing theorems

There are many important results concerning the vanishing of local cohomol-ogy modules. A few results of this type have already been presented earlierin the book: for example, Exercise 1.3.9(iv) is concerned with the fact that,if an R-module M is such that a contains an M -sequence of length n, thenHi

a(M) = 0 for all i < n; also, Theorem 3.3.1 shows that, if a can be gen-erated by t elements, then, for every R-module M , we have Hi

a(M) = 0 forall i > t; and we strengthened the latter result in Corollary 3.3.3, where weshowed that, for every R-module M , we have Hi

a(M) = 0 for all i > ara(a).In this chapter, we shall provide a further result of this type: we shall prove

Grothendieck’s Vanishing Theorem, which states that, if the R-module L (isnon-zero and) has (Krull) dimension n, then Hi

a(L) = 0 for all i > n. Weshall also prove that, when (R,m) is a local ring and the non-zero, finitelygenerated R-module M has dimension n, then Hn

m(M) = 0, so that, in viewof Grothendieck’s Vanishing Theorem, n = dimM is the greatest integer ifor which Hi

m(M) = 0. Also in this chapter, we shall explore in greater detailthe ideas of Exercise 1.3.9, and this investigation will lead to the result thatdepthM is the least integer i for which Hi

m(M) = 0. (Recall that depthMis the common length of all maximal M -sequences.) It will thus follow that,for such an M over such a local ring (R,m), it is only for integers i satisfyingdepthM ≤ i ≤ dimM that it is possible that Hi

m(M) could be non-zero,while this local cohomology module is definitely non-zero if i is at either ex-tremity of this range.

In §6.4, we shall exploit our earlier work to obtain a geometrical application:we shall establish a special case of Serre’s Affineness Criterion (see [77, §46,Corollaire 1]), concerning the following situation. Let V be an affine varietyover the algebraically closed field K. Let b be a non-zero ideal of O(V ), letV (b) denote the closed subset of V determined by b, and let U be the opensubset V \ V (b) of V . Thus U is a quasi-affine variety. It is of fundamen-

6.1 Grothendieck’s Vanishing Theorem 107

tal importance in algebraic geometry to be able to determine whether such aquasi-affine variety is itself affine. We shall show that U is affine if and onlyif Hi

b(O(V )) = 0 for all i ≥ 2. Thus this work is well suited to a chapter on‘vanishing theorems’!

In §6.5, we shall present two applications of local cohomology to local al-gebra. These applications are to results which have no mention of local coho-mology in their statements, but which have proofs that make non-trivial useof local cohomology. One concerns the Monomial Conjecture (of M. Hochster[37, Conjecture 1]) that whenever (ai)ni=1 is a system of parameters for the n-dimensional local ringR, then (a1 . . . an)

k ∈ (ak+11 , . . . , ak+1

n ) for all k ∈ N0.We shall present a proof (due to Hochster) of this conjecture in the case whereR has prime characteristic p; our proof makes use of the Frobenius action onHn

m(R) that was produced in 5.3.4 (and the fundamental fact thatHnm(R) = 0).

The other application is also to local rings of characteristic p, and concernstight closure; it too uses a Frobenius action.

6.1 Grothendieck’s Vanishing Theorem

Our first main aim in this chapter is to present a proof of Grothendieck’s Van-ishing Theorem. We preface this with a reminder about the dimension of anR-module.

6.1.1 Reminder. Let M be a non-zero R-module. The (Krull) dimension,dimM or dimRM , of M is the supremum of lengths of chains of prime idealsin the support ofM if this supremum exists, and∞ otherwise. In the case whenM is finitely generated, this is equal to dimR/(0 : M), the dimension of thering R/(0 :M), but this need not be the case if M is not finitely generated.

We adopt the convention that the dimension of the zero R-module is −1.

6.1.2 Grothendieck’s Vanishing Theorem. Let M be an R-module. ThenHi

a(M) = 0 for all i > dimM .

Proof. Since, for each p ∈ Spec(R),

SuppRp(Mp) = {qRp : q ∈ SuppM and q ⊆ p} ,

it follows from 4.3.3 that it is sufficient for us to prove this result under theadditional hypothesis that (R,m) is local. This is what we shall do.

When dimM = −1, there is nothing to prove, as then M = 0. The resultis also clear if a = R, as then Γa is the zero functor of 3.4.7. We thereforesuppose henceforth in this proof that M = 0 and a ⊆ m.

108 Fundamental vanishing theorems

We argue by induction on dimM . When dimM = 0, each non-zero ele-ment g ∈ M is annihilated by a power of m (and therefore by a power of a),because the ring R/(0 : g) is of dimension 0 and is therefore Artinian. Thus,in this case, M is a-torsion, and so it follows from 2.1.7(i) that Hi

a(M) = 0

for all i > 0 = dimM .Now suppose, inductively, that dimM = n > 0, and the result has been

proved for all R-modules of dimensions smaller than n. Since, by 3.4.10, foreach i ∈ N0, the local cohomology functor Hi

a commutes with direct limits,and M can be viewed as the direct limit of its finitely generated submodules,it is sufficient for us to prove that Hi

a(M′) = 0 for all i > n whenever M ′ is

a finitely generated submodule of M . Since such an M ′ must have dimensionnot exceeding n, we can therefore assume, in this inductive step, that M itselfis finitely generated. This we do.

By 2.1.7(iii), we have Hia(M) ∼= Hi

a(M/Γa(M)) for all i > 0. Also,M/Γa(M) has dimension not exceeding n, and is an a-torsion-freeR-module,by 2.1.2. In view of the inductive hypothesis, we can, and do, assume that Mis a (non-zero, finitely generated) a-torsion-free R-module.

We now use 2.1.1(ii) to deduce that a contains an element r which is a non-zerodivisor on M . Let t, i ∈ N with i > n. The exact sequence

0 −→Mrt−→M −→M/rtM −→ 0

(in which the second homomorphism is provided by multiplication by rt) in-

duces an exact sequence Hi−1a (M/rtM) −→ Hi

a(M)rt−→ Hi

a(M) of lo-cal cohomology modules. (The fact that the second homomorphism is againprovided by multiplication by rt follows from the fact that the functor Hi

a isR-linear.)

Now dim(M/rtM) < n since rt ∈⋃

p∈AssM p and every minimal memberof SuppM belongs to AssM (see [81, Theorem 9.39]). Hence, by the induc-tive hypothesis, Hi−1

a (M/rtM) = 0. Thus, for each t ∈ N, multiplication byrt provides a monomorphism of Hi

a(M) into itself. But r ∈ a and Hia(M)

is an a-torsion R-module, so that each element of it is annihilated by somepower of r. Therefore Hi

a(M) = 0. This completes the inductive step, and theproof.

The next theorem can be regarded as a companion to Grothendieck’s Van-ishing Theorem, because it shows that, in some circumstances, this VanishingTheorem is best possible. The method of proof employed here uses the pow-erful technique of reduction to the case where the local ring concerned is acomplete local domain. The argument is fairly sophisticated, as it relies onideas related to the structure theorems for complete local rings. Some readers


might like to be informed now that a different proof of the following theoremis provided in Chapter 7. We preface the theorem with an elementary remark.

6.1.3 Remark. Let R′ be a second commutative Noetherian ring and let f :

R→ R′ be a flat ring homomorphism. LetM be a non-zero, finitely generatedR-module, generated by m1, . . . ,mt; set c = (0 :R M).

There is an exact sequence

0 −→ c⊂−→ R

h−→ Rm1 ⊕ · · · ⊕Rmt

of R-modules and R-homomorphisms, where h(r) = (rm1, . . . , rmt) for allr ∈ R. Since R′ is a flat R-module, the induced sequence

0 � � �c⊗R R′ R⊗R R

′ (Rm1 ⊕ · · · ⊕Rmt)⊗R R′h⊗RR′

is again exact, and so it follows easily from the additivity of the tensor productfunctor that the sequence

0 −→ cR′⊆−→ R′

h′−→ (Rm1 ⊗R R

′)⊕ · · · ⊕ (Rmt ⊗R R′),

where h′(r′) = (m1⊗r′, . . . ,mt⊗r′) for all r′ ∈ R′, is exact. We thus deducethat cR′ = (0 :R′ (M ⊗R R

′)).It follows that, in particular, if (R,m) is local with (m-adic) completion R,

and M is a non-zero, finitely generated R-module, then

dimRM = dimR/c = dim R/cR = dimR(M ⊗R R).

6.1.4 The Non-vanishing Theorem. Assume that (R,m) is local, and let Mbe a non-zero, finitely generatedR-module of dimension n. ThenHn

m(M) = 0.

Proof. Let R denote the (m-adic) completion of R. Since the natural ring ho-momorphism R → R is (faithfully) flat, it follows from the Flat Base ChangeTheorem 4.3.2 that there is an R-isomorphism

Hnm(M)⊗R R ∼= Hn

mR(M ⊗R R),

and so it is enough for us to show that HnmR

(M ⊗R R) = 0. Of course, mR

is the maximal ideal of the local ring R, and M ⊗R R is a non-zero, finitelygenerated R-module; also 6.1.3 shows that this R-module has dimension n.Consequently, we can, and do, assume henceforth in this proof that R is com-plete.

Let p be a minimal member of SuppM for which dimR/p = n. Sincedim(pM) < n + 1, it follows from Grothendieck’s Vanishing Theorem 6.1.2that the natural epimorphism M →M/pM induces an epimorphism Hn

m(M)

−→ Hnm(M/pM), and so it is enough for us to show that Hn

m(M/pM) = 0.


Now, since√(0 :R M/pM) =

√p+ (0 :R M) = p (by [81, 9.23], for

example), we can deduce that (0 :R M/pM) = p. Hence dimR(M/pM) = n,and so, when we regard M/pM as an R/p-module in the natural way, it isfaithful, finitely generated, and of dimension n. By the Independence Theorem4.2.1, we have Hn

m(M/pM) ∼= Hnm/p(M/pM) as R-modules. Since R/p is

a complete local domain, it follows that it is enough for us to prove the resultunder the additional assumptions that R is a complete local domain and thatM is a faithful R-module.

At this point, we appeal to Cohen’s Structure Theorem for complete localrings: by [50, Theorem 29.4], there exists a complete regular local subring(R′,m′) ofR which is such thatR is finitely generated as anR′-module. SinceR is integral over R′, it follows that m is the one and only prime ideal of Rwhich has contraction toR′ equal to m′. Hence

√(m′R) = m, and so it follows

from 1.1.3 and the Independence Theorem 4.2.1 that Hnm(M) = Hn

m′R(M) ∼=Hn

m′(M) as R′-modules. It is thus sufficient for us to show that Hnm′(M) = 0.

Obviously, M is faithful and finitely generated as an R′-module. Moreover,dimR′ = dimR = n (since R is integral over R′). We can therefore replaceR by R′ and thus assume that R is a complete regular local ring during theremainder of this proof.

Next, let, for each R-module G,

τ(G) := {g ∈ G : there exists r ∈ R \ {0} such that rg = 0} ,

a submodule of G. Note that (0 : τ(M)) = 0, and so τ(M) has dimension lessthan n. Another use of Grothendieck’s Vanishing Theorem 6.1.2 shows thatthe natural epimorphismM →M/τ(M) induces an isomorphismHn

m(M)→Hn

m(M/τ(M)), and so it is sufficient for us to show that Hnm(M/τ(M)) = 0.

Note that τ(M/τ(M)) = 0 and that M/τ(M) still has dimension n (as itsannihilator is 0). Therefore we can, and do, assume that τ(M) = 0.

Let K be the field of fractions of R, and note that, since τ(M) = 0,the natural R-homomorphism M → M ⊗R K is injective. Let t denote the(torsion-free) rank of M as an R-module, that is, the vector space dimensiondimK(M ⊗RK). There exists r ∈ R \ {0} such that M , as an R-module, canbe embedded in the submodule

R 1r ⊕ · · · ⊕R

1r (t copies) of K ⊕ · · · ⊕K (t copies).

But R 1r∼= R, and so there is a free R-module F of rank t and an R-monomo-

rphism h : M → F ; also, since h ⊗R K : M ⊗R K → F ⊗R K must bea K-monomorphism between t-dimensional vector spaces over K, and there-fore an isomorphism, it follows that dim(Cokerh) < n. It therefore follows


from Grothendieck’s Vanishing Theorem 6.1.2 that h induces an exact se-quence Hn

m(M) −→ Hnm(F ) −→ 0. Now since R is a regular local ring, it

follows from 4.2.6 that Hnm(R) = 0; finally, we deduce from the additivity of

the functor Hnm that Hn

m(F ) = 0. Hence Hnm(M) = 0 and this completes the

proof.

6.1.5 Exercise. Suppose that M is a finitely generated R-module for whichM = aM . Show that there exists i ∈ N0 for which Hi

a(M) = 0.

In 3.3.4, we introduced the cohomological dimension cohd(a) of a.

6.1.6 Lemma. If a is proper, then ht a ≤ cohd(a).

Proof. Let p be a minimal prime ideal of a such that ht p = ht a =: h. Then,by 1.1.3 and 4.3.3, we have (Hh

a (R))p∼= Hh

aRp(Rp) = Hh

pRp(Rp), and this is

non-zero by the Non-vanishing Theorem 6.1.4. Therefore Hha (R) = 0, so that

ht a = h ≤ cohd(a).

6.1.7 �Exercise. Assume that (R,m) is local, and letM be a non-zero, finite-ly generated R-module of dimension n > 0. Show that Hn

m(M) is not finitelygenerated. (Here are some hints: use M/Γm(M) to see that one can make theadditional assumption that M is an m-torsion-free R-module; use 2.1.1(ii) tosee that, then, m contains an element r which is a non-zerodivisor on M ; notethat, if Hn

m(M) were finitely generated, then there would exist t ∈ N such thatrtHn

m(M) = 0; and consider the long exact sequence of local cohomology

modules induced by the exact sequence 0 −→Mrt−→M −→M/rtM −→ 0

in order to obtain a contradiction.)

6.1.8 Exercise. Provide an example to show that the result of Theorem 6.1.4is not always true if the hypothesis that M be finitely generated is omitted.

6.1.9 �Exercise. Let T : C(R) −→ C(R) be an R-linear covariant functor.For each R-module M , and each g ∈ M , let μg,M : R → M be the R-homomorphism for which μg,M (r) = rg for all r ∈ R.

Show that there is a natural transformation of functors

θ : ( • )⊗R T (R) −→ T

(from C(R) to C(R)) which is such that, for each R-module M ,

θM (g ⊗ z) = T (μg,M )(z) for all g ∈M and z ∈ T (R).

Show also that θF is an isomorphism whenever F is a finitely generated freeR-module, and deduce that, if T is right exact, then θM is an isomorphismwhenever M is a finitely generated R-module.


6.1.10 �Exercise. Let n be an integer such that Hia(M) = 0 for all i > n

and for all R-modules M . (For example, this condition would be satisfied ifn ≥ ara(a) (by 3.3.3), or if n ≥ dimR (by 6.1.2). Use Exercise 6.1.9 to provethat Hn

a is naturally equivalent to ( • )⊗R Hna (R).

We interpret the supremum of the empty set of integers as −∞.

6.1.11 Proposition. The cohomological dimension cohd(a) of a is equal tosup{i ∈ N0 : Hi

a(R) = 0}.

Proof. The result is clear when a = R, and so we suppose that a is proper.Denote by d the greatest integer i such that Hi

a(R) = 0. (Such an integerexists, by 3.3.4 and 6.1.6.) Clearly d ≤ cohd(a) =: c. By 6.1.10, the functorsHc

a and ( • ) ⊗R Hca(R) are naturally equivalent. Therefore Hc

a(R) = 0 andc ≤ d.

6.2 Connections with grade

It is now time for us to explore in detail connections, already hinted at in Exer-cise 1.3.9, between regular sequences and local cohomology. To set the scene,and clarify precisely what background information the reader will require forthis work, we begin by recalling the definition of ‘M -sequence’ and quoting,without proof, two results from Matsumura [50]; however, we shall use termi-nology different from his.

6.2.1 Definition. (See [50, p. 123].) Let a1, . . . , an ∈ R and let M be an R-module. We say that the sequence a1, . . . , an is a poor M -sequence preciselywhen

(i) a1 is a non-zerodivisor on M and, for each i = 2, . . . , n, the element aiis a non-zerodivisor on M/

∑i−1j=1 ajM .

Furthermore, a1, . . . , an is said to be an M -sequence, or an M -regular se-quence, precisely when it is a poor M -sequence, that is, it satisfies condition(i) above, and, in addition,

(ii)∑n

j=1 ajM =M .

6.2.2 Theorem. (See [50, Theorem 16.6].) Let M be a finitely generated R-module such that aM = M ; let n ∈ N. Then the following statements areequivalent:

(i) ExtiR(G,M) = 0 for all i < n and each finitely generated R-module Gfor which SuppG ⊆ Var(a);

6.2 Connections with grade 113

(ii) ExtiR(R/a,M) = 0 for all i < n;(iii) there is a finitely generated R-module G with SuppG = Var(a) for

which ExtiR(G,M) = 0 for all i < n; and(iv) there exists an M -sequence of length n contained in a. �

Ideas involved in Theorem 6.2.2 can be adapted to prove the result in thefollowing exercise, which is often useful.

6.2.3 �Exercise. Let N be an R-module (note that it is not assumed thatN is finitely generated) such that there exists a poor N -sequence of length ncontained in a. Show that ExtiR(G,N) = 0 for all i < n and each finitelygenerated R-module G for which SuppG ⊆ Var(a).

Observe that, with the notation of 6.2.2, every M -sequence contained in a

(even the empty one!) can be extended to a maximal one, since otherwise, asaM =M , there would exist an infinite sequence (bi)i∈N of elements of a suchthat b1, . . . , bn is an M -sequence for all n ∈ N, and this would lead to aninfinite strictly ascending chain

(b1) ⊂ (b1, b2) ⊂ · · · ⊂ (b1, . . . , bn) ⊂ · · ·

of ideals of R. We incorporate this observation into the next theorem, which isotherwise taken from Matsumura [50].

6.2.4 Theorem and Definition. (See [50, Theorem 16.7].) LetM be a finitelygenerated R-module such that aM =M .

(i) There exists an M -sequence contained in a which cannot be extendedto a longer one by the addition of an extra term (such a sequence willhenceforth be referred to as a maximal M -sequence contained in a).

(ii) Every M -sequence contained in a can be extended to a maximal one.(iii) All maximal M -sequences contained in a have the same length, namely

the least integer i such that ExtiR(R/a,M) = 0.

We shall refer to the common length of all maximal M -sequences containedin a as the M -grade of a, and we shall denote this non-negative integer bygradeM a. �

6.2.5 Notes. The following points should be noted.

(i) If the ideal a is proper, then the R-grade of a is defined: we shall followRees’s original terminology [68] and refer to this simply as the grade ofa, and we shall denote it by grade a.


(ii) Suppose that (R,m) is local, and that M is a non-zero finitely generatedR-module. Then it follows from Nakayama’s Lemma that mM = M ,and so gradeM m is defined: this is referred to as the depth of M , anddenoted by depthM or depthRM . Since every M -sequence must becontained in m, we see that depthM is equal to the common length ofall maximal M -sequences.

6.2.6 Exercise. Let M be an R-module and let i ∈ N0. Show that

Supp(Hi

a(M))⊆ SuppM ∩Var(a).

Deduce that, if M is finitely generated and aM = M , then Hja(M) = 0 for

all j ∈ N0.

6.2.7 Theorem. LetM be a finitely generatedR-module such that aM =M .Then gradeM a is the least integer i such that Hi

a(M) = 0.

Proof. Let g := gradeM a. We use induction on g. When g = 0, every ele-ment of a must be a zerodivisor on M , and so Γa(M) = 0 by 2.1.1(ii). Nowsuppose that g > 0 and that the result has been proved for each finitely gener-ated R-module N with aN = N and gradeN a < g.

There exists a1 ∈ a such that a1 is a non-zerodivisor on M . Set M1 :=

M/a1M , and observe that (aM1 =M1 and) gradeM1a = g−1. Therefore, by

the inductive hypothesis,Hia(M1) = 0 for all i < g−1, whileHg−1

a (M1) = 0.The exact sequence 0 −→M

a1−→M −→M1 −→ 0 induces, for each i ∈ N,an exact sequence

Hi−1a (M) −→ Hi−1

a (M1) −→ Hia(M)

a1−→ Hia(M).

This shows that, for i < g, the element a1 is a non-zerodivisor on Hia(M),

so that, since this module is a-torsion, it must be zero. We therefore have anexact sequence 0 −→ Hg−1

a (M1) −→ Hga (M), and since Hg−1

a (M1) = 0, itfollows that Hg

a (M) = 0.

6.2.8 Corollary. Assume that (R,m) is local, and letM be a non-zero, finite-ly generatedR-module. Then any integer i for whichHi

m(M) = 0 must satisfy

depthM ≤ i ≤ dimM,

while for i at either extremity of this range we do have Him(M) = 0.

Proof. This is immediate from Grothendieck’s Vanishing Theorem 6.1.2, theNon-vanishing Theorem 6.1.4 and the above 6.2.7, because M = mM byNakayama’s Lemma, and depthM = gradeM m.

6.2 Connections with grade 115

6.2.9 Corollary. Assume that (R,m) is local, and letM be a non-zero, finite-ly generated R-module. Then there is exactly one integer i for which Hi

m(M)

= 0 if and only if depthM = dimM , that is, if and only if M is a Cohen–Macaulay R-module (see [50, p. 134]).

In particular, if (R,m) is a regular local ring of dimension n, then n is theunique integer i for which Hi

m(R) = 0. �

6.2.10 Remark. Let M be a finitely generated R-module such that aM =M . It is immediate from Theorems 2.2.6(i)(c) and 6.2.7 that ηM : M −→Da(M) is an isomorphism if and only if gradeM a ≥ 2.

6.2.11 Exercise. Let T : C(R) −→ C(R) be an R-linear covariant functorwith the property that T (M) is a-torsion for every R-module M .

(i) Show that, for each R-module M , the result RiT (M) of applying thei-th right derived functor of T to M is again a-torsion.

(ii) Assume, in addition, that T is left exact and such that, for each finitelygenerated R-module M , the ideal a contains a non-zerodivisor on M ifand only if T (M) = 0. Let r ∈ N0, and let M be a finitely generatedR-module such that aM = M . Prove that RiT (M) = 0 for all integersi < r if and only if a contains an M -sequence of length r.

(iii) Use part (ii) applied to the functor HomR(R/a, • ) to reprove the equiv-alence of statements (ii) and (iv) in Theorem 6.2.2.

(iv) Use part (ii) applied to the functor Γa to prove the result of 6.2.7.

Corollary 6.2.8 raises the following question: if J is any prescribed finitenon-empty set of non-negative integers, does there exist a local ring (R′,m′)

having the property that Him′(R′) = 0 if and only if i ∈ J? An affirmative

answer to this question was provided by I. G. Macdonald [46], and the nexttwo exercises sketch the essence of his argument.

6.2.12 �Exercise. Let M be an R-module.

(i) Show that the Abelian groupR⊕M is a commutative ring (with identity)with respect to multiplication defined by

(r1,m1)(r2,m2) = (r1r2, r1m2 + r2m1)

for all (r1,m1), (r2,m2) ∈ R⊕M . This ring is called the trivial exten-sion of R by M , and we shall denote it by R ∝M .

(ii) Show that the nilradical of R ∝ M contains 0 × M , and show thatSpec(R ∝M) = {p×M : p ∈ Spec(R)}.

(iii) Show that the ring R ∝ M is Noetherian if and only if M is a finitelygenerated R-module.


(iv) Let S be a multiplicatively closed subset of R. Show that S ×M is amultiplicatively closed subset of R ∝ M and that there is a ring iso-morphism φ : (S ×M)−1(R ∝ M)

∼=−→ S−1R ∝ S−1M for whichφ((r,m)/(s,m′)) = (r/s, (sm − rm′)/s2) for all r ∈ R, s ∈ S andm,m′ ∈M . In particular, this will show that

(R ∝M)(p×M)∼= Rp ∝Mp for all p ∈ Spec(R).

(v) Show that, ifR is local andM is finitely generated, thenR ∝M is localwith maximal ideal m×M , and that dim(R ∝M) = dimR.

6.2.13 Exercise. Let h, n be integers such that 0 ≤ h ≤ n, and let J bean arbitrary set of integers such that {h, n} ⊆ J ⊆ {i ∈ N0 : h ≤ i ≤ n}.Let (R,m) be a regular local ring of dimension n, and let u1, . . . , un be nelements which generate the maximal ideal m. For each j = 0, . . . , n − 1, letpj = Ruj+1 + · · ·+Run. Set

M :=⊕

j∈J\{n}R/pj .

(i) Show that, for i ∈ N0, we have Him(M) = 0 if and only if i ∈ J \ {n}.

(ii) Let R′ := R ∝ M , the trivial extension of R by M of 6.2.12 above. Bythat exercise, R′ is a local ring, with maximal ideal m′ := m ×M . Thering homomorphism ψ : R ∝ M → R for which ψ((r,m)) = r for all(r,m) ∈ R ∝ M enables us to regard both R and M as R′-modules.

Use the exact sequence 0 −→ Mq−→ R ∝ M

ψ−→ R −→ 0 of R′-modules (in which q is the canonical injection) to show that, for i ∈ N0,we have Hi

m′(R′) = 0 if and only if i ∈ J .

6.2.14 Exercise. Let M be an R-module, and let a1, . . . , an ∈ R be suchthat M = (a1, . . . , an)M . Prove that a1, . . . , an is an M -sequence if and onlyif

Hi−1(a1,...,ai)

(M) = 0 for all i = 1, . . . , n.

(Here are some hints for the implication ‘⇐’. Use induction on n: for theinductive step, on the assumption that n > 1 and the result has been proved forsmaller values of n, use Exercise 1.3.9(iv) (in conjunction with the hypotheses)to help you prove that

Hi−2(a1,...,ai)

(M/a1M) = 0 for all i = 2, . . . , n;

then use the Independence Theorem 4.2.1 to deduce that

Hi−2(a2,...,ai)

(M/a1M) = 0 for all i = 2, . . . , n.)

6.3 Exactness of ideal transforms 117

6.2.15 Exercise. Let (R,m) be a regular local ring of dimension d ≥ 2, andlet u1, . . . , ud be d elements which generate m. Let t ∈ N0 with t < d − 1,and let p = Ru1 + · · ·+Rut, a prime ideal of R. Let M denote the R-moduleRp/R.

(i) Show that pM =M .(ii) Show that u1, . . . , ut is a maximal M -sequence contained in m.

(iii) Show that the least integer i such that Him(M) = 0 is d− 1.

Contrast this with the situation for finitely generated R-modules!

6.2.16 Exercise. Let (R,m) be a regular local ring of dimension d > 2, letK denote the quotient field of R, and let N := R⊕K/R.

Show that mN = N , that m consists entirely of zerodivisors on N , and thatηN : N → Dm(N) is an isomorphism. Compare this with the situation forfinitely generated R-modules described in 6.2.10.

6.3 Exactness of ideal transforms

This section prepares the ground for our presentation, in the next section, ofthe promised special case of Serre’s Affineness Criterion. Central to this prepa-ration are various necessary and sufficient conditions for the exactness of thea-transform functor Da; some of these were presented by P. Schenzel in [73].

6.3.1 Lemma. The following statements are equivalent:

(i) the a-transform functor Da is exact;(ii) Hi

a(R) = 0 for all i ≥ 2;(iii) H2

a(M) = 0 for each finitely generated R-module M ;(iv) H2

a(M) = 0 for each R-module M .

Proof. (i) ⇒ (ii) By 2.2.6(ii), for each i ∈ N, the i-th right derived functorRiDa of Da is naturally equivalent to Hi+1

a . Now exactness of Da impliesthatRiDa(R) = 0 for all i ∈ N; therefore Hi

a(R) = 0 for all i ≥ 2.(ii) ⇒ (iii) Let ara(a) = t. By 3.3.3, for every R-module M , we have

Hia(M) = 0 for all i > t. We now argue by descending induction on i. Suppose

that i ∈ N with i > 2, and we have proved that Hia(M

′) = 0 for each finitelygenerated R-module M ′; let M be an arbitrary finitely generated R-module.There exists an exact sequence

0 −→ N −→ F −→M −→ 0

of finitely generated R-modules and R-homomorphisms in which F is free.


This induces an exact sequence Hi−1a (F ) −→ Hi−1

a (M) −→ Hia(N). Since

i − 1 ≥ 2, it follows from the additivity of the (i − 1)-th local cohomologyfunctor and condition (ii) that Hi−1

a (F ) = 0; since Hia(N) = 0 by our in-

ductive assumption, we can deduce that Hi−1a (M) = 0. This completes the

inductive step. Hence statement (iii) is proved by descending induction.(iii)⇒ (iv) This is immediate from the fact (3.4.10) that H2

a commutes withdirect limits, because each R-module can be viewed as the direct limit of itsfinitely generated submodules.

(iv) ⇒ (i) By 2.2.6(ii), the first right derived functor R1Da of Da is nat-urally equivalent to H2

a . It therefore follows from statement (iv) (and the leftexactness of Da) that Da is exact.

6.3.2 Exercise. Assume that ht p = 1 for every minimal prime ideal p of a,and that Rq is a UFD for each prime ideal q of R which contains a. Prove thatDa : C(R)→ C(R) is exact.

6.3.3 Remark. Let R′ be a second commutative Noetherian ring and let f :

R→ R′ be a ring homomorphism. Suppose that the a-transform functor Da isexact. Then since the restriction functor � R : C(R′) → C(R) is exact and, by2.2.24, there is a natural equivalence of functors

ε : DaR′( • )� R −→ Da( • � R),

it follows that the aR′-transform functor DaR′ : C(R′)→ C(R′) is also exact.

Recall from Exercise 2.2.5 that Da(R) has a natural structure as a commu-tative ring with identity, and from Exercise 2.2.12 that ηR : R → Da(R) is aring homomorphism. Thus we can regard Da(R) as an R-algebra by means ofηR. Our next proposition is concerned with this R-algebra structure.

6.3.4 Proposition. Suppose that aDa(R) = Da(R). ThenDa(R) is a finitelygenerated R-algebra.

Proof. Note that Γa(R) = R if and only if a is nilpotent, and that in this caseΓa is the identity functor, so that Da(R) = 0 by 2.2.6(i)(c) and the claim isclear in this case. Thus we suppose henceforth in this proof that Γa(R) = R.

Set R := R/Γa(R). We mentioned just before the statement of the propo-sition that ηR : R −→ Da(R) is a ring homomorphism. By 2.2.6(i)(c), wehave Ker(ηR) = Γa(R), and so ηR induces an injective ring homomorphismθR : R −→ Da(R). Moreover, as Coker ηR ∼= H1

a(R) by 2.2.6(i)(c), theR-module Coker θR is a-torsion, where a := aR, the extension of a to R.

Therefore, by 2.2.17, the unique R-homomorphism ψ : Da(R) −→ Da(R)



RθR

Da(R)�

ηa,Rψ

Da(R)

��

commutes is a ring homomorphism. Furthermore, by the Independence The-orem 4.2.1, we have Hi

a(Da(R)) ∼= Hia(Da(R)) for all i ∈ N0, and so

Hia(Da(R)) = 0 for i = 0, 1 by 2.2.10(iv). Hence, in view of 2.2.15(iii),

the map ψ is a ring isomorphism. Therefore aDa(R) = Da(R), and, if wecan show that Da(R) is a finitely generated R-algebra, then it will follow thatDa(R) is a finitely generated R-algebra. Since Γa(R) = 0 (by 2.1.2), wetherefore assume henceforth in this proof that Γa(R) = 0.

By 2.1.1(ii), this means that a contains a non-zerodivisor s on R. Note thatηR : R → Da(R) is injective, by 2.2.6(i)(c). We again use Rs to denotethe ring of fractions of R with respect to the multiplicatively closed subset{si : i ∈ N0

}. By 2.2.18, the subring D :=

⋃n∈N(R :Rs an) of Rs satisfies

aD = D, and it will be sufficient for us to show that D is a finitely generatedR-algebra.

Let a1, . . . , at be t elements which generate a. Since aD = D, there ex-ist y1, . . . , yt ∈ D such that 1 =

∑ti=1 aiyi. We aim to show that D =

R[y1, . . . , yt]. We achieve this by showing, by induction on n, that

(R :Rs an) ⊆ R[y1, . . . , yt] for every n ∈ N0.

This claim is clear for n = 0, and so we suppose that n > 0 and the claim hasbeen proved for smaller values of n. Let z ∈ (R :Rs an). Note that

aiz ∈ (R :Rs an−1) ⊆ R[y1, . . . , yt] for all i = 1, . . . , t.

Thus z = 1z =∑t

i=1 aiyiz =∑t

i=1(aiz)yi ∈ R[y1, . . . , yt], and the induc-tive step is complete.

Therefore D = R[y1, . . . , yt], as claimed, and Da(R) is a finitely generatedR-algebra.

In Lemma 6.3.1, we established several criteria for the exactness of the a-transform functor Da. We are now in a position to prove a further such crite-rion.

6.3.5 Proposition. The a-transform functor Da is exact if and only if

aDa(R) = Da(R).


Proof. (⇒) Assume first that Da is exact. By 6.1.9, we have

Da(R/a) ∼= (R/a)⊗R Da(R) ∼= Da(R)/aDa(R).

Since Da(R/a) = 0 by 2.2.10(i), we deduce that aDa(R) = Da(R).(⇐) Assume now that aDa(R) = Da(R). By Lemma 6.3.1, it is enough

for us to show that Hia(R) = 0 for all i ≥ 2; by Corollary 2.2.10(v), we have

Hia(R)

∼= Hia(Da(R)) for all i ≥ 2, and therefore it is enough for us to show

that Hia(Da(R)) = 0 for all i ≥ 2.

By Proposition 6.3.4, the commutative R-algebra Da(R) is finitely gener-ated, and therefore a Noetherian ring. Therefore, by the Independence Theorem4.2.1,

Hia(Da(R)) ∼= Hi

aDa(R)(Da(R)) for all i ∈ N0.

However, the assumption that aDa(R) = Da(R) means that ΓaDa(R) is thezero functor, and so Hi

aDa(R)(Da(R)) = 0 for all i ∈ N0. This completes theproof.

6.3.6 Corollary. Assume the a-transform functor Da is exact. Then ht p ≤ 1

for every minimal prime ideal p of a.

Proof. Suppose that p is a minimal prime ideal of a with ht p =: t ≥ 2, andlook for a contradiction. Since Da is exact, it follows from Lemma 6.3.1 thatHt

a(R) = 0. Therefore, by 1.1.3 and 4.3.3,

HtpRp

(Rp) = HtaRp

(Rp) ∼= (Hta(R))p = 0.

However, this contradicts Theorem 6.1.4, since dimRp = t.

6.3.7 �Exercise. Let c be a second ideal of R, and suppose that c ⊆ a. Showthat Dc(ηa,( • )) : Dc( • ) −→ Dc(Da( • )) is a natural equivalence of functors.(Here is a hint: you might find the argument used in the proof of 2.2.10(iii)helpful.)

6.3.8 Lemma. Let b be a second ideal ofR and letM be anR-module. Thenthe R-homomorphism

αa,ab,Db(M) : Da(Db(M)) −→ Dab(Db(M))

(which results from application of the natural transformation αa,ab of 2.2.23(i)to the R-module Db(M)) is an isomorphism.


Proof. The diagram

Db(M)ηa,Db(M)

Da(Db(M))�

αa,ab,Db(M)

Db(M)ηab,Db(M)

Dab(Db(M))��

commutes. By 2.2.10(iv), we have Hib(Db(M)) = 0 for i = 0, 1. Hence, by

the Mayer–Vietoris sequence 3.2.3, for i = 0, 1, there is an exact sequence

Hia(Db(M)) −→ Hi

ab(Db(M)) −→ Hi+1a+b(Db(M)),

so that Hiab(Db(M)) is a-torsion. Hence, by 2.2.6(i)(c) and 2.2.15, there is

a unique homomorphism θ : Dab(Db(M)) −→ Da(Db(M)) such that thediagram

Db(M)ηab,Db(M)

Dab(Db(M))�

θ

Db(M)ηa,Db(M)

Da(Db(M))��

commutes. In fact, it is immediate from the uniqueness aspects of 2.2.15 that

θ ◦ αa,ab,Db(M) = IdDa(Db(M)) and αa,ab,Db(M) ◦ θ = IdDab(Db(M)),

and so αa,ab,Db(M) is an isomorphism.

6.3.9 Exercise. Let b be a second ideal of R. Suppose that the functors Da

and Db are both exact. Let Lg−→ M

h−→ N be an exact sequence of R-modules and R-homomorphisms.

(i) Use 6.3.8 to show that the induced sequence

� �Dab(Db(L)) Dab(Db(M)) Dab(Db(N))Dab(Db(g)) Dab(Db(h))

is exact.(ii) Use 6.3.7 to show that the induced sequence

� �Dab(L) Dab(M) Dab(N)Dab(g) Dab(h)

is exact, and conclude that the functor Dab is exact.


6.4 An Affineness Criterion due to Serre

Although the main result of this section, 6.4.4, is, strictly speaking, only aspecial case of Serre’s Affineness Criterion (see [77, §46, Corollaire 1]), weshall nevertheless refer to it as ‘Serre’s Affineness Criterion’.

6.4.1 Notation. In our discussion of Serre’s Affineness Criterion, we shalluse the following notation. We shall use V to denote an affine variety over thealgebraically closed field K, and b will denote a non-zero ideal of O(V ); weshall use U to denote the quasi-affine variety V \ V (b), where V (b) is theclosed subset of V determined by b.

Also, for an affine or quasi-affine variety W over K and a point q ∈W , weshall frequently denote the maximal ideal {f ∈ O(W ) : f(q) = 0} of O(W )

by IW (q). The local ring of W at q will be denoted by OW,q .

6.4.2 Reminders. Let the notation be as in 6.4.1.

(i) The quasi-affine variety U is said to be affine precisely when there existsan affine variety W over K and an isomorphism of varieties U

∼=−→W .(ii) Let W be an affine variety over K. If α : U → W is a morphism of

varieties, we shall denote by α∗ : O(W ) → O(U) the homomorphismof K-algebras induced by α. Recall, from [30, Chapter I, Proposition3.5] for example, that the correspondence α �→ α∗ provides a bijectivemap from the set of all morphisms of varieties U → W to the set of allK-algebra homomorphisms O(W )→ O(U).

(iii) Let T be a closed subvariety of V , so that T is a closed, irreducible (andso necessarily non-empty) subset of V and the ideal

p := {f ∈ O(V ) : f(t) = 0 for all t ∈ T}

of O(V ) is prime. Recall that codimV T , the codimension of T in V , isgiven by

codimV T = dimV − dimT = dimO(V )− dimO(T ) = htO(V ) p.

(iv) Recall that a non-empty closed subset C of V is said to be of pure codi-mension r in V precisely when every irreducible component of C hascodimension r in V .

(v) Let u ∈ U . With the natural identifications, we have

O(V ) ⊆ O(U) ⊆ OU,u = O(U)IU (u) ⊆ K(U) = K(V ).

(vi) Finally, we recall that points on the quasi-affine variety U can be ‘sep-arated by regular functions’. More precisely, let u1, . . . , ur be r distinct

6.4 An Affineness Criterion due to Serre 123

points of U and let c1, . . . , cr ∈ K. Then there is a function f ∈ O(U)

such that f(ui) = ci for all i = 1, . . . , r. In particular, if u, u′ ∈ U aresuch that IU (u) = IU (u

′), then u = u′.

6.4.3 Lemma. Let the notation be as in 6.4.1, and let W be an affine varietyover K. Let α : U → W be a surjective morphism of varieties for which α∗ :O(W )→ O(U) is an isomorphism of K-algebras. Then α is an isomorphismof varieties.

Proof. Let w ∈ W . Since α is surjective, there exists u ∈ U with α(u) =

w. Now, for f ∈ O(W ), we have f(w) = f(α(u)) = 0 if and only if(α∗(f))(u) = 0. It follows that α∗(IW (w)) = IU (u). This shows that IU (u)is uniquely determined by w. By 6.4.2(vi), it follows that there is exactly oneu ∈ U for which α(u) = w. Hence α is bijective. It thus remains only forus to show that its inverse α−1 : W → U is a morphism of varieties. Forthis, let U ′ be a non-empty open subset of U , and let f ∈ O(U ′): it is enoughfor us to show that (α−1)−1(U ′) = α(U ′) is an open subset of W and thatf ◦ (α−1� α(U ′)) : α(U

′) −→ K is regular.Before establishing these two points, we make one preparatory observation.

For each h ∈ O(U), the regular function (α∗)−1(h) ∈ O(W ) has the propertythat, for each w ∈W ,(

(α∗)−1(h))(w) =((α∗)−1(h)

) (α(α−1(w))

)= α∗((α∗)−1(h))(α−1(w)) = h(α−1(w)).

We now turn our attention to U ′. There is a non-zero ideal c of O(V )

such that U ′ = V \ V (c). Thus, with the notation of 2.3.1, we have U ′ =⋃g∈c UV (g). However, for each g ∈ c, we have

α(UV (g)) = α ({v ∈ V : g(v) = 0})= α ({u ∈ U : (g� U )(u) = 0})= {w ∈W : (g� U )(α

−1(w)) = 0}= {w ∈W :

((α∗)−1(g� U )

)(w) = 0}

by the preceding paragraph; since (α∗)−1(g�U ) is a regular function onW , wesee that α(UV (g)) is an open subset of W . Hence α(U ′) =

⋃g∈c α(UV (g)) is

open too.Next, let w ∈ α(U ′). Since f ∈ O(U ′), there exists an open subset U ′′ of

U ′ with α−1(w) ∈ U ′′ and regular functions h, k ∈ O(U) such that k doesnot vanish on U ′′ and f(p) = h(p)/k(p) for all p ∈ U ′′. It follows fromthe immediately preceding paragraph that α(U ′′) is an open subset of W that


contains w, and from the paragraph before that that, for all q ∈ α(U ′′), wehave((α∗)−1(k)

)(q) = k(α−1(q)) = 0 and

(f ◦ (α−1� α(U ′))

)(q) = f(α−1(q)) =

h(α−1(q))

k(α−1(q))=

((α∗)−1(h)

)(q)

((α∗)−1(k)) (q).

Since (α∗)−1(h) and (α∗)−1(k) are regular functions on W , it follows thatf ◦ (α−1� α(U ′)) is a regular function on α(U ′), as required. This completesthe proof that α−1 :W → U is a morphism of varieties.

We are now ready to present Serre’s Affineness Criterion.

6.4.4 Serre’s Affineness Criterion. Let the notation be as in 6.4.1. Then thefollowing conditions are equivalent:

(i) U = V \ V (b) is affine;(ii) Db : C(O(V )) −→ C(O(V )) is exact;

(iii) Hib(O(V )) = 0 for all i ≥ 2;

(iv) H2b(M) = 0 for each finitely generated O(V )-module M ;

(v) H2b(M) = 0 for each O(V )-module M ;

(vi) bDb(O(V )) = Db(O(V )).

Proof. The equivalence of the last five conditions (ii) – (vi) was established in6.3.1 and 6.3.5. It only remains for us to establish that these are also equivalentto statement (i). Let ι : U → V denote the inclusion morphism of varieties.

(i) ⇒ (vi) Assume that U is affine, so that there is an affine variety W overK and an isomorphism of varieties α : U

∼=−→W . Set β := ι◦α−1 :W → V .In view of 2.3.2, we therefore have a commutative diagram

O(V )ηO(V )

Db(O(V ))�

ι∗ ν=νV,b∼=

O(U)O(W )

β∗

α∗

∼=

��

�

��

of K-algebra homomorphisms. Let n be a maximal ideal of Db(O(V )). Then

η−1O(V )(n) = ι∗−1(ν−1(n)) = β∗−1(α∗−1(ν−1(n))).

Now α∗−1(ν−1(n)) is a maximal ideal of the ring O(W ), and so there existsw ∈W such that α∗−1(ν−1(n)) = IW (w). Hence

η−1O(V )(n) = β∗−1(IW (w)) = IV (β(w)).

6.4 An Affineness Criterion due to Serre 125

However, β(w) = ι(α−1(w)) ∈ U = V \ V (b), and so

b ⊆ IV (β(w)) = η−1O(V )(n).

This is true for each maximal ideal n of Db(O(V )); therefore bDb(O(V )) =

Db(O(V )).(vi) ⇒ (i) Assume that bDb(O(V )) = Db(O(V )). Then Db(O(V )) is a

finitely generated O(V )-algebra, by 6.3.4. Moreover Db(O(V )) is an integraldomain, and so there exists an affine variety W over K for which O(W ) =

Db(O(V )). It follows from Theorem 2.3.2 that there is anO(V )-isomorphismν : O(U)

∼=−→ Db(O(V )) for which the diagram

O(V )ι∗

O(U)�

ηO(V )ν∼=

Db(O(V )) = O(W )

��

��

commutes. By 6.4.2(ii), there is a morphism of varieties γ : U → W suchthat γ∗ = ν−1, and there is a morphism of varieties β : W → V such thatβ∗ = ηO(V ). Since (β ◦ γ)∗ = γ∗ ◦ β∗ = ι∗, it also follows from 6.4.2(ii) thatβ ◦γ = ι. Our strategy is to use Lemma 6.4.3 to show that γ is an isomorphismof varieties, and so our immediate aim is to show that γ is surjective.

Let w ∈W . Note that β∗−1(IW (w)) = IV (β(w)). Since

(β∗−1(IW (w)))Db(O(V )) ⊆ IW (w) ⊂ Db(O(V )) = bDb(O(V )),

it follows that b ⊆ β∗−1(IW (w)) = IV (β(w)). Therefore β(w) ∈ U . Denoteβ(w) by u. We aim to show that γ(u) = w.

Suppose that γ(u) = w, and look for a contradiction. By 6.4.2(vi), thereexists a function g ∈ O(W ) such that g(w) = 0 and g(γ(u)) = 0. Also, sinceb ⊆ IV (u), there exists h ∈ b \ IV (u). Now Coker ηO(V )

∼= H1b(O(V )), by

2.2.6(i)(c), and so this cokernel is b-torsion. Therefore, there is n ∈ N suchthat β∗(hn)g = ηO(V )(h

n)g = hng ∈ ηO(V )(O(V )) = β∗(O(V )). Thusthere exists k ∈ O(V ) such that β∗(hn)g = β∗(k). We shall now calculatek(u) in two ways. First of all,

k(u) = k(β(w)) = (β∗(k))(w) = (β∗(hn)g)(w)

= ((β∗(h))(w))ng(w) = 0.


On the other hand,

k(u) = k(ι(u)) = k(β ◦ γ(u)) = k(β(γ(u))) = (β∗(k))(γ(u))

= (β∗(hn)g)(γ(u)) = ((β∗(h))(γ(u)))n g(γ(u))

= (h(β ◦ γ(u)))n g(γ(u)) = (h(u))ng(γ(u));

this is non-zero by choice of h and g. This contradiction shows that γ(u) = w.We have therefore proved that γ is surjective, and so we can now use Lemma

6.4.3 to deduce that γ is an isomorphism of varieties, so that U is affine.

6.4.5 Corollary. Let the notation be as in 6.4.1, and assume U = V \ V (b)

is affine. Then V \ U = V (b) is of pure codimension 1 in V .

Proof. By Serre’s Affineness Criterion 6.4.4, the functor Db is exact. There-fore, by 6.3.6, ht p ≤ 1 for every minimal prime ideal p of b. But b = 0 andO(V ) is an integral domain, so that ht p = 1 for every minimal prime ideal pof b. Hence every irreducible component of V (b) has codimension 1 in V .

Our work on Serre’s Affineness Criterion raises the following questions.We again use the notation of 6.4.1. First, if U = V \ V (b) is affine, sothat bDb(O(V )) = Db(O(V )), then it follows from 6.3.4 that Db(O(V ))

is a finitely generated O(V )-algebra: is the converse statement true, that is, ifDb(O(V )) is a finitely generated O(V )-algebra, is it necessarily the case thatU is affine? Second, is the converse of 6.4.5 true, that is, if V \ U = V (b) isof pure codimension 1 in V , is it necessarily the case that U is affine? Anotherexamination of the example studied in 2.3.7 and 3.3.5 will provide us withnegative answers to both questions.

6.4.6 Example. We consider again the affine variety V in A4 studied in 2.3.7and 3.3.5 and given by

V := VA4(X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

As in 3.3.5, let L := VA4(X2, X3, X4) ={(c, 0, 0, 0) ∈ A4 : c ∈ C

}and

L :={(c, 0) ∈ A2 : c ∈ C

}. Our argument uses the morphism of varieties

α : A2 → V of 2.3.7 for which α((c, d)) = (c, cd, d(d − 1), d2(d − 1)) forall (c, d) ∈ A2. It was shown in 3.3.5 that the restriction of α provides anisomorphism of (quasi-affine) varieties A2 \ ({(0, 1)} ∪ L)

∼=−→ V \ L. NowL is of pure codimension 1 in V . (As L is actually a subvariety of V , thisstatement is equivalent to the statement that codimV L = 1.) If we can showthat U := A2 \ ({(0, 1)}∪L) is not affine and thatO(U) is a finitely generatedC-algebra, then it will follow that both questions posed just after 6.4.5 have

6.5 Applications to local algebra in prime characteristic 127

negative answers. Let q := (0, 1) ∈ A2. As A2 \ U = L ∪ {q} is not of purecodimension 1 in A2, it follows from 6.4.5 that U is not affine.

Next, L = VA2(Y ), so that A2 \L is affine, and, by 2.3.2 and 2.2.19, we canidentify O(A2 \ L) with the subring C[X,Y ]Y = C[X,Y, Y −1] of C(X,Y ).

Note that IA2\L(q) = (X,Y − 1)C[X,Y, Y −1]. (Observe that the maximal

ideal (X,Y − 1) of C[X,Y ] does not meet{Y i : i ∈ N0

}.) Since I

A2\L(q)

contains a C[X,Y, Y −1]-sequence X,Y − 1 of length 2, it follows from 2.2.8that

ηC[X,Y,Y −1] : C[X,Y, Y−1] −→ DIA2\L(q)(C[X,Y, Y

−1])

is an isomorphism. But, by 2.3.2, and as U = (A2 \ L) \ {q}, there is anisomorphism

DIA2\L(q)(C[X,Y, Y−1]) ∼= O((A2 \ L) \ {q}) = O(U)

of O(A2 \ L)-algebras. Hence O(U) is a finitely generated C-algebra. Also,the fact that the map ηC[X,Y,Y −1] is an isomorphism shows that

(X,Y − 1)DIA2\L(q)(C[X,Y, Y−1]) = DIA2\L(q)(C[X,Y, Y

−1]),

and so we see again (this time from Serre’s Affineness Criterion 6.4.4) that thequasi-affine variety U is not affine.

Thus both questions posed just after 6.4.5 have negative answers.

6.4.7 Exercise. Let the notation be as in 6.4.1. Also, let b′ be a second non-zero ideal ofO(V ) and letU ′ denote the quasi-affine variety V \V (b′). Deducefrom Exercise 6.3.9 and Serre’s Affineness Criterion 6.4.4 that, if U and U ′ areaffine, then U ∩ U ′ is also affine.

6.5 Applications to local algebra in prime characteristic

In this section, we present some applications of local cohomology to the studyof algebra over local rings of prime characteristic. These applications concernresults that do not involve local cohomology in their statements, but whichhave proofs that make non-trivial use of local cohomology.

In this work, we shall use some techniques that assist calculation with reg-ular sequences, and our first exercises in the section are concerned with these.The reader should recall the definition of poor M -sequence (where M is anR-module) given in 6.2.1


6.5.1 �Exercise. Let M be an R-module, and let r1, . . . , rn be a poor M -sequence, where n ≥ 2. Show that r1 is a non-zerodivisor onM/(r2, . . . , rn)M .

6.5.2 �Exercise. Let M be an R-module, let r2, . . . , rn, a, b ∈ R, wheren ≥ 2, and suppose that a, r2, . . . , rn is a poor M -sequence.

(i) Assume that

abm1 + r2m2 + · · ·+ rnmn = am′1 + r2m′2 + · · ·+ rnm

′n,

where m1, . . . ,mn,m′1, . . . ,m

′n ∈M . Show that

m′1 ∈ (b, r2, . . . , rn)M.

(ii) Deduce that, if b, r2, . . . , rn is also a poor M -sequence, thenab, r2, . . . , rn is a poor M -sequence.

6.5.3 �Exercise. Let M be an R-module, and r1, . . . , rn ∈ R, where n ∈ N.

(i) Let i ∈ {1, . . . , n}, and suppose that ri can be written as ri = ab, wherea, b ∈ R. Show that r1, . . . , ri−1, ri, ri+1, . . . , rn is a poor M -sequenceif and only if

r1, . . . , ri−1, a, ri+1, . . . , rn and r1, . . . , ri−1, b, ri+1, . . . , rn

are poor M -sequences.(ii) Let t1, . . . , tn be arbitrary positive integers. Show that r1, . . . , rn is a

poor M -sequence if and only if rt11 , . . . , rtnn is a poor M -sequence.

6.5.4 Notation for the section. Throughout the section, M will denote anR-module, n will denote a positive integer, and Ln(R) will denote the set ofn × n lower triangular matrices with entries in R. For H ∈ Ln(R), we shalluse |H| to denote the determinant of H , that is, the product of the diagonalentries of H . We shall use T to denote matrix transpose; displayed matriceswill be shown between rectangular brackets.

Let d1, . . . , dn ∈ R. We shall use diag(d1, . . . , dn) to denote the diagonalmatrix in Ln(R) whose (i, i)-th entry is di (for each i = 1, . . . , n).

6.5.5 Remark. Suppose that x1, . . . , xn, y1, . . . , yn ∈ R and H = [hij ] ∈Ln(R) are such that [y1 · · · yn]T = H [x1 · · · xn]T . The fact that the adjointmatrix AdjH satisfies (AdjH)H = |H|In ensures that

|H|(x1, . . . , xn)M ⊆ (y1, . . . , yn)M.

Multiplication by |H| therefore induces an R-homomorphism

M/(x1, . . . , xn)M −→M/(y1, . . . , yn)M.


Let k ∈ {1, . . . , n}. Since H is lower triangular, the k × k submatrix Hk ofH obtained by deleting the (k+1)-th,. . ., n-th rows and columns ofH satisfies[y1 · · · yk]T = Hk [x1 · · · xk]T . We deduce from the above paragraph that|Hk|(x1, . . . , xk)M ⊆ (y1, . . . , yk)M , so that

|H|(x1, . . . , xk)M ⊆ (y1, . . . , yk)M

because |Hk| is a factor of |H|.

6.5.6 Theorem. (L. O’Carroll [64, Theorem 3.2]) Suppose that x1, . . . , xn,y1, . . . , yn ∈ R and H ∈ Ln(R) are such that

(i) [y1 · · · yn]T = H [x1 · · · xn]T , and(ii) y1, . . . , yn is a poor M -sequence.

Then the R-homomorphism α : M/(x1, . . . , xn)M −→ M/(y1, . . . , yn)M

induced by multiplication by |H| is a monomorphism, and x1, . . . , xn is alsoa poor M -sequence.

Proof. We use induction on n. When n = 1, the result is immediate from6.5.3(i). We therefore suppose that n > 1 and that the result has been provedfor sequences of length smaller than n. Let hij denote the (i, j)-th entry of H(for all i, j = 1, . . . , n).

Now [x1 y2 · · · yn]T = H ′ [x1 x2 · · · xn]T , where H ′ = [h′ij ] ∈ Ln(R) isspecified as follows:

h′ij =

{1 if i = j = 1,

hij otherwise.

Let β : M/(x1, x2, . . . , xn)M −→ M/(x1, y2, . . . , yn)M be the R-homom-orphism induced by multiplication by |H ′|. Note also that

[y1 y2 · · · yn]T = D [x1 y2 · · · yn]T where D := diag(h11, 1, . . . , 1).

Let γ : M/(x1, y2, . . . , yn)M −→ M/(y1, y2, . . . , yn)M be the R-homom-orphism induced by multiplication by |D| = h11. Since |H| = |D||H ′|, wehave γ ◦ β = α. In order to show that α is a monomorphism, it is thereforesufficient for us to show that both γ and β are monomorphisms.

Since y1 = h11x1, we see by 6.5.3(i) that h11, y2, . . . , yn and x1, y2, . . . , ynare poor M -sequences. Let m ∈ M be such that h11m ∈ (y1, y2, . . . , yn)M ;thus h11m = y1m1 + · · · + ynmn for some m1, . . . ,mn ∈ M . Since y1 =

h11x1, we obtain h11(m − x1m1) ∈ (y2, . . . , yn)M . Since h11, y2, . . . , yn isa poor M -sequence, we can use 6.5.1 to see that h11 is a non-zerodivisor onM/(y2, . . . , yn)M ; hence m − x1m1 ∈ (y2, . . . , yn)M . It follows that γ is amonomorphism.


We now turn our attention to β. Set R = R/x1R and M = M/x1M ;for r ∈ R, denote the natural image of r in R by r. We noted in the lastparagraph that x1, y2, . . . , yn is a poor M -sequence. Therefore y2, . . . , yn isa poor M -sequence in R. Moreover [y2 · · · yn]T = G [x2 · · · xn]T , whereG = [gij ] ∈ Ln−1(R) is given by gij = hi+1,j+1 for all i, j ∈ {1, . . . , n− 1}.

Let β : M/(x2, . . . , xn)M −→ M/(y2, . . . , yn)M denote the R-homom-orphism induced by multiplication by |G| = |H ′|. By the inductive hypothesis,β is a monomorphism. An easy calculation then shows that β is a monomor-phism. Also, the facts that y1 = h11x1 and y1 is a non-zerodivisor on M

ensure that x1 is a non-zerodivisor on M . The inductive hypothesis yields thatx2, . . . , xn is a poor M -sequence, and so we can conclude that x1, . . . , xn is apoor M -sequence.


6.5.7 Corollary. Suppose that a1, . . . , an are n elements of R that generatea, and let M be an R-module. Set a := a1 . . . an. Use the notation of 5.3.1to denote natural images of elements of Ma in the n-th cohomology module ofthe Cech complex of M with respect to a1, . . . , an.

Suppose that a1, . . . , an is a poor M -sequence, and that m ∈M and i ∈ Nare such that [m/ai] = 0. Then m ∈ (ai1, . . . , a

in)M .

Proof. By 5.3.2, there is k ∈ N0 such that k ≥ i and ak−im ∈∑n

u=1 akuM .

Now diag(ak−i1 , . . . , ak−i

n )[ai1 · · · ain

]T=[ak1 · · · akn

]T and ak1 , . . . , akn is a

poor M -sequence, by 6.5.3(ii). Since

| diag(ak−i1 , . . . , ak−i

n )|m = ak−im ∈ (ak1 , . . . , akn)M,

it follows from O’Carroll’s Theorem 6.5.6 that m ∈ (ai1, . . . , ain)M .

6.5.8 Remark. Suppose that (R,m) is local with dimR = n > 0. Recallthat a system of parameters for R is a sequence (ri)

ni=1 of n elements of m

such that the ideal (r1, . . . , rn) of R generated by the terms of the sequence ism-primary. By a subsystem of parameters forRwe mean a sequence (r′i)

ti=1 of

t elements of m, with t ≤ n, which can be extended to a system of parametersfor R by the addition of n− t extra terms.

Let (ai)ni=1 be a system of parameters for R.When R is Cohen–Macaulay, so that a1, . . . , an is an R-sequence, we must

have (a1 . . . an)k ∈ (ak+1

1 , . . . , ak+1n ) for all k ∈ N0, as we now show. If

(a1 . . . an)k ∈ (ak+1

1 , . . . , ak+1n ), then, since ak+1

1 , . . . , ak+1n is anR-sequence

by 6.5.3(ii), we can use O’Carroll’s Theorem 6.5.6 in conjunction with theequation

diag(ak1 , . . . , akn) [a1 · · · an]

T=[ak+11 · · · ak+1

n

]T


to deduce that 1 ∈ (a1, . . . , an), a contradiction.This observation leads to the following famous conjecture.

6.5.9 The Monomial Conjecture. (See M. Hochster [37, Conjecture 1].)Suppose that (R,m) is local (but not necessarily Cohen–Macaulay) and thatdimR = n > 0. The conjecture that (a1 . . . an)k ∈ (ak+1

1 , . . . , ak+1n ) for all

k ∈ N0 and all systems of parameters (ai)ni=1 for R is known as the MonomialConjecture.

6.5.10 Theorem. (See M. Hochster [37, pp. 33–34].) Suppose that (R,m) islocal with dimR = n > 0, and has prime characteristic p. Then the conclu-sion of the Monomial Conjecture is true in R.

Proof. Let (ai)ni=1 be a system of parameters for R. Suppose there existsk ∈ N0 such that (a1 . . . an)k ∈ (ak+1

1 , . . . , ak+1n ), and seek a contradiction.

Set a := a1 . . . an, and represent elements of Hnm(R) using a1, . . . , an and

a in the manner described in 5.3.1. Let F denote the Frobenius action onHn

m(R) of 5.3.4 and 5.3.6. Let y := [1/a] ∈ Hnm(R); then the supposition

that ak ∈ (ak+11 , . . . , ak+1

n ) means that there exist r1, . . . , rn ∈ R such thatak = ak+1

1 r1 + · · ·+ ak+1n rn, so that, by the comments in 5.3.1,

y =

[1

a

]=

[ak

ak+1

]=

[ak+11 r1 + · · ·+ ak+1

n rnak+1

]= 0.

But an arbitrary element z of Hnm(R) can be expressed as z = [r/at] for some

r ∈ R and t ∈ N0. Choose h ∈ N such that ph ≥ t. On use of 5.3.6 we nowsee that

z =[ rat

]=

[ap

h−tr

aph

]= ap

h−tr

[1

aph

]= ap

h−trFh(y) = 0.

We have therefore shown that Hnm(R) = 0, contrary to the Non-vanishing

Theorem 6.1.4.

Hochster’s theorem above provides one example of an important result inlocal algebra whose statement makes no mention of local cohomology but forwhich local cohomology provides a proof. Below we present another such re-sult, from the theory of tight closure. Some definitions are needed.

6.5.11 Definitions. Suppose that R has prime characteristic p. We use R◦

to denote the complement in R of the union of the minimal prime ideals ofR. An element r ∈ R belongs to the tight closure a∗ of a if and only if thereexists c ∈ R◦ such that crp

n ∈ a[pn] for all n � 0. We say that a is tightly

closed precisely when a∗ = a. The theory of tight closure was invented by M.


Hochster and C. Huneke [38], and many applications have been found for thetheory: see [41] and [42], for example.

The next exercise establishes some basic properties of tight closure.

6.5.12 �Exercise. Suppose that R has prime characteristic p.

(i) Show that the tight closure a∗ of a is an ideal of R.(ii) Let b be a second ideal of R with a ⊆ b. Show that a∗ ⊆ b∗.

(iii) Show that (a∗)∗ = a∗, so that tight closure really is a ‘closure operation’.

6.5.13 Exercise. Suppose that R has prime characteristic p, and let a, b beideals of R.

(i) Show that (a+ b)∗ = (a∗ + b∗)∗ and (ab)∗ = (a∗b∗)∗.(ii) Show that 0∗ =

√0, and conclude that

√0 ⊆ c∗ for every ideal c of R.

(iii) Let π : R −→ R/√0 be the natural ring homomorphism. Show that

a∗ = π−1((π(a)(R/

√0))∗)

.

(iv) Show that, if a is tightly closed, then so too is (a : b).(v) Show that every maximal ideal of R is tightly closed.

6.5.14 Exercise. Let K be a field of prime characteristic p, and let R be thesubring of the ring of formal power series K[[X]] given by

R :={∑∞

i=0 aiXi ∈ K[[X]] : a1 = 0, a0, a2, a3, . . . ∈ K

}.

In R, calculate (X2R)∗, (X3R)∗, (X2R)∗ ∩ (X3R)∗, (X2R ∩ X3R)∗,((X2R)∗ : (X3R)∗) and (X2R : X3R)∗.

6.5.15 Proposition. Suppose that (R,m) is local, of prime characteristic p,and Cohen–Macaulay with dimR = n > 0. Let (ai)ni=1 be a system of pa-rameters for R, and suppose that the ideal q := (a1, . . . , an) is tightly closed.Then the ideal (at11 , . . . , a

tnn ) is tightly closed for all t1, . . . , tn ∈ N.

Proof. We argue by induction on t :=∑n

i=1 ti. Suppose that t > n and thatthe desired result has been proved for smaller values of t.

Without loss of generality, we can, and do, assume that t1 > 1. By our induc-tive hypothesis, (at1−1

1 , at22 , . . . , atnn ) is tightly closed. Let r ∈ (at11 , . . . , a

tnn )∗.

Then r ∈ (at1−11 , at22 , . . . , a

tnn )∗ = (at1−1

1 , at22 , . . . , atnn ), and so there exist

r1, . . . , rn ∈ R such that

r = r1at1−11 + r2a

t22 + · · ·+ rna

tnn .

Now r2at22 + · · ·+ rna

tnn ∈ (at11 , . . . , a

tnn ) ⊆ (at11 , . . . , a

tnn )∗, and so

r1at1−11 = r − (r2a

t22 + · · ·+ rna

tnn ) ∈ (at11 , . . . , a

tnn )∗.


Therefore, there exists c ∈ R◦ such that

c(r1a

t1−11

)pj

∈(at1p

j

1 , . . . , atnpj

n

)for all j � 0.

Now at1pj

1 , . . . , atnpj

n (for j ∈ N) is an R-sequence, and the diagonal matrix

D := diag(a(t1−1)pj

1 , 1, . . . , 1) ∈ Ln(R)

satisfiesD[ap

j

1 at2pj

2 · · · atnpj

n

]T=[at1p

j

1 at2pj

2 · · · atnpj

n

]T.We can there-

fore use O’Carroll’s Theorem 6.5.6 to deduce that

crpj

1 ∈(ap

j

1 , at2p

j

2 , . . . , atnpj

n

)for all j � 0.

Therefore r1 ∈ (a1, at22 , . . . , a

tnn )∗. By the inductive hypothesis, the ideal

(a1, at22 , . . . , a

tnn ) is tightly closed, so that r1 ∈ (a1, a

t22 , . . . , a

tnn ). Therefore

r = r1at1−11 + r2a

t22 + · · ·+ rna

tnn ∈ (at11 , a

t22 , . . . , a

tnn ).

Thus (at11 , at22 , . . . , a

tnn ) is tightly closed. This completes the inductive step,

and the proof.

6.5.16 Theorem. (See R. Fedder and K.-i. Watanabe [17, Proposition 2.2].)Suppose that (R,m) is local, of prime characteristic p, and Cohen–Macaulaywith dimR = n > 0. Suppose that there is one system of parameters (ai)ni=1

for R which generates a tightly closed ideal. Then the ideal of R generated byeach system of parameters for R is tightly closed, as is the ideal generated byeach subsystem of parameters for R.

Proof. First of all, by 6.5.15, the ideal (at11 , . . . , atnn ) is tightly closed for all

t1, . . . , tn ∈ N. Let F denote the Frobenius action on Hnm(R) of 5.3.4 and

5.3.6. We plan to prove that, if y ∈ Hnm(R) is such that there exists c ∈ R◦

with cF j(y) = 0 for all j � 0, then y = 0.Represent elements of Hn

m(R) = Hn(a1,...,an)

(R) using a1, . . . , an and a :=

a1 . . . an in the manner described in 5.3.1. Thus y ∈ Hnm(R) can be written

as y = [r/at] for some r ∈ R and t ∈ N0. Suppose that there exists c ∈ R◦

with cF j(y) = 0 for all j � 0. By 5.3.6, this means that [crpj

/atpj

] = 0

for all j � 0. Since a1, . . . , an is an R-sequence, we can deduce from 6.5.7that crp

j ∈ (atpj

1 , . . . , atpj

n ) for all j � 0. Therefore r ∈ (at1, . . . , atn)∗.

Since (at1, . . . , atn) is tightly closed, we see that r ∈ (at1, . . . , a

tn), so that

y = [r/at] = 0 by the comments in 5.3.1.Let (bi)ni=1 be an arbitrary system of parameters for R. We show now that

the ideal (b1, . . . , bn) is tightly closed. Let s ∈ (b1, . . . , bn)∗, so that there

exists c′ ∈ R◦ such that c′spj ∈ (bp

j

1 , . . . , bpj

n ) for all j � 0. Represent


elements of Hnm(R) = Hn

(b1,...,bn)(R) using b1, . . . , bn and b := b1 . . . bn in

the manner described in 5.3.1, and consider z := [s/b] ∈ Hnm(R). At this

point, it is important to recall from 5.3.5 that the Frobenius action F onHnm(R)

does not depend on any choice of generators for any m-primary ideal of R.Therefore, using 5.3.6 and 5.3.1 again, we can deduce that

c′F j(z) =

[c′sp

j

bpj

]= 0 for all j � 0

because c′spj ∈ (bp

j

1 , . . . , bpj

n ) for all j � 0. The claim proved in the im-mediately preceding paragraph therefore shows that z = [s/b] = 0, so thats ∈ (b1, . . . , bn) by 6.5.7 because b1, . . . , bn is an R-sequence. Therefore(b1, . . . , bn) is tightly closed.

To complete the proof, let i ∈ {0, 1, . . . , n − 1} and set c := (b1, . . . , bi);we have to show that c is tightly closed. (Interpret c as 0 when i = 0.) By theimmediately preceding paragraph, the ideal (b1, . . . , bi, bti+1, . . . , b

tn) is tightly

closed for all t ∈ N. Let v ∈ c∗; then v belongs to the tight closure of everyideal of R that contains c (by 6.5.12(ii)). Hence

v ∈⋂t∈N

(b1, . . . , bi, bti+1, . . . , b

tn)∗ =⋂t∈N

(b1, . . . , bi, bti+1, . . . , b

tn)

⊆⋂t∈N

((b1, . . . , bi) +mt

)=⋂t∈N

(c+mt)= c

by Krull’s Intersection Theorem. Therefore c is tightly closed, and the proof iscomplete.

6.5.17 Remark. Suppose that (R,m) is local and of prime characteristic.We say that R is F -rational if and only if every proper ideal c of R whichcan be generated by ht c elements is tightly closed. The Theorem 6.5.16 ofFedder and Watanabe shows that a Cohen–Macaulay local ring of prime char-acteristic is F -rational if one single system of parameters generates a tightlyclosed ideal. It can be proved that, if R is excellent, or a homomorphic imageof a Cohen–Macaulay local ring, and is F -rational, then it must be Cohen–Macaulay; however, that result is beyond the scope of this book. We have in-cluded 6.5.16 because, firstly, it gives another example of a significant resultin local algebra whose statement makes no mention of local cohomology, butfor which local cohomology provides a proof, and, secondly, because it givessome hints about the important role that local cohomology can play in tightclosure theory.

7

Artinian local cohomology modules

In this chapter, we shall show that certain local cohomology modules areArtinian, that is, satisfy the descending chain condition on submodules, andwe shall use our results to provide a different proof of a theorem in Chapter 6.

Suppose, temporarily, that (R,m) is local. In the Non-vanishing Theorem6.1.4, we proved that, if M is a non-zero, finitely generated R-module of di-mension n, then Hn

m(M) = 0. One consequence of our work in this chapteris that we can give an alternative proof of this result, and, at the same time,obtain more information than we deduced in Chapter 6. Our approach in thischapter exploits the facts that the R-module Hn

m(M) is actually Artinian, andthat, for Artinian modules over commutative rings, there is available a theoryof secondary representation that is, in several respects, dual to the theory ofprimary decomposition of Noetherian modules over commutative rings.

7.1 Artinian modules

We begin with some arguments due to L. Melkersson ([51], [52]) which willenable us to show that, for a non-zero, finitely generated module M of dimen-sion n over the local ring (R,m), the n-th local cohomology module Hn

a (M)

of M with respect to a is Artinian, and all the local cohomology modulesHi

m(M) (i ∈ N0) of M with respect to m are Artinian.We revert to our standard hypotheses concerning R and a (although it is

worth pointing out that the first two results below (due to Melkersson) actuallyhold under weaker hypotheses).

7.1.1 Lemma. (L. Melkersson [51, Lemma 2.1]). Let a ∈ R and let M bean aR-torsion R-module. Suppose that N,N ′ are submodules of M such thatN ′ ⊆ N and ai(0 :N ai+1) = ai(0 :N ′ ai+1) for all i ∈ N0. Then N = N ′.

136 Artinian local cohomology modules

Proof. Since each element of N is annihilated by some power of a, it isenough for us to show that (0 :N ai) ⊆ N ′ for all i ∈ N. We prove thisby induction on i. By hypothesis, (0 :N a) = (0 :N ′ a) ⊆ N ′, and so weassume that, for i ∈ N, we have proved that (0 :N ai) ⊆ N ′.

Let z ∈ (0 :N ai+1). Then aiz ∈ ai(0 :N ai+1) = ai(0 :N ′ ai+1), and sothere exists z′ ∈ (0 :N ′ ai+1) such that aiz = aiz′. Hence z − z′ ∈ (0 :N ai),and so z−z′ ∈ N ′ by the inductive assumption. Therefore z = (z−z′)+z′ ∈N ′, and the inductive step is complete.

7.1.2 Theorem. (L. Melkersson [51, Theorem 1.3]). Assume that M is ana-torsion R-module for which (0 :M a) is Artinian. Then M is Artinian.

Proof. We suppose that a can be generated by t elements and proceed byinduction on t. When t = 0, we have a = 0 and (0 :M a) = M , so that thereis nothing to prove in this case.

Now suppose that t = 1 and a = Ra for a ∈ R. Let

L1 ⊇ L2 ⊇ · · · ⊇ Ln ⊇ Ln+1 ⊇ · · ·

be a descending chain of submodules of M . Observe that, for each i ∈ N0 andeach submodule L of M , we have ai(0 :L a

i+1) ⊆ (0 :M a). In fact, for eachn ∈ N,

(0 :Ln a) ⊇ · · · ⊇ ai(0 :Ln ai+1) ⊇ ai+1(0 :Ln a

i+2) ⊇ · · ·

is a descending chain of submodules of the Artinian R-module (0 :M a), andso is eventually stationary: let En denote its eventual stationary value, so thatthere is kn ∈ N such that ai(0 :Ln a

i+1) = En for all i ≥ kn.Since

En = akn+kn+1(0 :Ln akn+kn+1+1)

⊇ akn+kn+1(0 :Ln+1 akn+kn+1+1) = En+1,

we see that E1 ⊇ E2 ⊇ · · · ⊇ En ⊇ En+1 ⊇ · · · is a descending chain ofsubmodules of the ArtinianR-module (0 :M a), and so there exists t ∈ N suchthat En = Et for all n ≥ t. Now for all i, n ∈ N with n ≥ t and i ≥ kt, wehave

Et = akt(0 :Lt akt+1) ⊇ ai(0 :Lt a

i+1) ⊇ ai(0 :Ln ai+1) ⊇ En = Et.

Thus ai(0 :Ln ai+1) = Et for all n ≥ t and i ≥ kt.

We plan to use Lemma 7.1.1. With this in mind, we consider, for each integeri = 0, 1, 2, . . . , kt − 1, the descending chain

ai(0 :L1 ai+1) ⊇ · · · ⊇ ai(0 :Ln a

i+1) ⊇ ai(0 :Ln+1 ai+1) ⊇ · · ·

7.1 Artinian modules 137

of submodules of the Artinian R-module (0 :M a): there exists u ∈ N withu ≥ t such that ai(0 :Ln ai+1) = ai(0 :Lu ai+1) for all n ≥ u and alli = 0, 1, 2, . . . , kt − 1.

We now have that, for each integer n ≥ u,

ai(0 :Ln ai+1) = ai(0 :Ln+1 a

i+1) for all i ∈ N0:

this equation is true for all i = 0, 1, 2, . . . , kt − 1 by the last paragraph and forall i ≥ kt by the paragraph before that. Therefore, by Lemma 7.1.1, we haveLn = Ln+1 for all n ≥ u . It follows that M is Artinian.

Now suppose, inductively, that t > 1 and the result has been proved for ide-als that can be generated by fewer than t elements. Suppose that a is generatedby t elements a1, . . . , at. Set b = Ra1+ · · ·+Rat−1 andN = (0 :M b). ThenN is Rat-torsion and (0 :N at) = (0 :M a). It therefore follows from what wehave already proved in the case where t = 1 that N = (0 :M b) is Artinian.Since M is, of course, b-torsion and b can be generated by t− 1 elements, wecan apply the inductive hypothesis to deduce that M is Artinian. The inductivestep is complete.

We immediately exploit Theorem 7.1.2 to prove that certain local cohomol-ogy modules are Artinian. The proofs presented below of the next two theo-rems are due to L. Melkersson (see [52, Theorems 2.1 and 2.2]), although theresults themselves are somewhat older.

7.1.3 Theorem. Assume that (R,m) is local, and let M be a finitely gener-ated R-module. Then the R-module Hi

m(M) is Artinian for all i ∈ N0.

Proof. We use induction on i. First, Γm(M) is a finitely generated R-moduleannihilated by a power of m; hence H0

m(M) has finite length.Now suppose, inductively, that i > 0 and we have shown that Hi−1

m (M ′) isArtinian for every finitely generated R-module M ′. Now

Him(M) ∼= Hi

m(M/Γm(M))

by 2.1.7(iii). Also, 2.1.2 shows thatM/Γm(M) is m-torsion-free. We thereforeassume in addition that M is an m-torsion-free R-module.

We now use 2.1.1(ii) to deduce that m contains an element r which is anon-zerodivisor on M . The exact sequence

0 −→Mr−→M −→M/rM −→ 0

induces an exact sequence Hi−1m (M/rM) −→ Hi

m(M)r−→ Hi

m(M) of lo-cal cohomology modules. Since M/rM is a finitely generated R-module, itfollows from the inductive hypothesis that Hi−1

m (M/rM) is Artinian, so that,


in view of the above exact sequence, the R-module (0 :Him(M) r) is Artinian.

Since Him(M) is m-torsion and therefore Rr-torsion, it follows from Theorem

7.1.2 that Him(M) is Artinian. The inductive step is complete.

The following exercise generalizes Theorem 7.1.3 to non-local situations.

7.1.4 �Exercise. Assume that R/a is Artinian, and let M be a finitely gener-ated R-module. Show that Hi

a(M) is Artinian for all i ∈ N0.

7.1.5 Exercise. Suppose that m is a maximal ideal of R, and that q is anm-primary ideal of R. Let M be an R-module, and let i ∈ N0.

(i) Let s ∈ R \m. Show that multiplication by s provides an automorphismof Hi

q(M), so that this local cohomology module has a natural structureas an Rm-module.

(ii) Show that a subset of Hiq(M) is an Rm-submodule if and only if it is an

R-submodule.(iii) Deduce from (i), (ii) and Theorem 7.1.3 that, when M is finitely gener-

ated, Hiq(M) is an Artinian R-module.

7.1.6 Theorem. Assume (R,m) is local, and let M be a non-zero, finitelygenerated R-module of dimension n. Then the R-module Hn

a (M) is Artinian.

Proof. We use induction on n. If n = 0, then√(0 : M) = m, so that M is

annihilated by a power of m and so has finite length; therefore its submoduleΓa(M) also has finite length.

Now suppose, inductively, that n > 0 and we have established the resultfor (non-zero, finitely generated) R-modules of dimension smaller than n.Now Hn

a (M) ∼= Hna (M/Γa(M)) by 2.1.7(iii). If dim(M/Γa(M)) < n, then

Hna (M/Γa(M)) = 0 by Grothendieck’s Vanishing Theorem 6.1.2, and there

is nothing to prove. Since M/Γa(M) is a-torsion-free (by 2.1.2), we thereforemake the additional assumption that M is an a-torsion-free R-module.

The argument now proceeds like that used in the proof of Theorem 7.1.3above. By 2.1.1(ii), the ideal a contains an element r which is a non-zerodivisoron M . The exact sequence 0 −→ M

r−→ M −→ M/rM −→ 0 induces anexact sequence Hn−1

a (M/rM) −→ Hna (M)

r−→ Hna (M). Since r is not in

any minimal member of SuppM , we have dim(M/rM) ≤ n − 1, so that,by the inductive hypothesis (or Grothendieck’s Vanishing Theorem 6.1.2), theR-module Hn−1

a (M/rM) is Artinian. Therefore, in view of the above exactsequence, (0 :Hn

a (M) r) is Artinian. Since Hna (M) is Rr-torsion, it follows

from Theorem 7.1.2 that Hna (M) is Artinian. The inductive step is complete.

7.2 Secondary representation 139

It is perhaps worth pointing out that, in our applications of Melkersson’sTheorem 7.1.2 in the proofs of 7.1.3 and 7.1.6, we have only used the specialcase of 7.1.2 in which a is principal. However, this is perhaps not surprising,as most of our proof of 7.1.2 was devoted to the case when a is principal.

The following exercise generalizes Theorem 7.1.6 to non-local situations.

7.1.7 Exercise. Let M be a non-zero, finitely generated R-module of finitedimension n. Show that the R-module Hn

a (M) is Artinian.

7.2 Secondary representation

We intend to exploit Theorem 7.1.3 to provide another proof of the Non-vanishing Theorem 6.1.4, which states that, if M is a non-zero, finitely gener-ated module of dimension n over the local ring (R,m), then Hn

m(M) = 0. Weare going to use the theory of secondary representations of Artinian modules.As this theory, although mentioned in Matsumura [50, Section 6, Appendix], isnot as well known as the theory of primary decomposition, we shall guide thereader through the main points by means of a series of exercises. Although weshall maintain our standard hypothesis that R is Noetherian, the reader mightbe interested to learn that this condition is not strictly necessary for the devel-opment of a worthwhile theory: see D. Kirby [44], I. G. Macdonald [45] or D.G. Northcott [62].

7.2.1 Definitions and �Exercise. Let S be an R-module. We say that S issecondary precisely when S = 0 and, for each r ∈ R, either rS = S orthere exists n ∈ N such that rnS = 0. Show that, when this is the case,p :=

√(0 :R S) is a prime ideal of R: in these circumstances, we say that S is

a p-secondary R-module.Show that a non-zero homomorphic image of a p-secondary R-module is

again p-secondary.Show that, if S1, . . . , Sn (where n ∈ N) are p-secondary submodules of an

R-module M , then so too is∑n

i=1 Si.

7.2.2 Definitions and �Exercise. Let M be an R-module. A secondary rep-resentation of M is an expression for M as a sum of finitely many secondarysubmodules of M . Such a secondary representation

M = S1 + · · ·+ Sn with Si pi-secondary (1 ≤ i ≤ n)

of M is said to be minimal precisely when

(i) p1, . . . , pn are n different prime ideals of R; and


(ii) for all j = 1, . . . , n, we have Sj ⊆n∑

i=1i�=j

Si.

We say that M is a representable R-module precisely when it has a sec-ondary representation. As the sum of the empty family of submodules of anR-module is zero, we shall regard a zero R-module as representable.

Show that a representable R-module has a minimal secondary representa-tion.

7.2.3 �Exercise. The First Uniqueness Theorem. Let M be a representableR-module and let


and

M = S′1 + · · ·+ S′n′ with S′i p′i-secondary (1 ≤ i ≤ n′)

be two minimal secondary representations of M . Prove that n = n′ and

{p1, . . . , pn} = {p′1, . . . , p′n} .

(Here is hint: show that, for p ∈ Spec(R), it is the case that p is one ofp1, . . . , pn if and only if there is a homomorphic image of M which is p-secondary.)

7.2.4 Definition. Let M be a representable R-module and let


be a minimal secondary representation of M . Then the n-element set

{p1, . . . , pn} ,

which is independent of the choice of minimal secondary representation of Mby 7.2.3, is called the set of attached prime ideals of M and denoted by AttM

or AttRM . The members of AttM are referred to as the attached prime idealsor the attached primes of M .

7.2.5 �Exercise. Suppose that the R-module M is representable, and let p ∈Spec(R). Use the Noetherian property of R to show that p ∈ AttM if andonly if there is a homomorphic image of M which has annihilator equal to p.

7.2.6 Exercise. Let 0 −→ L −→ M −→ N −→ 0 be an exact sequence ofrepresentable R-modules and R-homomorphisms. Prove that

AttN ⊆ AttM ⊆ AttL ∪AttN.

7.2 Secondary representation 141

7.2.7 �Exercise. The Second Uniqueness Theorem. Let M be a represent-able R-module and let


be a minimal secondary representation of M .Suppose that pj is a minimal member of {p1, . . . , pn} with respect to inclu-

sion. Prove that Sj =⋂

r∈R\pjrM .

In the light of the First Uniqueness Theorem 7.2.3, this means that, in aminimal secondary representation of M , each secondary term correspondingto a minimal member of AttM is uniquely determined by M and independentof the choice of minimal secondary representation.

In order to make use of this theory of secondary representation, we shallneed the fact that every Artinian R-module is representable. This fact is thesubject of the next two exercises.

7.2.8 Definition and �Exercise. We say that anR-moduleN is sum-irreduc-ible precisely when it is non-zero and cannot be expressed as the sum of twoproper submodules of itself.

Prove that an Artinian sum-irreducible R-module is secondary.

7.2.9 �Exercise. Let A be an Artinian R-module. Show that A can be ex-pressed as a sum of finitely many sum-irreducible submodules, and deducefrom Exercise 7.2.8 above that A is representable.

We can therefore form the finite set AttA. Note that AttA = ∅ if and onlyif A = 0.

Actually, the class of R-modules which possess secondary representationsis, in general, larger than the class of Artinian R-modules: this is illustrated bythe next exercise.

7.2.10 Exercise. Let E be an injective R-module.

(i) Suppose thatQ is anR-module with the property that its zero submoduleis a p-primary submodule of Q. Prove that HomR(Q,E), if non-zero, isp-secondary.

(ii) Let M be a finitely generated R-module. Prove that HomR(M,E) isrepresentable, and that AttR(HomR(M,E)) ⊆ AssRM .

(iii) Deduce that E is representable and that AttRE ⊆ AssR = ass 0.(iv) The injective R-module E is said to be an injective cogenerator for R

precisely when HomR(N,E) = 0 for every non-zero R-module N .Prove that, when this is the case, AttR(HomR(M,E)) = AssRM forevery finitely generated R-module M .


7.2.11 Proposition. Let A be an Artinian R-module, and let r ∈ R. Then

(i) rA = A if and only if r ∈ R \⋃

p∈AttA p; and(ii)

√(0 : A) =

⋂p∈AttA p.

Proof. Clearly, we can assume that A = 0, since Att 0 = ∅. Let

A = S1 + · · ·+ Sn with Si pi-secondary (1 ≤ i ≤ n)

be a minimal secondary representation of M .(i) Suppose that r ∈ R \

⋃p∈AttA p; then rSi = Si for all i = 1, . . . , n,

and so rA = A. On the other hand, if r ∈ pj for some j with 1 ≤ j ≤ n, thenrhSj = 0 for a sufficiently large integer h, and so

rhA = rhS1 + · · ·+ rhSn ⊆n∑

i=1i�=j

Si ⊂ A.

(ii) To prove this, just note that√(0 : A) =

⋂ni=1

√(0 : Si) =

⋂ni=1 pi.

Part (i) of 7.2.11 provides an Artinian analogue of the well-known fact that,if N is a Noetherian R-module and r ∈ R, then r is a non-zerodivisor on N ifand only if r lies outside all the associated prime ideals of N .

There is more to the theory of secondary representation than the brief out-line presented above: the interested reader should consult the references citedearlier for this topic, especially [45]. We have presented little more than thepart of the theory we shall need to use. Its relevance for us lies in the fact that,when (R,m) is local, and M is a finitely generated R-module, then, for eachi ∈ N0, the local cohomology moduleHi

m(M) is, by 7.1.3, Artinian, and so wecan form the finite set of prime ideals Att(Hi

m(M)) and use the theory of sec-ondary representation for these modules. In the subsequent work, we shall beinterested in whether, for particular j, the local cohomology module Hj

m(M)

is finitely generated (and so, since it is in any case Artinian, of finite length).With this in mind, we present here one more result about secondary represen-tation before going on to apply the theory to local cohomology modules.

7.2.12 Corollary. Let (R,m) be local and let A be an Artinian R-module.ThenA is finitely generated, and so of finite length, if and only if AttA ⊆ {m}.

Proof. (⇒) WhenA is of finite length, there exists h ∈ N such that mhA = 0,so that A is either 0 or m-secondary.

(⇐) If AttA ⊆ {m}, then, by 7.2.11(ii), there exists h ∈ N such thatmhA = 0, and it then follows from, for example, [81, 7.30] that A has finitelength.

7.3 The Non-vanishing Theorem again 143

7.3 The Non-vanishing Theorem again

Our first lemma of this section is in preparation for the main theorem of thischapter. Recall our convention that the dimension of the zero R-module is −1.

7.3.1 Lemma. Let (R,m) be local and letM be a non-zero, finitely generatedR-module of dimension n. Then the set

Σ := {N ′ : N ′ is a submodule of M and dimN ′ < n}

has a largest element with respect to inclusion, N say. Set G :=M/N . Then

(i) dimG = n;(ii) G has no non-zero submodule of dimension less than n;

(iii) AssG = {p ∈ AssM : dimR/p = n}; and(iv) Hn

m(G)∼= Hn

m(M).

Proof. Since M is a Noetherian R-module, the set Σ has a maximal member,N say. Since the sum of any two members of Σ is again in Σ, it follows that Ncontains every member of Σ, and so is the largest element of Σ.

(i) It follows from the canonical exact sequence

0 −→ N −→M −→ G −→ 0

that SuppG ⊆ SuppM , and that any p ∈ AssM with dimR/p = n mustbelong to SuppG since it cannot belong to SuppN . Hence dimG = n and{p ∈ AssM : dimR/p = n} ⊆ AssG.

(ii) Suppose that L is a submodule of M such that N ⊆ L ⊆ M anddim(L/N) < n. Consideration of the canonical exact sequence

0 −→ N −→ L −→ L/N −→ 0

shows that dimL < n; hence L ⊆ N and L/N = 0.(iii) Let p ∈ AssG. By (ii), dimR/p = n, so that, since SuppG ⊆

SuppM , we must have p ∈ AssM . Thus

AssG ⊆ {p ∈ AssM : dimR/p = n} .

Since the reverse inclusion was established in our proof of (i) above, we havecompleted the proof of (iii).

(iv) Since dimN < n, it follows from Grothendieck’s Vanishing Theorem6.1.2 that Hn

m(N) = Hn+1m (N) = 0. The claim therefore follows from the

long exact sequence of local cohomology modules (with respect to m) thatresults from the exact sequence 0 −→ N −→M −→ G −→ 0.


The next theorem is the main result of this chapter. With it, we not onlyoffer an alternative proof of the result of the Non-vanishing Theorem 6.1.4 (aspromised just before 6.1.3), but we also provide more information than wasgiven in Theorem 6.1.4. (We should perhaps point out that our proof below in7.3.2 does quote from Theorem 6.2.7 the fact that, if M is a non-zero, finitelygenerated module of depth d over the local ring (R,m), then Hd

m(M) = 0;however, no use was made of 6.1.4 in our proof of 6.2.7, and, indeed, theargument given in the proof of 6.2.7 is elementary.)

7.3.2 Theorem. Assume (R,m) is local, and let M be a non-zero, finitelygenerated R-module of dimension n. Then Hn

m(M) = 0 and

Att(Hnm(M)) = {p ∈ AssM : dimR/p = n} .

Proof. (This proof is due to I. G. Macdonald and R. Y. Sharp [47, Theorem2.2].) Throughout this proof, we make tacit use of the fact, proved in Theorem7.1.3, that, for each i ∈ N0, the module Hi

m(M) is Artinian, so that, by 7.2.9,it has a secondary representation and we can form the set Att(Hi

m(M)).We use induction on n. When n = 0, the module M has finite length and so

is annihilated by some power of m. Hence H0m(M) ∼= Γm(M) = M = 0. By

7.2.9 and 7.2.12,

Att(H0m(M)) = AttM = {m} = AssM = {p ∈ AssM : dimR/p = 0} .

The result has been proved in this case.Assume, inductively, that n > 0 and that the result has been proved for non-

zero, finitely generated R-modules of dimension n− 1. By 7.3.1, we can, afterfactoring out, if necessary, the largest submodule of M of dimension smallerthan n, assume that M has no non-zero submodule of dimension smaller thann. We shall make this assumption for the remainder of the inductive step, andwith this assumption our aim is to show that Att(Hn

m(M)) = AssM .Since n > 0, we have m ∈ AssM , and so there exists r ∈ m which is a non-

zerodivisor on M . We suppose that Hnm(M) = 0, and look for a contradiction.

If n = 1, we have 1 ≤ gradeM m = depthM ≤ dimM = 1, so thatgradeM m = 1 and we have a contradiction to 6.2.7. Thus, in our search for acontradiction, we can, and do, assume that n > 1.

Now, for each r ∈ m which is a non-zerodivisor on M , the module M/rM

(is non-zero and finitely generated and) has dim(M/rM) = n − 1, and theexact sequence 0 −→ M

r−→ M −→ M/rM −→ 0 induces a long exactsequence of local cohomology modules

Hn−1m (M)

r−→ Hn−1m (M) −→ Hn−1

m (M/rM) −→ 0

7.3 The Non-vanishing Theorem again 145

in view of our supposition that Hnm(M) = 0. Thus, for each r ∈ m which is

a non-zerodivisor on M , we have Hn−1m (M)/rHn−1

m (M) ∼= Hn−1m (M/rM),

and this is non-zero by the inductive hypothesis. Therefore Hn−1m (M) = 0.

Our next step is to prove that m ∈ Att(Hn−1m (M)). We suppose that m ∈

Att(Hn−1m (M)) and look for a contradiction. Then, by the Prime Avoidance

Theorem,

m ⊆(⋃

p∈Att(Hn−1m (M)) p

)⋃(⋃q∈AssM q

),

so that, in the light of 7.2.11(i), there exists r1 ∈ m which is a non-zerodivisoron M and such that Hn−1

m (M) = r1Hn−1m (M); this contradicts the fact that

Hn−1m (M/r1M) = 0.Thus m ∈ Att(Hn−1

m (M)): let p1, . . . , pt be the remaining members ofAtt(Hn−1

m (M)). Again by the Prime Avoidance Theorem, there exists

r2 ∈ m \(⋃t

i=1 pi

)⋃(⋃q∈AssM q

).

Since r2 ∈ m and r2 is a non-zerodivisor on M , we again have

Hn−1m (M)/r2H

n−1m (M) ∼= Hn−1

m (M/r2M),

and, by the inductive hypothesis, Hn−1m (M/r2M) = 0 and

Att(Hn−1m (M/r2M)) ⊆ {p ∈ Ass(M/r2M) : dimR/p = n− 1} .

But, by 7.2.5,

Att(Hn−1m (M)/r2H

n−1m (M)) ⊆

{p ∈ Att(Hn−1

m (M)) : r2 ∈ p},

and m is the only member of the latter set. Since n > 1, we have obtained acontradiction.

Thus we have proved that Hnm(M) = 0. To complete the inductive step,

since M now has no non-zero submodule of dimension smaller than n, it re-mains for us to prove that Att(Hn

m(M)) = AssM .We know that gradeM m ≥ 1; also, for each r ∈ m which is a non-

zerodivisor on M , we have dim(M/rM) = n − 1, so that Hnm(M/rM) = 0

by Grothendieck’s Vanishing Theorem 6.1.2, and the exact sequence of localcohomology modules induced by the exact sequence

0 −→Mr−→M −→M/rM −→ 0

yields that Hnm(M) = rHn

m(M). It therefore follows from 7.2.11(i) that

m \⋃

p∈AssM p ⊆ m \⋃

q∈Att(Hnm(M)) q.

Let q ∈ Att(Hnm(M)): it follows from the above inclusion relation and the


Prime Avoidance Theorem that q ⊆ p for some p ∈ AssM . Since Hnm is an

R-linear functor, it follows that

(0 :M) ⊆ (0 : Hnm(M)) ⊆ q ⊆ p.

As n = dimR/(0 :M) = dimR/p, it follows that q = p. Hence

Att(Hnm(M)) ⊆ AssM.

To establish the reverse inclusion, let p ∈ AssM , so that dimR/p = n. Bythe theory of primary decomposition, there exists a p-primary submodule Q ofM ; thus M/Q is a non-zero finitely generated R-module with Ass(M/Q) =

{p}. Note that M/Q cannot have any non-zero submodule of dimension lessthan n (or else it would have an associated prime ideal other than p). Thus, bythe work in the preceding six paragraphs applied to M/Q rather than M , wehave Hn

m(M/Q) = 0 and

∅ = Att(Hnm(M/Q)) ⊆ Ass(M/Q) = {p} .

Hence Att(Hnm(M/Q)) = {p}. Since dimQ < n + 1, Grothendieck’s Van-

ishing Theorem 6.1.2 tells us that Hn+1m (Q) = 0; therefore, the canonical

exact sequence 0 −→ Q −→ M −→ M/Q −→ 0 induces an epimorphismHn

m(M) −→ Hnm(M/Q). It now follows from 7.2.5 that

{p} = Att(Hnm(M/Q)) ⊆ Att(Hn

m(M)).

Hence AssM ⊆ Att(Hnm(M)), and therefore AssM = Att(Hn

m(M)). Thiscompletes the inductive step.

We show next how to deduce the result of Exercise 6.1.7 very quickly fromTheorem 7.3.2.

7.3.3 Corollary. Assume (R,m) is local, and let M be a non-zero, finitelygenerated R-module of dimension n > 0. Then Hn

m(M) is not finitely gener-ated.

Proof. By 7.1.3 and 7.3.2, the local cohomology module Hnm(M) is a non-

zero Artinian module which has a non-maximal attached prime ideal. HenceHn

m(M) is not finitely generated, by 7.2.12.

7.3.4 �Exercise. Let M be a finitely generated R-module for which the ideala+ (0 : M) is proper. Let p be a minimal prime ideal of a+ (0 : M), and lett := dimRp

Mp. Prove that Hta(M) = 0, and that, if t > 0, then Ht

a(M) is notfinitely generated.

8

The Lichtenbaum–Hartshorne Theorem

In this chapter, we take up again the main theme of Chapter 6, and establishanother vanishing theorem for local cohomology modules, namely the localLichtenbaum–Hartshorne Vanishing Theorem (see R. Hartshorne [29, Theo-rem 3.1], and C. Peskine and L. Szpiro [66, chapitre III, theoreme 3.1]). Whiletwo important vanishing theorems in Chapter 6, namely Grothendieck’s Van-ishing Theorem 6.1.2, and Theorem 6.2.7, which shows that, for a finitelygenerated R-module M such that aM = M , we have Hi

a(M) = 0 for alli < gradeM a, can be regarded as ‘algebraic’ in nature, the Lichtenbaum–Hartshorne Theorem is of ‘analytic’ nature, in the sense that it is intimatelyrelated with ‘formal’ methods and techniques, that is, with passage to comple-tions of local rings and with the structure theory for complete local rings.

Results of a ‘formal’ nature in algebraic geometry sometimes provide pow-erful tools: the Lichtenbaum–Hartshorne Theorem is one such example, for weshall see in Chapter 19 that it can play a crucial role in the study of connectivityin algebraic varieties.

The local Lichtenbaum–Hartshorne Vanishing Theorem gives necessary andsufficient conditions, over an n-dimensional local ring (R,m), for the vanish-ing of n-th local cohomology modules with respect to a, where a is proper. Thesufficiency of these conditions is rather harder to prove than the necessity, andso we shall just discuss the sufficiency in this introduction. Let R denote thecompletion of R. It turns out that Hn

a (M) = 0 for every R-module M if, foreach P ∈ Spec(R) satisfying dim R/P = n, we have dim R/(aR+P) > 0.

In the case when R is an n-dimensional complete local domain, the state-ment simplifies: then, Hn

a (M) = 0 for every R-module M if dimR/a > 0.The general statement can be deduced from this special case by means of stan-dard reductions, and we now outline our strategy for proof of the special case.This strategy is similar to that used by M. Brodmann and C. Huneke in [4].

Of course, the fact (3.4.10) that the local cohomology functorHia commutes

148 The Lichtenbaum–Hartshorne Theorem

with direct limits means that it is enough to prove that Hna (M

′) = 0 for eachfinitely generated R-module M ′ in order to obtain the above result; in fact,Lemma 8.1.7 below will show that it is enough to prove merely that Hn

a (R) =

0. Then we shall use the Noetherian property ofR to reduce to the case where ais a prime ideal p with dimR/p = 1. As in our proof in Chapter 6 of Theorem6.1.4, we again use ideas related to the structure theorems for complete localrings to reduce to the case where R is a Gorenstein ring.

The arguments outlined in the last paragraph will reduce our proof to thefollowing: given a complete Gorenstein local domain R of dimension n andp ∈ Spec(R) with dimR/p = 1, how can we show that Hn

p (R) = 0? Ourapproach to this will use the ideas concerning the symbolic powers of p ofExercise 3.1.4 (for which we shall provide a solution!) to see that

Hnp (R)

∼= lim−→j∈N

ExtnR(R/p(j), R);

we shall then use the facts that R is Gorenstein and depthR/p(j) > 0 to seethat ExtnR(R/p

(j), R) = 0 for all j ∈ N.

8.1 Preparatory lemmas

Our intention is to use Noetherian induction to reduce part of the proof ofthe Lichtenbaum–Hartshorne Theorem to a case where (R is a complete localdomain and) a is a prime ideal p of R such that dimR/p = 1. To achieve this,we wish to make our first application of a very useful proposition (8.1.2 below)that relates local cohomology with respect to a+Rb, where b is an element ofR, to local cohomology with respect to a. Our proof of 8.1.2 uses the followinglemma about injective modules.

8.1.1 Lemma. Let b ∈ R, and let I be an injective R-module. Then thesequence of R-modules and R-homomorphisms

0 −→ ΓRb(I)ξI−→ I

τI−→ Ib −→ 0,

in which ξI is the inclusion map and τI is the natural homomorphism, issplit exact. Consequently, when Ib is regarded as an R-module in the natu-ral way, it is injective; furthermore, application of the additive functor Γa tothe above split exact sequence yields a further split exact sequence of injectiveR-modules

0 −→ Γa+Rb(I) −→ Γa(I) −→ Γa(Ib) −→ 0.

8.1 Preparatory lemmas 149

Proof. Since I is injective, H1Rb(I) = 0; therefore, the exactness of the first

sequence in the statement of the lemma is immediate from 2.2.20. By 2.1.4, theR-module ΓRb(I) is injective, and so this exact sequence splits, and Ib, as R-module, is injective. The second part is then immediate once one recalls from1.1.2 that Γa(ΓRb(I)) = Γa+Rb(I), and from 2.1.4 that Γa(I

′) is injectivewhenever I ′ is an injective R-module.

8.1.2 Proposition. Let b ∈ R and let f : M −→ N be a homomorphism ofR-modules.

(i) There is a long exact sequence of R-modules and R-homomorphisms

0 H0a+Rb(M) H0

a(M) H0a(Mb)

H1a+Rb(M) H1

a(M) H1a(Mb)

· · · · · ·

Hia+Rb(M) Hi

a(M) Hia(Mb)

Hi+1a+Rb(M) · · ·

� � �

� � �

�

� � �

� �


Hia+Rb(M) Hi

a(M) Hia(Mb) Hi+1

a+Rb(M)� � �

Hia+Rb(N) Hi

a(N) Hia(Nb) Hi+1

a+Rb(N)� � �

Hia+Rb(f) Hi

a(f) Hia(fb) Hi+1

a+Rb(f)

� � � �

commutes for all i ∈ N0.(ii) For each i ∈ N0, there is a commutative diagram

0 H1Rb(H

ia(M)) Hi+1

a+Rb(M) ΓRb(Hi+1a (M)) 0� � � �

0 H1Rb(H

ia(N)) Hi+1

a+Rb(N) ΓRb(Hi+1a (N)) 0� � � �

Hi+1a+Rb(f)H1

Rb(Hia(f)) ΓRb(H

i+1a (f))

� � �

with exact rows. The top row is referred to as the comparison exact sequencefor M .

Proof. Let

I• : 0d−1

−→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·


be an injective resolution ofM , so that there is anR-homomorphism α :M →I0 such that the sequence

0 −→Mα−→ I0

d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

is exact. Similarly, let the exact sequence

0 −→ Nβ−→ J0 e0−→ J1 −→ · · · −→ J i ei−→ J i+1 −→ · · ·

provide an injective resolution for N . By the (dual of) the Comparison Theo-rem [71, 6.9], there exists a chain map φ• = (φi)i∈N0 : I• −→ J• for whichthe diagram

Mα

I0�

φ0f

J0Nβ

��

�

commutes.It is immediate from Lemma 8.1.1 that there is a commutative diagram

0 Γa+Rb(I•) Γa(I

•) Γa((I•)b) 0� � � �

0 Γa+Rb(J•) Γa(J

•) Γa((J•)b) 0� � � �

Γa(φ•)Γa+Rb(φ

•) Γa((φ•)b)

� � �

of complexes of R-modules and chain maps of such complexes such that, foreach i ∈ N0, the sequence 0 → Γa+Rb(I

i) → Γa(Ii) → Γa((I

i)b) → 0 isexact, and a similar property holds for the lower row. Thus the above commu-tative diagram of complexes gives rise to a long exact sequence of cohomologymodules of the complexes in the top row, a similar long exact sequence for thebottom row, and a chain map of the first long exact sequence into the second.Since, by 8.1.1, (I•)b provides an injective resolution forMb asR-module and(J•)b provides an injective resolution for Nb as R-module, all the claims in (i)follow.

(ii) For each R-module K, let τK : K −→ Kb be the natural map.For an R-submodule L of Ii, we use the Rb-monomorphism Lb −→ (Ii)b

induced by inclusion to identify Lb as an Rb-submodule of (Ii)b. With thisconvention, it is routine to check that

Ker(Γa((di)b)) = (Ker(Γa(d

i)))b, Im(Γa((di−1)b)) = (Im(Γa(d

i−1)))b.


Hence

(Hia(M))b =

(Ker(Γa(d

i))/ Im(Γa(di−1)))b

∼= (Ker(Γa(di)))b/(Im(Γa(d

i−1)))b

= Ker(Γa((di)b))/ Im(Γa((d

i−1)b)) = Hia(Mb).

This natural isomorphism (Hia(M))b ∼= Hi

a(Mb), and the corresponding onefor N , enable us to deduce from (i) that there is a commutative diagram

Hia+Rb(M) Hi

a(M) (Hia(M))b Hi+1

a+Rb(M)� � �τHi

a(M)

τHia(N)

Hia+Rb(N) Hi

a(N) (Hia(N))b Hi+1

a+Rb(N)� � �

Hia+Rb(f) Hi

a(f) (Hia(f))b Hi+1

a+Rb(f)

� � � �

with exact rows. The result now follows from 2.2.20, which shows that

Ker τHi+1a (M) = ΓRb(H

i+1a (M)),

that Ker τHi+1a (N) = ΓRb(H

i+1a (N)) and that there is a commutative diagram

Hia(M) (Hi

a(M))b H1Rb(H

ia(M)) 0� � �

Hia(N) (Hi

a(N))b H1Rb(H

ia(N)) 0� � �

Hia(f) (Hi

a(f))b H1Rb(H

ia(f))

� � �

τHia(M)

τHia(N)

with exact rows.

8.1.3 Remark. In 3.3.4 we introduced cohd(a), the cohomological dimen-sion of a. We deduce from 8.1.2(i) that, if b ∈ R, then

cohd(a+Rb) ≤ cohd(a) + 1.

8.1.4 Exercise. Let b be a second ideal of R and let i ∈ N0. Show that therestrictions of the functorsHi

a+b andHia to the full subcategory of C(R) whose

objects are the b-torsion R-modules are naturally equivalent.

8.1.5 Exercise. Let f : M −→ N be a homomorphism of R-modules and


let b ∈ R. Use 8.1.2(ii) and 8.1.4 to show that there is a commutative diagram

0 H1a+Rb(H

ia(M)) Hi+1

a+Rb(M) Γa+Rb(Hi+1a (M)) 0� � � �

0 H1a+Rb(H

ia(N)) Hi+1

a+Rb(N) Γa+Rb(Hi+1a (N)) 0� � � �

Hi+1a+Rb(f)H1

a+Rb(Hia(f)) Γa+Rb(H

i+1a (f))

� � �

with exact rows.

We make use of Proposition 8.1.2 in the following lemma.

8.1.6 Lemma. Let (R,m) be a local integral domain of dimension n; supposethat the set

B := {b : b is an ideal of R, Hnb (R) = 0 and dimR/b > 0}

is non-empty. Let p be a maximal member of B. (The fact that R is Noetherianensures that such a p exists.)

Then p is prime and dimR/p = 1.

Proof. Suppose that the statement ‘p is prime and dimR/p = 1’ is false.This means that there exists q ∈ Spec(R) such that dimR/q = 1 and p ⊂ q.Let b ∈ q \ p. Then, by 8.1.2(i), there is an exact sequence

Hnp+Rb(R) −→ Hn

p (R) −→ Hnp (Rb).

By the Independence Theorem 4.2.1, there is an isomorphism of R-modulesHn

p (Rb) ∼= HnpRb

(Rb). Since dimRb < n (as b ∈ m), we can deduce fromGrothendieck’s Vanishing Theorem 6.1.2 that Hn

pRb(Rb) = 0. It therefore fol-

lows from the above exact sequence thatHnp+Rb(R) = 0, and since p ⊂ p+Rb

and dimR/(p+Rb) > 0, this contradicts the maximality of p.

Our next lemma will come as no surprise to any reader who has solved Ex-ercises 6.1.9 and 6.1.10. However, we provide a more direct argument, whichavoids use of 6.1.9.

8.1.7 Lemma. Assume that dimR = n and Hna (R) = 0. Then Hn

a (M) = 0

for every R-module M .

Proof. SinceHia commutes with direct limits (by 3.4.10) and eachR-module

is the direct limit of its finitely generated submodules, it is enough for us toprove that Hn

a (M) = 0 whenever M is finitely generated. However, in thatcase, there exists an exact sequence 0 −→ N −→ F −→ M −→ 0 offinitely generated R-modules and R-homomorphisms in which F is free. This


induces an exact sequence Hna (F ) −→ Hn

a (M) −→ Hn+1a (N). It follows

from the additivity of the local cohomology functor Hna and the assumption

that Hna (R) = 0 that Hn

a (F ) = 0; since Hn+1a (N) = 0 by Grothendieck’s

Vanishing Theorem 6.1.2, we can deduce that Hna (M) = 0, as required.

8.1.8 Exercise. Assume that dimR = n and that a is proper, having minimalprime ideals p1, . . . , pt. Let M be an R-module for which Hn

pi(M) = 0 for all

i = 1, . . . , t. Show that Hna (M) = 0.

In the course of our proof of the local Lichtenbaum–Hartshorne VanishingTheorem, we shall reduce to the case where R is a complete local domainand then appeal to Cohen’s Structure Theorem for complete local rings: weshall again quote [50, Theorem 29.4] to deduce that there exists a completeregular local subring R′ of R which is such that R is finitely generated as anR′-module. The next lemma will be applied to this situation.

8.1.9 Lemma. Suppose that (R,m) is local, and that (R′,m′) is a local sub-ring of R which is such that R is finitely generated as an R′-module. Letp ∈ Spec(R) be such that dimR/p = 1. There exists b ∈ R such that, ifB denotes the subring R′[b] of R, then

√(p ∩B)R = p.

Proof. Of course,R is integral overR′. By the Incomparability Theorem (see[81, 13.33], for example), p must be a minimal prime ideal of (p ∩ R′)R: letp2, . . . , pt be the other minimal prime ideals of (p∩R′)R (there may be none).By the Prime Avoidance Theorem, there exists b ∈ p \

⋃ti=2 pi. Then B :=

R′[b], necessarily a Noetherian ring, is finitely generated as an R′-module, andso is integral over R′.

To complete the proof, it is enough for us to show that p is the one and onlyminimal prime ideal of (p∩B)R. It certainly is one, since R is integral over Bso that we can appeal to the Incomparability Theorem again, as we did at thebeginning of this proof. Let us suppose that q is another minimal prime idealof (p ∩B)R, different from p.

Obviously q = m. Since R is integral over R′, it follows that q ∩ R′ = m′

(by [81, 13.31], for example). Since p ∩ R′ ⊆ p ∩ B ⊆ q, it follows thatp∩R′ ⊆ q∩R′ ⊂ m′. But R/p is an integral extension of R′/(p∩R′), and sodimR′/(p∩R′) = dimR/p = 1. We can therefore deduce that p∩R′ = q∩R′.Another use of the Incomparability Theorem as before shows that q must bea minimal prime ideal of (p ∩ R′)R, and so must be one of p2, . . . , pt. Butb ∈ p ∩ R′[b] ⊆ (p ∩ B)R ⊆ q, and this is a contradiction because b waschosen outside p2 ∪ · · · ∪ pt.

In our application of Lemma 8.1.9 in which R′ is a complete regular local


ring, it will be important for us to know that the ringB = R′[b] is a Gorensteinring. This is the motivation behind the next lemma. In fact, the first part of thislemma is actually a special case of [81, Proposition 13.40], but we include aproof for the convenience of the reader.

8.1.10 Lemma. Let R be a subring of the integral domain S, and supposethat R is integrally closed and S = R[b], where b ∈ S is integral over R. LetK be the field of fractions of R. Then b is algebraic over K and its minimalpolynomial f over K belongs to R[X]. Also, the surjective ring homomor-phism φ : R[X] → R[b] given by evaluation at b has kernel fR[X], so that φinduces an isomorphism of R-algebras

φ : R[X]/fR[X]∼=−→ R[b] = S.

Proof. Since b is integral over R, it is certainly algebraic over K. Let itsminimal polynomial over K be

f = Xh + ah−1Xh−1 + · · ·+ a1X + a0 ∈ K[X].

We aim to show that a0, . . . , ah−1 ∈ R.There exists a field extension L of the field of fractions of S such that

f splits into linear factors in L[X]: let s = s1, . . . , sh ∈ L be such thatf = (X − s1)(X − s2) . . . (X − sh) in L[X]. Equate coefficients to seethat each of a0, . . . , ah−1 can be written as a ‘homogeneous polynomial’ (infact, a ‘symmetric function’) in s1, . . . , sh with coefficients ±1; in particular,a0, . . . , ah−1 ∈ R[s1, . . . , sh].

Since b is integral over R, there exists

g = Xn + rn−1Xn−1 + · · ·+ r1X + r0 ∈ R[X]

such that g(b) = 0. Therefore g ∈ Kerφ ⊆ fK[X], and so there existsg1 ∈ K[X] such that g = fg1 in K[X]. For each i = 1, . . . , h, evaluateboth sides of this equation at si ∈ L to see that g(si) = 0. Thus all thesi (i = 1, . . . , h) are integral over R, and so, by [81, 13.21] for example,the ring R[s1, . . . , sh] is a finitely generated R-module; it therefore followsthat a0, . . . , ah−1 are all integral over R, since they belong to R[s1, . . . , sh].But a0, . . . , ah−1 ∈ K and R is integrally closed; hence a0, . . . , ah−1 ∈ R.

Finally, Kerφ = fK[X] ∩ R[X], and it is elementary to deduce from thefacts that f is monic and has all its coefficients in R that fK[X] ∩ R[X] =

fR[X].

8.1.11 Lemma. Suppose that (R,m) is local and of dimension n, and alsothatR is a Gorenstein ring. LetM be a non-zero, finitely generatedR-module.Then depthM > 0 if and only if ExtnR(M,R) = 0.


Proof. Recall from [50, Theorem 18.1] (or from [3]) that R has injectivedimension n as a module over itself.

(⇒) There exists r ∈ m such that r is a non-zerodivisor on M . The exactsequence 0 −→M

r−→M −→M/rM −→ 0 induces an exact sequence

ExtnR(M/rM,R) −→ ExtnR(M,R)r−→ ExtnR(M,R) −→ 0

since Extn+1R (M/rM,R) = 0. Application of Nakayama’s Lemma to the

finitely generated R-module ExtnR(M,R) now shows that this module is zero.(⇐) By [50, Theorem 18.1], we have ExtnR(R/m, R) = 0. It follows that

m ∈ AssM , for otherwise there would be an exact sequence

0 −→ R/m −→M −→ C −→ 0

of R-modules, and the induced exact sequence

ExtnR(M,R) −→ ExtnR(R/m, R) −→ Extn+1R (C,R) = 0

would provide a contradiction.

The next exercise extends Lemma 8.1.11 in a precise manner.

8.1.12 Exercise. Suppose that (R,m) is a Gorenstein local ring of dimensionn. Let M be a non-zero, finitely generated R-module. Show that depthM isthe least integer i such that Extn−i

R (M,R) = 0.

The next lemma provides the promised solution for Exercise 3.1.4.

8.1.13 Lemma. Let (R,m) be a complete local domain, and let p be a primeideal of R of dimension 1, that is, such that dimR/p = 1.

(i) For each n ∈ N, there exists t ∈ N such that p(t) ⊆ mn.(ii) The family (p(n))n∈N is a system of ideals of R over N in the sense

of 2.1.10, and the (negative strongly) connected sequences of covariantfunctors (from C(R) to C(R))(

Hip

)i∈N0

and

(lim−→j∈N

ExtiR(R/p(j), • )

)i∈N0

are isomorphic.

Proof. Let ψ : R→ Rp denote the natural homomorphism.(i) Recall that p(n) := ψ−1(pnRp) = ψ−1 ((pRp)

n). Hence⋂n∈N p(n) = ψ−1

(⋂n∈N(pRp)

n)= ψ−1(0) = Kerψ,

by Krull’s Intersection Theorem [50, Theroem 8.9]. Since R is a domain, itfollows that

⋂n∈N p(n) = 0.


We can now use Chevalley’s Theorem ([59, Section 5.2, Theorem 1] or [89,Chapter VIII, Section 5, Theorem 13]) to deduce that, for each n ∈ N, thereexists t ∈ N such that p(t) ⊆ mn.

(ii) Let n ∈ N. Note that, by [81, 5.47(i)] for example, p is the uniqueminimal prime ideal of pn, and p(n) is the (uniquely determined) p-primaryterm in any minimal primary decomposition of pn. Since dimR/p = 1, theonly possible associated prime ideal of pn in addition to p is m. Thus eitherpn = p(n), or there exists h ∈ N such that p(n) ∩ mh ⊆ pn. It now followsfrom part (i) that there exists t ∈ N such that p(t) ⊆ pn.

Of course, pn ⊆ p(n) for all n ∈ N. Since it is clear that (p(n))n∈N is aninverse family of ideals of R over N in the sense of 1.2.10, it now follows fromProposition 3.1.1(iii) that (p(n))n∈N is actually a system of ideals of R over N,and from part (ii) of the same proposition that the negative strongly connectedsequences of covariant functors(

lim−→j∈N

ExtiR(R/pj , • )

)i∈N0

and

(lim−→j∈N

ExtiR(R/p(j), • )

)i∈N0

are isomorphic. The proof can now be completed by an appeal to 1.3.8.

8.2 The main theorem

We are now in a position to put all our various lemmas of §8.1 together toproduce a proof of the main theorem of this chapter.

8.2.1 The Lichtenbaum–Hartshorne Vanishing Theorem. (See R. Hartsh-orne [29, Theorem 3.1].) Suppose that (R,m) is local of dimension n, and alsothat a is proper. Then the following statements are equivalent:

(i) Hna (R) = 0;

(ii) for each (necessarily minimal) prime ideal P of R, the completion of R,satisfying dim R/P = n, we have dim R/(aR+P) > 0.

Proof. (i)⇒ (ii) Assume that Hna (R) = 0 and that there exists a prime ideal

P of R such that dim R/P = n but dim R/(aR +P) = 0. Since the naturalring homomorphism R → R is flat, it follows from the Flat Base ChangeTheorem 4.3.2 that there is an R-isomorphism Hn

a (R)⊗R R ∼= HnaR

(R), and

so HnaR

(R) = 0.

Now mR is the maximal ideal of the local ring R, and our assumptionsmean that (R/P,mR/P) is an n-dimensional local ring and (aR +P)/P is

8.2 The main theorem 157

an (mR/P)-primary ideal of this ring. It therefore follows from 1.1.3 and The-orem 6.1.4 (or Theorem 7.3.2) that Hn

(aR+P)/P(R/P) = 0. We now deduce

from the Independence Theorem 4.2.1 that HnaR

(R/P) = 0. Therefore, by

8.1.7, we must have HnaR

(R) = 0, and this is a contradiction.

(ii) ⇒ (i) Assume that dim R/(aR + P) > 0 for each P ∈ Spec(R) forwhich dim R/P = n. We suppose also that Hn

a (R) = 0, and we again lookfor a contradiction. We again use the Flat Base Change Theorem 4.3.2 to seethat there is an R-isomorphism Hn

a (R) ⊗R R ∼= HnaR

(R), from which we

deduce that HnaR

(R) = 0 (because the natural ring homomorphism R → R isfaithfully flat). We can therefore assume, in our search for a contradiction, thatR is complete: we make this assumption in what follows.

There is an ascending chain 0 = b0 ⊂ b1 ⊂ · · · ⊂ bh−1 ⊂ bh = R ofideals of R such that, for each i = 1, . . . , h, there exists pi ∈ Spec(R) withbi/bi−1

∼= R/pi. It follows from the half-exactness ofHna thatHn

a (R/pi) = 0

for at least one i between 1 and h. The Independence Theorem 4.2.1 now en-ables us to deduce that Hn

(a+pi)/pi(R/pi) = 0. Note that, by Grothendieck’s

Vanishing Theorem 6.1.2, the complete local domain R/pi has dimension n.Therefore, our assumptions imply that dim(R/pi)/((a+ pi)/pi) > 0. There-fore we can, and do, assume henceforth in our search for a contradiction thatR is an n-dimensional complete local domain and the proper ideal a satisfiesHn

a (R) = 0 and dimR/a > 0. We shall show that such a situation is impossi-ble.

By Lemma 8.1.6, there exists p ∈ Spec(R) such that Hnp (R) = 0 and

dimR/p = 1. We now appeal to Cohen’s Structure Theorem for completelocal rings: by [50, Theorem 29.4], there exists a complete regular local subring(R′,m′) of R which is such that R is finitely generated as an R′-module. ByLemma 8.1.9, there exists b ∈ R such that, if B denotes the subring R′[b] ofR, then

√(p ∩B)R = p. Note that, since R is integral over B, the latter is

also local and of dimension n, with unique maximal ideal m ∩ B. Note alsothat, since B is a finitely generated R′-module, it is a complete local ring.

Since R′ is integrally closed, it follows from Lemma 8.1.10 that there is amonic polynomial f ∈ R′[X] and an isomorphism of R′-algebras

R′[X]/fR′[X] ∼= R′[b] = B.

By [50, Theorem 19.5], the ring R′[X] is regular. As f is monic, it is a non-zerodivisor in R′[X]. We now deduce that B is a Gorenstein local ring asfollows. There exists exactly one maximal ideal Q of R′[X] that contains f ;


we have

B ∼= (R′[X]/fR′[X])Q/fR′[X]∼= R′[X]Q/(f/1)R

′[X]Q;

and the latter is a Gorenstein local ring by [50, Exercise 18.1].Since√

(p ∩B)R = p, we can deduce from 1.1.3 that Hn(p∩B)R(R) = 0,

and thence by the Independence Theorem 4.2.1 that Hn(p∩B)(R) = 0. There-

fore Hn(p∩B)(B) = 0 by Lemma 8.1.7. Let q := p ∩ B; observe that, since R

is integral over B, we have dimB/q = dimR/p = 1.We have nearly arrived at a contradiction: B is a complete n-dimensional

Gorenstein local domain, q ∈ Spec(B) has dimB/q = 1, and Hnq (B) = 0.

This is, in fact, impossible, because, by Lemma 8.1.13(ii),

Hnq (B) ∼= lim

−→j∈N

ExtnB(B/q(j), B)

and the latter module is zero by Lemma 8.1.11, since depthB/q(j) > 0 for allj ∈ N because each q(j) is a q-primary ideal.

8.2.2 Corollary. Suppose that (R,m) is local and has dimension n, and thatdim R/(aR +P) > 0 for every prime ideal P of R such that dim R/P = n.Then Hi

a(M) = 0 for all i ≥ n and for every R-module M .

Proof. This follows from Grothendieck’s Vanishing Theorem 6.1.2, Lemma8.1.7, and the Lichtenbaum–Hartshorne Vanishing Theorem 8.2.1.

8.2.3 Exercise. Suppose that (R,m) is local and thatM is a non-zero finitelygenerated R-module of dimension n. Assume that

dim R/(aR+P) > 0

for every prime ideal P of SuppR(M ⊗R R) such that dim R/P = n. Provethat Hi

a(M) = 0 for all i ≥ n.

8.2.4 �Exercise. Suppose that (R,m) is local and that A is an Artinian R-module. Show that, for each a ∈ A, there exists t ∈ N such that mta = 0.Deduce that A has a natural structure as a module over R, the completion ofR, that a subset of A is an R-submodule if and only if it is an R-submodule,and that the map φ : A → A ⊗R R defined by φ(a) = a ⊗ 1 for all a ∈ A isan isomorphism of R-modules.

The next two exercises involve the theory of secondary representation andattached primes of Artinian modules, discussed in §7.2.

8.2.5 Exercise. Let R′ denote a second commutative Noetherian ring and letf : R → R′ be a ring homomorphism. Suppose that the R′-module M ′ has


a secondary representation (7.2.2). Show that, when M ′ is viewed as an R-module by means of f , it has a secondary representation as an R-module, andthat

AttRM′ ={f−1(p′) : p′ ∈ AttR′ M ′} .

Is it possible for M ′ to have fewer attached prime ideals as an R-module thanit has as an R′-module? Justify your response.

8.2.6 Exercise. Suppose that (R,m) is local and has dimension n, and alsothat a is proper. By Theorem 7.1.6, the R-module Hn

a (R) is Artinian.

(i) Use the Lichtenbaum–Hartshorne Vanishing Theorem 8.2.1 to prove that

AttR(Hna (R)) = {P ∩R : P ∈ Spec(R), dim R/P = n,

and dim R/(aR+P) = 0}.

(It is perhaps worth pointing out that the result of this exercise (whichis due to R. Y. Sharp [80, Corollary 3.5]) amounts, in effect, to a refine-ment of the statement of the local Lichtenbaum–Hartshorne VanishingTheorem, because the set on the right-hand side of the above display isempty if and only if Hn

a (R) = 0.)(ii) Deduce that, if n > 0, thenHn

a (R), if non-zero, is not finitely generated.

8.2.7 Remark. Let n ∈ N and let K be a field. This comment is concernedwith the ring K[X1, . . . , Xn] of polynomials over K.

Each f ∈ K[X1, . . . , Xn]\ (X1, . . . , Xn) is a unit in K[[X1, . . . , Xn]], andso there is a natural injective ring homomorphism

φ : K[X1, . . . , Xn](X1,...,Xn) −→ K[[X1, . . . , Xn]].

It is not difficult to see that we can use φ to regard K[[X1, . . . , Xn]] as thecompletion of the regular local ring K[X1, . . . , Xn](X1,...,Xn). Let m denotethe maximal ideal (X1, . . . , Xn) of K[X1, . . . , Xn].

It follows that, if b is an ideal of K[X1, . . . , Xn] contained in m, and weuse xi to denote the natural image of Xi in K[X1, . . . , Xn]/b (for each i =1, . . . , n), then the injective ring homomorphism

K[x1, . . . , xn](x1,...,xn) −→ K[[X1, . . . , Xn]]/bK[[X1, . . . , Xn]]

obtained by composing the natural ring isomorphism

K[x1, . . . , xn](x1,...,xn) = (K[X1, . . . , Xn]/b)m/b

∼=−→ K[X1, . . . , Xn]m/bK[X1, . . . , Xn]m


with the injective ring homomorphism

K[X1, . . . , Xn]m/bK [X1, . . . , Xn]m

−→ K[[X1, . . . , Xn]]/bK[[X1, . . . , Xn]]

induced by φ provides the completion of K[x1, . . . , xn](x1,...,xn).

8.2.8 Exercise. Consider the special case of the situation of Remark 8.2.7 inwhich n = 3 and b = (X1X2). With these choices, let

R = K[x1, x2, x3](x1,x2,x3) = (K[X1, X2, X3]/(X1X2))(X1,X2,X3)/(X1X2).

Let p be the prime ideal (x1/1, x3/1) of R. Use the exact sequence

0 −→ R −→ R/(x1/1)⊕R/(x2/1) −→ R/(x1/1, x2/1) −→ 0

of Lemma 3.2.1 to show that H2p(R) is not finitely generated. Use the descrip-

tion of the completion R of R provided by 8.2.7 to find a prime ideal P of Rsuch that dim R/P = 2 but dim R/(pR+P) = 0.

8.2.9 Exercise. Consider the special case of the situation of Remark 8.2.7in which K has characteristic 0, n = 3 and b = (X2

2 − X21 − X3

1 ). SetR′ := K[X1, X2, X3] and m′ := (X1, X2, X3). Let

c := (X1 +X2 −X2X3)R′ + ((X3 − 1)2(X1 + 1)− 1)R′.

(i) Show that b ∈ Spec(R′), that b ⊆ c, and also that cR′m′ ∈ Spec(R′m′).(Here is a hint: do not forget that R′m′ is a regular local ring.)

(ii) Follow the procedure of 8.2.7, and set

R := (R′/b)m′/b and R = K[[X1, X2, X3]]/bK[[X1, X2, X3]];

regard R as the completion of R in the manner described in 8.2.7. It will beconvenient to use xi to denote the natural image of Xi in R′/b and in R (fori = 1, 2, 3). Show that R is a 2-dimensional local domain, and that

p := (x1 + x2 − x2x3)R+ ((x3 − 1)2(x1 + 1)− 1)R

is a prime ideal of R with dimR/p = 1.(iii) Let u ∈ R be given by

u := 1+1

2x1−

1

8x21+

1

16x31−

5

128x41+ · · ·+

(−1)n−12(2n− 2)!

4nn!(n− 1)!xn1 + · · · ,

so that u is a unit of R and u2 = 1 + x1. Show that

pR = (x1 + x2 − x2x3)R+ (x3 − 1 + u−1)R,

and that this is a prime ideal of R.


(iv) Use the exact sequence

0→ R −→ R/(x2−x1u)⊕R/(x2+x1u) −→ R/(x2−x1u, x2+x1u)→ 0

of Lemma 3.2.1 to show thatH2pR

(R) is not finitely generated, and deduce that

H2p(R) is not finitely generated.

(v) Find a prime ideal P of R such that dim R/P = 2 but

dim R/(pR+P) = 0.

8.2.10 Definition and �Exercise. Suppose that (R,m) is local. We say thatRis analytically irreducible precisely when R, the completion ofR, is an integraldomain.

Note that an analytically irreducible local ring must itself be a domain.Suppose that (R,m) is analytically irreducible, and that a is proper. Let

dimR = n. Show that Hna (R) = 0 if and only if dimR/a > 0, and that,

when this is the case, Hia(M) = 0 for all i ≥ n and for every R-module M .

8.2.11 Exercise. Suppose (R,m) is an analytically irreducible (see 8.2.10above) local domain of dimension 2, and that dimR/a > 0. Show that thea-transform functor Da is exact.

8.2.12 Exercise. Show by means of an example that, if the phrase ‘analyt-ically irreducible’ is omitted from the hypotheses in Exercise 8.2.11 above,then the resulting statement is no longer always true.

8.2.13 Exercise. For this exercise, we recommend that the reader refers tothe example studied in 2.3.7, 3.3.5 and 4.3.7, and applies the ideas of Remark8.2.7 in the special case in which n = 3 and

b =√(X1X4 −X2X3, X

21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

Let S′ denote the subring C[X,XY, Y (Y − 1), Y 2(Y − 1)] of C[X,Y ].Consider the ideals

n := (X,XY, Y (Y −1), Y 2(Y −1)) and r := (XY, Y (Y −1), Y 2(Y −1))

of S′.

(i) Show that rS′q is a principal ideal of S′q for every maximal ideal q of S′

different from n.(ii) Show that S := S′n is a local domain of dimension 2, and that p := rS′n

is a prime ideal of S such that dimS/p = 1.(iii) Use a calculation made at the end of 3.3.5 to show that H2

p(S) = 0.


(iv) Use the description of the completion (S, M) of S provided by Remark8.2.7, together with calculations made in 4.3.7, to see that S has exactlytwo minimal prime ideals, P1 and P2 say, and that

P1 +P2 = M and Pi + pS = M for i = 1 or 2.

Show that H2pS(S) is not finitely generated, and deduce that H2

p(S) isnot finitely generated.

8.2.14 Exercise. Let R be the localization C[X,Y 2, XY, Y 3](X,Y 2,XY,Y 3)

of the subring C[X,Y 2, XY, Y 3] of C[X,Y ]. Show that H2a(R) = 0 for every

ideal a of R which is not primary to the maximal ideal.

8.2.15 Definition and Exercise. Let V be a quasi-affine variety over an al-gebraically closed field K, and let p ∈ V .

We say that V is analytically irreducible at p precisely when the completionOV, p of the local ring OV, p of V at p is an integral domain, that is, if andonly if OV, p is analytically irreducible (see 8.2.10); otherwise, we say that Vis analytically reducible at p.

(i) Now let V be an affine surface overK (that is, an affine algebraic varietyoverK of dimension 2), and letC ⊂ V be a (not necessarily irreducible)curve (that is, a non-empty closed subset of pure codimension 1 in V (see6.4.2(iv))) which avoids all the points of V at which V is analyticallyreducible. Use Serre’s Affineness Criterion 6.4.4 to show that the quasi-affine variety V \ C is, in fact, affine.

(ii) By considering the example studied in 2.3.7, 3.3.5, 4.3.7 and (especially)6.4.6, show that the condition that C ‘avoids all the points of V at whichV is analytically reducible’ cannot be omitted from the hypotheses ofthe result established in part (i) above.

8.2.16 Exercise. Let V be an affine surface over an algebraically closed fieldK; let C ⊂ V be a (not necessarily irreducible) curve (see 8.2.15(i)). Let b bethe ideal of C in O(V ) and let p ∈ C. We say that C is fully branched in Vat p precisely when dim OV, p/(bOV, p +P) > 0 for all minimal primes P ofOV, p satisfying dim OV, p/P = 2.

(i) Show that C is fully branched in V at p if and only if

H2bOV, p

(OV, p) = 0.

(ii) Show that V \C is affine if and only if C is fully branched in V at all itspoints.


8.2.17 Exercise. This exercise is concerned with the situation of Exercise8.2.9, and so we use the notation of that exercise. It was not necessary in 8.2.9for the reader to know that the ideal c is actually prime, but, in fact, it is andthe first three parts of this exercise sketch a route to a proof of this fact.

(i) Note that K[X1, X2, X3] = K[X1, X2, X3], where X1 = X1 + 1 andX3 = X3 − 1. Note also that X1, X2, X3 are algebraically independentover K.

(ii) Set A := K[X1, X2, X3]/(X23 X1 − 1); consider A0 := K[X3]X3

and

K[X2, X3]X3as subrings of K(X2, X3) in the natural ways. Show that

A ∼= K[X2, X3]X3= A0[X2].

(iii) Show that

K[X1, X2, X3]/(X1−X2X3 − 1, X23 X1 − 1)

∼= K[X2, X3]X3/(X2 + X−1

3 − X−33 ) ∼= A0,

and deduce that c is a prime ideal of K[X1, X2, X3].(iv) Now assume that K is algebraically closed.

Let C := VA3(K)(X1 +X2 −X2X3, (X3 − 1)2(X1 + 1) − 1) andV := VA3(K)(X

22 −X2

1 −X31 ). Show that C ⊆ V and (with or without

the aid of Exercise 8.2.16 above) that V \ C is not affine.

9

The Annihilator and Finiteness Theorems

There have been several examples earlier in this book of non-finitely gener-ated local cohomology modules of finitely generated modules: for example,in 6.1.7, and also in 7.3.3, we saw that, if (R,m) is local, and N is a non-zero, finitely generated R-module of dimension n > 0, then Hn

m(N) is notfinitely generated. Since H0

m(N), being isomorphic to a submodule of N , iscertainly finitely generated, it becomes of interest to identify the least integeri for which Hi

m(N) is not finitely generated. This integer is referred to as thefiniteness dimension fm(N) of N relative to m. Our work in this chapter onGrothendieck’s Finiteness Theorem will enable us to see that, in this situation,and under mild restrictions on R,

fm(N) = min{depthRp

Np + dimR/p : p ∈ SuppN \ {m}}.

However, our approach will not restrict attention to the case where the idealwith respect to which local cohomology is calculated is the maximal ideal ofa local ring; also, the approach we shall take will show that questions aboutsuch finiteness dimensions are intimately related to questions about preciselywhich ideals annihilate local cohomology modules. We shall, in fact, establishGrothendieck’s Finiteness Theorem as a special case of Faltings’ AnnihilatorTheorem, which is concerned with the following question: given a second idealb of R, and given a finitely generated R-module M , is there a greatest integeri for which Hi

a(M) is annihilated by some power of b, and, if so, what is it?

9.1 Finiteness dimensions

Our first two results will provide motivation for the formal introduction of theconcept of finiteness dimension. Let M be a finitely generated R-module. It is

9.1 Finiteness dimensions 165

clear that, if, for some i ∈ N, the local cohomology module Hia(M) is finitely

generated, then auHia(M) = 0 for some u ∈ N, and so

a ⊆ √(0 : Hia(M)).

It is not quite so clear that there is a sort of converse to this result: we showbelow, in our first proposition of this chapter, that, if, for some t ∈ N, it isthe case that a ⊆ √(0 : Hi

a(M)) for all i < t, then it follows that Hia(M) is

finitely generated for all i < t.

9.1.1 Lemma. Let Lf−→M

g−→ N be an exact sequence ofR-modules andR-homomorphisms, and suppose that a ⊆ √(0 : L) and a ⊆ √(0 : N). Thena ⊆ √(0 :M) also.

Proof. Let r ∈ a. Then, for some t ∈ N, we have rtg(M) = 0, and sortM ⊆ Ker g = Im f . However, there exists u ∈ N such that ruL = 0, and soru(Im f) = 0. Hence ru+tM = 0.

9.1.2 Proposition. Let M be a finitely generated R-module, and let t ∈ N.Then the following statements are equivalent:

(i) Hia(M) is finitely generated for all i < t;

(ii) a ⊆ √(0 : Hia(M)) for all i < t.

Proof. (i) ⇒ (ii) As we remarked immediately before the statement of 9.1.1,this implication is clear.

(ii) ⇒ (i) We use induction on t. When t = 1, there is nothing to prove,since H0

a(M) = Γa(M) is a submodule of M , and so is finitely generated. Sosuppose that t > 1 and that the result has been proved for smaller values of t.By this assumption, Hi

a(M) is finitely generated for i = 0, 1, . . . , t− 2, and itonly remains for us to prove that Ht−1

a (M) is finitely generated.It follows from 2.1.7(iii) that Hi

a(M) ∼= Hia(M/Γa(M)) for all i > 0.

Also, M/Γa(M) is an a-torsion-free R-module, by 2.1.2. Hence we can, anddo, assume that M is an a-torsion-free R-module.

We now use 2.1.1(ii) to deduce that a contains an element r which is a non-zerodivisor on M . Since a ⊆ √(0 : Ht−1

a (M)), there exists u ∈ N such thatruHt−1

a (M) = 0. The exact sequence

0 −→Mru−→M −→M/ruM −→ 0

166 The Annihilator and Finiteness Theorems

induces a long exact sequence

0 H0a(M)

ru

H0a(M) H0

a(M/ruM)

H1a(M)

ru

H1a(M) H1

a(M/ruM)

· · · · · ·

Hia(M)

ru

Hia(M) Hi

a(M/ruM)

Hi+1a (M) · · · .

� � �

� � �

�

� � �

� �

It follows from this long exact sequence and 9.1.1 that

a ⊆ √(0 : Hia(M/ruM)) for all i < t− 1,

so that, by the inductive hypothesis, Hia(M/ruM) is finitely generated for all

i < t − 1. In particular, we see that Ht−2a (M/ruM) is finitely generated;

since ruHt−1a (M) = 0, it follows from the above long exact sequence that

Ht−1a (M) is a homomorphic image of Ht−2

a (M/ruM), and so is finitely gen-erated. This completes the inductive step.

Proposition 9.1.2 provides some motivation for the following definition.Here, and throughout the book, we adopt the convention that the infimum ofthe empty set of integers is to be taken as∞.

9.1.3 Definition. Let M be a finitely generated R-module. In the light ofProposition 9.1.2, we define the finiteness dimension fa(M) of M relative toa by

fa(M) = inf{i ∈ N : Hi

a(M) is not finitely generated}

= inf{i ∈ N : a ⊆ √(0 : Hi

a(M))}.

Note that fa(M) is either a positive integer or∞, and that, since H0a(M) is

finitely generated,

fa(M) = inf{i ∈ N0 : Hi

a(M) is not finitely generated}.

9.1.4 Exercise. For a finitely generated R-module M , show that fa(M) > 1

if and only if the a-transform Da(M) is finitely generated.

In the situation of 9.1.3, it is reasonable to regard the condition that a ⊆√(0 : Hi

a(M)) as asserting that Hia(M) is ‘small’ in a sense, because if

this condition holds for all i less than some positive integer t, then Hia(M)

is finitely generated for all i < t (by 9.1.2). However, sometimes it is more

9.1 Finiteness dimensions 167

realistic to hope for a weaker condition than ‘a ⊆ √(0 : Hia(M))’: we intro-

duce a second ideal b of R, and, when b ⊆ a, think of Hia(M) as being ‘small’

relative to b if b ⊆ √(0 : Hia(M)).

9.1.5 Definition. Let M be a finitely generated R-module and let b be asecond ideal of R. We define the b-finiteness dimension fba (M) of M relativeto a by

fba (M) := inf{i ∈ N0 : b ⊆ √(0 : Hi

a(M))}.

Note that fba (M) is either∞ or a non-negative integer not exceeding dimM

and that faa (M) = fa(M) because a ⊆ √(0 : Γa(M)). Note that, if b ⊆ √a,then b ⊆ √(0 : Γa(M)), so that we can then write

fba (M) := inf{i ∈ N : b ⊆ √(0 : Hi

a(M))}.

However, in general we shall not assume that b ⊆ √a.

9.1.6 �Exercise. Let the situation be as in 9.1.5, and let S be a multiplica-tively closed subset of R. Show that

fba (M) ≤ fS−1b

S−1a (S−1M).

9.1.7 �Exercise. Let R′ be a second commutative Noetherian ring and letf : R → R′ be a ring homomorphism; assume that R′, when regarded as anR-module by means of f , is finitely generated.

Let M ′ be a finitely generated R′-module and let b be a second ideal of R.Prove that

fba (M′) = fbR

′aR′ (M ′).

9.1.8 Lemma. Let b, c be further ideals of R, and let M be a finitely gener-ated R-module.

(i) If b ⊆ c, then fba (M) = fba (M/Γc(M)).(ii) If fba (M) > 0, then fba (M) = fba (M/Γa(M)).

Proof. Suppose that there exists n ∈ N such that bnΓc(M) = 0. ThenbnHj

a(Γc(M)) = 0 for all j ∈ N0. Now the exact sequence

0 −→ Γc(M) −→M −→M/Γc(M) −→ 0

induces, for each i ∈ N0, an exact sequence

Hia(Γc(M)) −→ Hi

a(M) −→ Hia(M/Γc(M)) −→ Hi+1

a (Γc(M)),

and, since b ⊆ √(0 : Hja(Γc(M))) for all j ∈ N0, it follows from 9.1.1 that

b ⊆ √(0 : Hia(M)) if and only if b ⊆ √(0 : Hi

a(M/Γc(M))),


so that fba (M) = fba (M/Γc(M)).(i) There exists n ∈ N such that cnΓc(M) = 0. If b ⊆ c, then bnΓc(M) = 0,

and the desired result follows from the above.(ii) Assume that fba (M) > 0. This means that bnΓa(M) = 0 for some

n ∈ N0. Now use the first paragraph of this proof with a taken as c.

9.1.9 Exercise. LetM be a finitely generatedR-module for whichM = aM .Show that fRa (M) = gradeM a.

9.2 Adjusted depths

The b-finiteness dimension fba (M) of 9.1.5 is one of the invariants used inFaltings’ Annihilator Theorem. We now motivate the introduction of anotherof the ingredients. This motivation concerns the situation of Exercise 7.3.4,and we provide now a solution to the second part of that exercise.

Let M be a finitely generated R-module for which the ideal a + (0 : M)

is proper. Let p be a minimal prime of a + (0 : M), and suppose that t :=

dimRpMp > 0. Then, by 4.3.3, 4.2.2, and 1.1.3 used in conjunction with the

fact that p is a minimal prime of a+ (0 :M), we have

(Hta(M))p ∼= Ht

aRp(Mp) ∼= Ht

aRp+(0:RpMp)(Mp)

= Ht(a+(0:M))Rp

(Mp) = HtpRp

(Mp).

By 7.3.3, this Rp-module is not finitely generated, since t := dimRpMp > 0;

hence Hta(M) is not finitely generated. We can therefore write fa(M) ≤ t.

There exists a minimal prime q of (0 : M) such that q ⊂ p and ht p/q = t.Note that depthRq

Mq = 0, that a ⊆ q, and that p is a minimal prime of a+ q;furthermore, we can write fa(M) ≤ depthRq

Mq + ht p/q.Next suppose that r ∈ R is a non-zerodivisor on M and that there is a min-

imal prime p′ of a + (0 : M/rM) such that t′ := dimRp′ (M/rM)p′ > 0.Another application of Exercise 7.3.4 yields that Ht′

a (M/rM) is not finitelygenerated. It therefore follows from the long exact sequence of local cohomol-ogy modules induced by the exact sequence

0 −→Mr−→M −→M/rM −→ 0

that either Ht′a (M) or Ht′+1

a (M) is not finitely generated. The reasoning inthe preceding paragraph applied to M/rM (rather than M ) shows that thereexists a minimal prime q′ of (0 : M/rM) such that q′ ⊂ p′ and ht p′/q′ = t′.Again, note that depthRq′ Mq′ = 1, that a ⊆ q′, that p′ is a minimal prime of

9.2 Adjusted depths 169

a+ q′, and that

fa(M) ≤ 1 + t′ = depthRq′ Mq′ + ht p′/q′.

It is hoped that, after this little discussion, the reader will not be too dis-mayed to find that we now begin to consider the values of the expressions

depthRsMs + ht(a+ s)/s

for prime ideals s ∈ SuppM . (Here, htR/s is to be interpreted as ∞.) Notethat, for such an s, we have ht(a + s)/s > 0 if and only if (a + s)/s is anon-zero ideal of the integral domain R/s, that is, if and only if s ∈ Var(a).We can think, loosely, of ht(a + s)/s as the ‘distance’ from s to Var(a) inSpec(R), as it is, at least in the case when R is catenary (see [50, p. 31]), theminimum length of a saturated ascending chain of prime ideals starting with s

(as its smallest term) and ending in Var(a).

9.2.1 Notation and Conventions. Let M be a finitely generated R-module.For p ∈ SuppM we shall abbreviate depthRp

Mp by depthMp; we shalladopt the convention that the depth of a zero module over a local ring is ∞,and accordingly, for p ∈ Spec(R) \ SuppM , we shall write depthMp =∞.

We interpret the height of the improper ideal R of R as∞; accordingly, fora proper ideal d of R, we write htR/d =∞.

9.2.2 Definitions. For a p ∈ Spec(R) and a finitely generated R-module M ,we define the a-adjusted depth of M at p, denoted adja depthMp, by

adja depthMp := depthMp + ht(a+ p)/p.

Note that this is ∞ unless p ∈ SuppM and a + p ⊂ R, and then it is anon-negative integer not exceeding dimM .

Let b be a second ideal of R. We define the b-minimum a-adjusted depth ofM , denoted by λba(M), by

λba(M) = inf {adja depthMp : p ∈ Spec(R) \Var(b)}= inf {depthMp + ht(a+ p)/p : p ∈ Spec(R) \Var(b)} .

Thus λba(M) is either∞ or a non-negative integer not exceeding dimM .

Faltings’ Annihilator Theorem, the main result of this chapter, asserts that,under mild restrictions on R, the invariants λba(M) and fba (M) (where M is afinitely generated R-module and b is a second ideal of R) are equal. Faltings’original proof in [14] is rather different from the one we shall present below.

The special case of Faltings’ Annihilator Theorem in which b = a reducesto Grothendieck’s Finiteness Theorem (see [26, Expose VIII, Corollaire 2.3]),


which is another fundamental result of local cohomology. This theorem pro-vides information about the finiteness dimension fa(M) of M relative to a

(again under mild restrictions on R).Our approach to the proof of Faltings’ Annihilator Theorem begins with an

investigation of some of the properties of the invariant λba(M) introduced in9.2.2.

9.2.3 Remarks. Let b, c and d be further ideals of R such that c ⊆ b anda ⊆ d. Let M be a finitely generated R-module. Then

(i) λba(M) = λb√a(M) = λ√b√a(M);

(ii) λba(M) ≤ λbd(M); and(iii) λba(M) ≤ λca(M).

9.2.4 Lemma. Let b, c be further ideals of R, and let M be a finitely gener-ated R-module.

(i) If b ⊆ c, then λba(M) = λba(M/Γc(M)).(ii) If λba(M) > 0, then λba(M) = λba(M/Γa(M)).

Proof. Suppose that, for each p ∈ Spec(R)\Var(b), we have (Γc(M))p = 0.Then (M/Γc(M))p ∼=Mp and

depth(M/Γc(M))p + ht(a+ p)/p = depthMp + ht(a+ p)/p,

so that λba(M) = λba(M/Γc(M)).(i) Suppose that b ⊆ c. Then, for each p ∈ Spec(R)\Var(b) we have c ⊆ p,

so that (Γc(M))p = 0. The claim therefore follows from the above paragraph.(ii) Assume that λba(M) > 0. Let p ∈ Spec(R) \ Var(b). We claim that

(Γa(M))p = 0. This is clearly the case if a ⊆ p, so suppose that a ⊆ p.If (Γa(M))p were not zero, then there would be an associated prime idealq of Γa(M) with q ⊆ p, and, necessarily, a ⊆ q. But then we would haveq ∈ Spec(R) \ Var(b) with adja depthMq = depthMq + ht(a+ q)/q = 0,contrary to the assumption that λba(M) > 0. Thus (Γa(M))p = 0 in all cases.

The desired result now follows from the first paragraph of this proof with a

taken as c.

9.2.5 Lemma. Let b be a second ideal of R, let M be a finitely generatedR-module, and let S be a multiplicatively closed subset of R. Then

λba(M) ≤ λS−1b

S−1a(S−1M).

9.3 The first inequality 171

Proof. Let P ∈ Spec(S−1R) \ Var(S−1b). Then there exists p ∈ Spec(R)

such that p ∩ S = ∅ and P = S−1p. Now p ∈ Var(b), and so

λba(M) ≤ depthMp + ht(a+ p)/p.

Now (S−1M)P = (S−1M)S−1p∼= Mp as Rp-modules (when (S−1M)S−1p

is regarded as an Rp-module by means of the natural isomorphism Rp −→(S−1R)S−1p). Also

ht(a+ p)/p ≤ ht(S−1a+ S−1p)/S−1p,

and so λba(M) ≤ depth(S−1M)P+ht(S−1a+P)/P. The result follows.

The next lemma shows that λba(M) is, in a certain sense, independent of thebase ring.

9.2.6 Lemma. Let b be a second ideal of R, let M be a finitely generatedR-module, and let c be an ideal of R such that c ⊆ (0 :M). Then

λba(M) = λ(b+c)/c(a+c)/c(M).

Proof. Let p ∈ Spec(R) \ Var(b). Either p ⊇ c or p ⊇ c. If p ⊇ c, thenMp = 0 and depthMp + ht(a+ p)/p =∞. This means that

λba(M) = inf {depthMp′ + ht(a+ p′)/p′ : p′ ∈ Var(c) \Var(b)} .

However, if p ∈ Var(c) \Var(b), then p/c ∈ Spec(R/c) \Var((b+ c)/c), andit is an elementary exercise to check that depthMp/c = depthMp and

ht(((a+ c)/c) + (p/c))/(p/c) = ht(a+ p)/p.

The claim follows from these observations.

9.3 The first inequality

Our proof of Faltings’ Annihilator Theorem consists of demonstrations that,with the notation of 9.1.5 and 9.2.2, we have fba (M) ≤ λba(M) and, undermild restrictions on R, λba(M) ≤ fba (M). We start with the first inequality.We prepare the ground with several lemmas.

9.3.1 Lemma. Let b be a second ideal ofR, and letM be a finitely generatedR-module. Suppose that p ∈ AssM \ Var(b) has ht(a + p)/p = 1. Thenb ⊆ √(0 : H1

a(M)).


Proof. It suffices to show that a∩b ⊆ √(0 : H1a(M)). Since ht(a+p)/p = 1,

we must have that a ⊆ p, so that p ∈ Var(a ∩ b). Therefore we may replace b

by a ∩ b; we therefore assume henceforth in this proof that b ⊆ a.As p ∈ AssM , there is an exact sequence

0 −→ R/p −→M −→ N −→ 0

of R-modules and R-homomorphisms; this induces an exact sequence

H0a(N) −→ H1

a(R/p) −→ H1a(M).

Since N is finitely generated, b ⊆ a ⊆ √(0 : H0a(N)). Therefore, by 9.1.1,

it is enough for us to show that b ⊆ √(0 : H1a(R/p)). By the Independence

Theorem 4.2.1, there is an R-isomorphism H1a(R/p)

∼= H1(a+p)/p(R/p), and

so it is enough for us to show that

(b+ p)/p ⊆√(

0 : H1(a+p)/p(R/p)

).

It is therefore enough for us to prove that, if R is a domain, ht a = 1 and0 = b ⊆ a, then b ⊆ √(0 : H1

a(R)). We shall achieve this by showing that, inthese circumstances, rH1

a(R) = 0 for every r ∈ a \ {0}.Suppose that, on the contrary, rH1

a(R) = 0 for some r ∈ a \ {0}. Let q bea minimal prime of a with ht q = 1. Let R′ := Rq and m′ := qRq, so that(R′,m′) is a 1-dimensional local domain. We deduce from 1.1.3 and 4.3.3 that

rH1m′(R′) = rH1

aRq(Rq) ∼= r(H1

a(R))q = 0.

Of course, r ∈ m′ \ {0}. Since (R′,m′) is a 1-dimensional local domain, wehave m′ =

√rR′. Therefore, by 1.1.3 and 2.2.21(i),

H1m′(R′) = H1

rR′(R′) ∼= R′r/R′.

Thus rR′r ⊆ R′, so that r(1/r2) ∈ R′. Hence r is a unit of R′. This contradic-tion completes the proof.

9.3.2 Lemma. Let b be a second ideal of R, and let M be a finitely gen-erated R-module. Suppose that p ∈ SuppRM ∩ Var(a) \ Var(b) and lett := gradeMp

(aRp). Then b ⊆ √(0 : Hta(M)).

Proof. The hypotheses ensure thatMp = aRpMp. In view of 4.3.3 and 6.2.7,we have (Ht

a(M))p ∼= HtaRp

(Mp) = 0. By hypothesis, there exists b ∈ b \ p.It follows that (Ht

a(M))b = 0, so that b ∈ √(0 : Hta(M)).

9.3.3 Exercise. Show that, for a finitely generated R-module M for whichM = aM , we have λRa (M) = gradeM a. (Here is a hint: observe that

gradeM a = inf{depthMp : p ∈ Var(a)}.)


9.3.4 Lemma. Let M be a finitely generated R-module, and p, s ∈ Spec(R)

with p ⊆ s. Then

depthMs ≤ depthMp + ht s/p.

Proof. We use induction on h := ht s/p, there being nothing to prove whenh = 0.

Consider the case in which h = 1. Clearly, we can assume p ∈ SuppM .Now Mp is Rp-isomorphic to the localization of Ms at the prime ideal pRs

(when that localization is regarded as an Rp-module by means of the naturalisomorphism Rp

∼=−→ (Rs)pRs). In order to establish the desired result when

h = 1, it is therefore enough for us to show that, if (R,m) is local, M is afinitely generated R-module, and p ∈ SuppM is such that dimR/p = 1, thendepthM ≤ depthMp + 1. This we do.

Let gradeM p = t (see 6.2.4), and let a1, . . . , at be a maximal M -sequencecontained in p. Then a1/1, . . . , at/1 is an Mp-sequence contained in pRp. LetN :=M/

∑tj=1 ajM . Then

depthN = depthM − t, depthNp = depthMp − t

and gradeN p = 0. Thus we can achieve our aim by showing that depthN ≤depthNp + 1. We can assume that depthN > 0, since otherwise there isnothing to prove. This assumption means that m ∈ AssN . But p consistsentirely of zerodivisors on N , and so p ⊆

⋃p′∈AssN p′. It therefore follows

from the Prime Avoidance Theorem and the fact that dimR/p = 1 that p ∈AssN , and so depthN ≤ dimR/p = 1 by [50, Theorem 17.2].

The claim in the lemma has therefore been proved when h = 1; supposenow that h > 1 and the claim has been proved for smaller values of h. Sinceht s/p = h, there exists q ∈ Spec(R) such that p ⊂ q ⊂ s, ht q/p = 1 andht s/q = h − 1. We can now use the inductive hypothesis and the (alreadyestablished) truth of the claim in the case when h = 1 to see that

depthMs ≤ depthMq + h− 1

≤ depthMp + 1 + h− 1 = depthMp + h.


9.3.5 Lemma. Let M be a finitely generated R-module. Suppose that p, q ∈Spec(R) are such that q is a minimal prime of a+p and ht q/p = ht(a+p)/p.Then

adja depthMs ≤ adja depthMp

for all s ∈ Spec(R) with p ⊆ s ⊆ q.


Proof. It is clear that q ⊇ a+ s, and so ht q/s ≥ ht(a+ s)/s. Hence, on useof 9.3.4, we have

adja depthMs = depthMs + ht(a+ s)/s

≤ depthMs + ht q/s

≤ depthMp + ht s/p+ ht q/s

≤ depthMp + ht q/p

= depthMp + ht(a+ p)/p = adja depthMp,

as required.

9.3.6 Lemma. Let s, q ∈ Spec(R) with s ⊂ q be such that ht q/s > 1. Leta ∈ q \ s. Then there exists t ∈ Spec(R) with s ⊂ t ⊂ q such that a ∈ t.

Proof. Let R := R/s; for c ∈ R, denote the natural image of c in R by c.Let the minimal primes of the proper principal ideal Ra be p1/s, . . . , ph/s,

where p1, . . . , ph ∈ Spec(R). Now ht pi/s = 1 for all i = 1, . . . , h, by Krull’sPrincipal Ideal Theorem. Therefore, since ht q/s > 1, it follows from thePrime Avoidance Theorem that q/s ⊆

⋃hi=1 pi/s. Hence, there exists b ∈ R

such that

b ∈ q/s \h⋃

i=1

pi/s.

Since Rb ⊆ q/s, there exists t ∈ Spec(R) with s ⊂ t ⊆ q such that t/s is aminimal prime of Rb. Since ht q/s > 1 and ht t/s = 1, it follows that t ⊂ q.Since b ∈ t \

⋃hi=1 pi, we can deduce that t is different from all of p1, . . . , ph.

Hence a ∈ t.

9.3.7 Theorem. Let b be a second ideal of R, and let M be a finitely gener-ated R-module. Then

fba (M) ≤ λba(M).

Proof. Set λ := λba(M). If λ = ∞, then there is nothing to prove; we there-fore suppose that λ is finite, and argue by induction on λ.

When λ = 0, there exists p ∈ Spec(R) \Var(b) with

adja depthMp = depthMp + ht(a+ p)/p = 0.

This means that depthMp = 0 and p ∈ Var(a), so that

pRp ∈ AssRpMp ∩Var(aRp)

and gradeMp(aRp) = 0. Therefore b ⊆ √

(0 : H0a(M)) by 9.3.2, so that

fba (M) = 0.


Now suppose that λ > 0 and assume, inductively, that the desired inequalityhas been proved for smaller values of λ. We can assume that fba (M) > 0.By 9.1.8(ii) and 9.2.4(ii), we have fba (M) = fba (M/Γa(M)) and λba(M) =

λba(M/Γa(M)). We may therefore replace M by M/Γa(M), and, in view of2.1.2, assume that M is a-torsion-free for the remainder of this proof.

Choose p ∈ Spec(R) \Var(b) with

adja depthMp = depthMp + ht(a+ p)/p = λ.

Assume first that p ∈ Var(a). Then t := gradeMpaRp ≤ depthMp = λ, so

that fba (M) ≤ t ≤ λ by 9.3.2.Assume now that p ∈ Var(a). If λ = 1, then

adja depthMp = depthMp + ht(a+ p)/p = 1.

These conditions mean that ht(a + p)/p = 1 and depthMp = 0. Hencep ∈ AssM , and so it follows from Lemma 9.3.1 that b ⊆ √

(0 : H1a(M)).

Hence fba (M) ≤ 1 = λ.So we may assume that λ > 1 (and p ∈ Var(a)). Since p ∈ Var(a ∩ b),

there exists a ∈ a∩b\p. Let q be a minimal prime of a+p such that ht q/p =

ht(a+ p)/p. Then p belongs to the set

Σ := {s′ ∈ Spec(R) : p ⊆ s′ ⊆ q and a ∈ s′} ;

let s be a maximal member of Σ. Now a ∈ a ∩ b ⊆ a ⊆ q, and so s ⊂ q.Lemma 9.3.6 shows that ht q/s = 1. Note that the fact that a ∈ s ensures thats ∈ Spec(R) \Var(b).

We can now deduce from 9.3.5 and the definition of λ that

λ ≤ adja depthMs ≤ adja depthMp = λ.

Therefore adja depthMs = λ. Since a ∈ a \ s, we have a ⊆ s. It followsthat q is a minimal prime of a + s such that ht q/s = ht(a + s)/s = 1. Thuswe can replace p by s and so make the additional assumption that ht q/p =

ht(a+ p)/p = 1.We now propose to localize at q. Note that pRq ∈ Spec(Rq) \ Var(bRq),

that qRq is a minimal prime of aRq + pRq, that

ht qRq/pRq = ht(aRq + pRq)/pRq = 1,

that Mq is a finitely generated aRq-torsion-free Rq-module, and that

adjaRqdepth(Mq)pRq

= depth(Mq)pRq+ ht(aRq + pRq)/pRq

= depthMp + 1 = λ.


We can therefore deduce from 9.2.5 that λbRq

aRq(Mq) = λ. Also, by 9.1.6, we

have fba (M) ≤ fbRq

aRq(Mq). These considerations mean that it is enough for us

to prove the desired result under the additional assumption that (R, q) is local,and we make this assumption in what follows.

Our next aim is to show that p contains a non-zerodivisor on M . Supposethat this is not the case, so that p ⊆ s for some s ∈ AssM . Since M is a-torsion-free, a contains a non-zerodivisor on M , by 2.1.1(ii), so that, as q ⊇ a,we see that s ⊂ q. As ht q/p = 1, it follows that p = s ∈ AssM . This meansthat depthMp = 0, and λ = depthMp+ht(a+p)/p = 1. This contradictionshows that there exists r ∈ p which is a non-zerodivisor on M .

Now

λba(M/rM) ≤ adja depth(M/rM)p = depth(M/rM)p + 1

= depthMp − 1 + 1 < λ.

Therefore, by the induction hypothesis, fba (M/rM) ≤ λba(M/rM) < λ. Setu := fba (M/rM), so that b ⊆ √(0 : Hu

a (M/rM)). But the exact sequence

0 −→Mr−→M −→M/rM −→ 0

induces an exact sequence

Hua (M) −→ Hu

a (M/rM) −→ Hu+1a (M),

and so it follows from 9.1.1 that b ⊆ √(0 : Hia(M)) for i = u or i = u + 1.

Thus

fba (M) ≤ u+ 1 = fba (M/rM) + 1 ≤ λ.


9.4 The second inequality

We now embark on the more difficult part of our proof of Faltings’ AnnihilatorTheorem, namely the proof that, when R is a homomorphic image of a regularring, then, with the notation of 9.1.5 and 9.2.2, we have λba(M) ≤ fba (M).(Recall (see [50, p. 157]) that a commutative Noetherian ring is said to beregular precisely when its localizations at all of its prime ideals are regularlocal rings.)

9.4.1 Lemma. Let M be a finitely generated R-module, and let

p ∈ Spec(R) \ SuppM.

9.4 The second inequality 177

Then there exists s ∈ R \ p such that, for every ideal b of R, we have

sHib(M) = 0 for all i ∈ N0.

Proof. Since M is finitely generated, there exists s ∈ (0 : M) \ p, and this shas the desired properties because the functors Hi

b (i ∈ N0) are R-linear.

9.4.2 Lemma. LetM be a finitely generatedR-module, and let p ∈ Spec(R)

be such that Mp is a non-zero free Rp-module. Then there exist t ∈ N and anR-homomorphism π : Rt −→M such that (Kerπ)p = (Cokerπ)p = 0.

Proof. Set t := rankRpMp. There exist m1, . . . ,mt ∈ M such that m1/1,

. . . ,mt/1 form a base for the free Rp-module Mp. Let π : Rt −→ M be theR-homomorphism for which π((r1, . . . , rt)) =

∑ti=1 rimi for all (r1, . . . , rt)

∈ Rt. Then the localization of π at p is an isomorphism, and so (Kerπ)p =

(Cokerπ)p = 0.


be such thatMp is a non-zero freeRp-module. Then there exists s ∈ R\p suchthat, for every proper ideal b of R, we have

sHib(M) = 0 for all i < grade b.

Proof. By 9.4.2, there exist t ∈ N and an R-homomorphism π : Rt −→ M

such that (Kerπ)p = (Cokerπ)p = 0. By 9.4.1, there exist u, v ∈ R \ p suchthat, for every ideal b of R, we have

uHib(Kerπ) = vHi

b(Cokerπ) = 0 for all i ∈ N0.

Set s := uv. Let b be a proper ideal of R and let i ∈ N0 with i < grade b.Now the exact sequence 0 −→ Kerπ −→ Rt −→ Imπ −→ 0 induces an

exact sequence Hib(R

t) −→ Hib(Imπ) −→ Hi+1

b (Kerπ). Since i < grade b

we have Hib(R) = 0, by 6.2.7, so that Hi

b(Rt) = 0 in view of the additivity of

the functor Hib. By the immediately preceding paragraph, uHi+1

b (Kerπ) = 0.Therefore uHi

b(Imπ) = 0.Next, the exact sequence 0 −→ Imπ −→ M −→ Cokerπ −→ 0 induces

an exact sequence Hib(Imπ) −→ Hi

b(M) −→ Hib(Cokerπ). As uHi

b(Imπ)

= 0 and vHib(Cokerπ) = 0, it follows that sHi

b(M) = uvHib(M) = 0.

9.4.4 Conventions. Let M be an R-module. We shall denote the projectivedimension of M by proj dimM or, occasionally, by proj dimRM if it is es-sential to specify the underlying ring concerned. In particular, the reader iswarned that, when S is a multiplicatively closed subset of R, we shall alwayswrite proj dimS−1M rather than proj dimS−1R S

−1M . We adopt the con-vention that a zero module has projective dimension −∞.



be such that proj dimMp <∞. Then there exists s ∈ R\p such that, for everyproper ideal b of R, we have

sHib(M) = 0 for all i < grade b− proj dimMp.

Note. In the case when Mp = 0, one should interpret the ‘− − ∞’ in theabove statement as ‘∞’.

Proof. Set h := proj dimMp. We use induction on h. If h = −∞, thenMp = 0 and the result is clear from Lemma 9.4.1. When h = 0, the desiredresult follows from Lemma 9.4.3, since thenMp is a non-zero freeRp-module.

We therefore assume, inductively, that h > 0 and the result has been provedfor smaller values of h. There is a non-zero, finitely generated free R-moduleF and an exact sequence 0 −→ N −→ F −→ M −→ 0 of R-modules andhomomorphisms. Localization yields an exact sequence

0 −→ Np −→ Fp −→Mp −→ 0.

Therefore proj dimNp = h − 1 (since h > 0), and so, by the inductive hy-pothesis, there exists s ∈ R \ p such that, for every proper ideal b of R, wehave

sHib(N) = 0 for all i < grade b− h+ 1.

Thus sHi+1b (N) = 0 for all i < grade b− h. Let b be a proper ideal of R and

let i ∈ N0 with i < grade b− h.The exact sequence 0 −→ N −→ F −→ M −→ 0 induces a further

exact sequence Hib(F ) −→ Hi

b(M) −→ Hi+1b (N). Since i < grade b we

have Hib(F ) = 0, by 6.2.7 and the additivity of the functor Hi

b. By the imme-diately preceding paragraph, sHi+1

b (N) = 0. Therefore sHib(M) = 0. This

completes the inductive step.


be such that proj dimMp < ∞. Then there exists s ∈ R \ p such thatproj dimMs = proj dimMp.

Proof. Set h := proj dimMp. We use induction on h. If h = −∞, thenMp = 0 and there exists s ∈ R \ p such that sM = 0. Hence Ms = 0 andproj dimMs = −∞ = h.

When h = 0, then, by Lemma 9.4.2, there exist t ∈ N and an R-homomor-phism π : Rt −→M such that (Kerπ)p = (Cokerπ)p = 0. Then there existss ∈ R \ p such that sKerπ = sCokerπ = 0. It follows that (Kerπ)s =

(Cokerπ)s = 0 and πs : (Rt)s −→ Ms is an isomorphism. Since Ms = 0,we have proj dimMs = 0 = h.


We therefore assume, inductively, that h > 0 and the result has been provedfor smaller values of h. There is a non-zero, finitely generated free R-moduleF and an exact sequence 0 −→ N −→ F −→ M −→ 0 of R-modules andhomomorphisms. Localization yields an exact sequence

0 −→ Np −→ Fp −→Mp −→ 0.

Note that proj dimNp = h − 1, and so, by the inductive hypothesis, thereexists s ∈ R \ p such that proj dimNs = proj dimNp = h− 1. But then theexact sequence

0 −→ Ns −→ Fs −→Ms −→ 0

shows that proj dimMs ≤ h. Since p∩{sn : n ∈ N0} = ∅, we see that Mp∼=

(Ms)pRs (when the latter is regarded as an Rp-module by means of the naturalisomorphism Rp −→ (Rs)pRs). Hence h = proj dimMp ≤ proj dimMs ≤h, so that proj dimMs = proj dimMp. This completes the inductive step.

9.4.7 Corollary. Let M be a finitely generated R-module. Then for eacht ∈ N0 ∪ {−∞}, the set

Ut(M) := {p ∈ Spec(R) : proj dimMp ≤ t}

is an open subset of Spec(R) (in the Zariski topology).

Proof. Let p ∈ Ut(M), so that h := proj dimMp ≤ t. By 9.4.6, there existss ∈ R \ p such that proj dimMs = h. Therefore, for each q in the open neigh-bourhood Spec(R)\Var(sR) of p, we have proj dimMq ≤ proj dimMs ≤ t

(because Mq∼= (Ms)qRs when the latter is regarded as an Rq-module by

means of the natural isomorphism Rq −→ (Rs)qRs), so that q ∈ Ut(M).

9.4.8 Notation and Remarks. Let M be a finitely generated R-module. Foreach t ∈ N0 ∪ {−∞}, let

Ut(M) := {p ∈ Spec(R) : proj dimMp ≤ t} ,

let Ct(M) = Spec(R) \ Ut(M), and let ct(M) =⋂

p∈Ct(M) p. Then

(i) Ut(M) is an open, and so Ct(M) is a closed, subset of Spec(R), for allt ∈ N0 (by 9.4.7);

(ii) Spec(R) \ SuppM = U−∞(M) and

U−∞(M) ⊆ U0(M) ⊆ U1(M) ⊆ · · · ⊆ Ut(M) ⊆ · · · ;

(iii) SuppM = C−∞(M) ⊇ C0(M) ⊇ C1(M) ⊇ · · · ⊇ Ct(M) ⊇ · · · ;(iv)

√(0 :M) = c−∞(M) ⊆ c0(M) ⊆ c1(M) ⊆ · · · ⊆ ct(M) ⊆ · · · ;


(v) for each t ∈ N0 ∪ {−∞}, the ideal ct(M) is radical (that is, is equal toits own radical), and Var(ct(M)) = Ct(M); and

(vi) ifR is regular, so that, for all p ∈ Spec(R), the localizationRp has finiteglobal dimension and proj dimMp is finite, then⋃

t∈N0∪{−∞}Ut(M) = Spec(R) and

⋂t∈N0∪{−∞}

Ct(M) = ∅.

9.4.9 �Exercise. Let M be a finitely generated R-module and let S be amultiplicatively closed subset of R. Show that, with the notation of 9.4.8,

ct(S−1M) = S−1(ct(M)) for all t ∈ N0 ∪ {−∞} .

9.4.10 Proposition. Let M be a finitely generated R-module, and let t ∈N0 ∪ {−∞}. We use the notation of 9.4.8. There exists n ∈ N such that, forevery proper ideal d of R, we have

ct(M)nHid(M) = 0 for all i < grade d− t.

Note. In the case when t = −∞, one should interpret the ‘− − ∞’ in theabove statement as ‘∞’.

Proof. Let p ∈ Ut(M), so that proj dimMp ≤ t. By 9.4.5, there existssp ∈ R \ p such that, for every proper ideal d of R, we have spHi

d(M) = 0 forall i < grade d− proj dimMp, and therefore for all i < grade d− t.

Set g :=∑

p′∈Ut(M) sp′R, and observe that, for every proper ideal d of R,we have gHi

d(M) = 0 for all i < grade d − t. Note that no prime ideal inUt(M) contains g. The latter statement implies that Var(g) ⊆ Ct(M), so thatct(M) ⊆ √g. Hence there exists n ∈ N such that ct(M)n ⊆ g, and the resultfollows from this.

9.4.11 Exercise. Assume that R is a homomorphic image of a regular (com-mutative Noetherian) ring, and let M be a finitely generated R-module withthe property that SuppM has exactly one minimal member. Assume that cis an ideal of R such that Mp is a Cohen–Macaulay Rp-module for all p ∈Spec(R) \Var(c). Prove that there exists n ∈ N such that

cnHiq(M) = 0 for all q ∈ Var(c) and all i < dimRq

Mq.

9.4.12 Exercise. Let the situation be as in Proposition 9.4.10. Prove that thereexists n ∈ N such that, for every choice of flat ring homomorphism f : R →R′ of commutative Noetherian rings and every choice of proper ideal B′ ofR′, we have

ct(M)nHiB′(M ⊗R R

′) = 0 for all i < gradeB′ − t.


(Here is a hint: make appropriate modifications to the preparatory results, suchas 9.4.1, 9.4.3 and 9.4.5, that were used in our approach to the proof of Propo-sition 9.4.10.)

9.4.13 Exercise. (The result of this exercise is due to C. L. Huneke.) As-sume that R is a homomorphic image of a regular (commutative Noetherian)ring for which there exists a non-zerodivisor c ∈ R such that the ring Rc isCohen–Macaulay. Assume that either (a) R is an integral domain, or (b) R isan equidimensional (see [50, p. 250]) local ring.

Use Exercise 9.4.12 above to prove that there exists n ∈ N such that, forall choices of r ∈ N0 and all choices of an ideal B′ of the polynomial ringR[X1, . . . , Xr] (interpret this as R in the case when r = 0), we have

cnHiB′(R[X1, . . . , Xr]) = 0 for all i < htB′.

9.4.14 Lemma. Let (R,m) be a regular local ring of dimension d, let Mbe a finitely generated R-module, and let b be an ideal of R. Then, with thenotation of 9.4.8, we have b ⊆ cd−λb

m(M)(M).

Proof. Set t := d− λbm(M). It is enough to prove that, for each p ∈ Ct(M),we must have b ⊆ p. Suppose that, for one such p, we have b ⊆ p, and lookfor a contradiction. Then (p ∈ SuppM and)

∞ > adjm depthMp = depthMp + ht(m+ p)/p ≥ λbm(M).

Since R is a catenary domain, ht(m+ p)/p = htm/p = d− ht p. As Rp is aregular local ring, it follows from the Auslander–Buchsbaum–Serre Theoremthat depthMp + proj dimMp = dimRp = ht p. Therefore

ht p− proj dimMp + d− ht p ≥ λbm(M),

so that proj dimMp ≤ t. This contradiction completes the proof.

We remind the reader that our present major aim is a proof of the fact that,when R is a homomorphic image of a regular ring, then, with the notation of9.1.5 and 9.2.2, we have λba(M) ≤ fba (M). In view of 9.1.7 and 9.2.6, it willbe enough to do this in the case when R itself is regular. In the light of this, thenext result already proves our desired inequality in a special case.

9.4.15 Proposition. Assume that R is regular, and that dimR/a = 0. Let bbe a second ideal of R, and let M be a finitely generated R-module. Then

λba(M) ≤ fba (M).

Proof. Since dimR/a = 0, we have Var(a) = ass a is a finite set of maximalideals of R: let its members be m1, . . . ,mh. Consider an integer j with 1 ≤


j ≤ h. Set tj := htmj − λbRmj

mjRmj(Mmj ). Since htmj = grademjRmj , we

see that grademjRmj − tj = λbRmj

mjRmj(Mmj ). By 9.4.10, there exists nj ∈ N

such that

ctj (Mmj )njHi

mjRmj(Mmj ) = 0 for all i < λ

bRmj

mjRmj(Mmj ).

By 9.4.14, we have bRmj ⊆ ctj (Mmj ).Set n := max {n1, . . . , nh}. We can use 1.1.3 and 4.3.3 to see that

HimjRmj

(Mmj ) = HiaRmj

(Mmj )∼= (Hi

a(M))mj for all i ∈ N0.

Also

λbRmj

mjRmj(Mmj ) = λ

bRmj

aRmj(Mmj ) ≥ λba(M),

by 9.2.3(i) and 9.2.5. It therefore follows that

(bnHia(M))mj

∼= (bRmj )nHi

mjRmj(Mmj ) = 0 for 1 ≤ j ≤ h, i < λba(M).

Since a local cohomology module with respect to a is a-torsion, the sup-port of such a local cohomology module must be contained in Var(a) =

{m1, . . . ,mh}. Hence

Supp(bnHi

a(M))= ∅ for all i < λba(M).

In view of the definition of fba (M) (see 9.1.5), this completes the proof.

9.4.16 Theorem. Assume that R is a homomorphic image of a regular (com-mutative Noetherian) ring. Let b be a second ideal ofR, and letM be a finitelygenerated R-module. Then

λba(M) ≤ fba (M).

Proof. In view of 9.1.7 and 9.2.6, we can, and do, assume that R itself isregular.

We suppose that λba(M) > fba (M) and look for a contradiction. Let I(R)denote the set of all ideals ofR. SinceR is Noetherian, we can, and do, assumethat a is a maximal member of the set{

a′ ∈ I(R) : λba′(M) > fba′(M)}.

Note that dimR/a > 0, by 9.4.15. Set λ := λba(M), and let q be a minimalprime of a such that ht q = ht a. By 9.4.14, 9.4.9, 9.2.5, 9.2.3(i) and 9.4.8(iv),

bRq ⊆ cht a−λ

bRqqRq

(Mq)(Mq) =

(cht a−λ

bRqaRq

(Mq)(M)

)q

⊆ (cht a−λ(M))q.

Therefore, there exists u ∈ R \ q such that bRu ⊆ (cht a−λ(M))Ru. Since

9.5 The main theorems 183

dimR/a > 0, there exists s ∈ Var(a) which is not a minimal prime of a: letv ∈ s \ q and put s := uv. Note that (by 9.4.9 again)

a ⊂ a+Rs ⊂ R and bRs ⊆ (cht a−λ(M))s = cht(aRs)−λ(Ms).

Since ht aRs = grade aRs, it follows from 9.4.10 that there exists n ∈ Nsuch that (

cht(aRs)−λ(Ms))nHi

aRs(Ms) = 0 for all i < λ.

Hence bnHiaRs

(Ms) = 0 for all i < λ (when HiaRs

(Ms) is regarded as anR-module in the natural way). Therefore, by the Independence Theorem 4.2.1,we have bnHi

a(Ms) = 0 for all i < λ.Since a ⊂ a + Rs, it follows from the ‘maximality’ assumption about a

made in the second paragraph of this proof that λba+Rs(M) ≤ fba+Rs(M).Now λ = λba(M) ≤ λba+Rs(M), by 9.2.3(ii), and so there exists n′ ∈ N suchthat bn

′Hi

a+Rs(M) = 0 for all i < λ.By 8.1.2(i), there is, for each i < λ, an exact sequence

Hia+Rs(M) −→ Hi

a(M) −→ Hia(Ms).

It now follows from 9.1.1 that b ⊆ √(0 : Hia(M)) for all i < λ. This shows

that λ = λba(M) ≤ fba (M), and this contradiction completes the proof.

9.5 The main theorems

We can now put together Theorems 9.3.7 and 9.4.16 to prove Faltings’ Anni-hilator Theorem.

9.5.1 Faltings’ Annihilator Theorem. (See G. Faltings [14].) Assume thatR is a homomorphic image of a regular (commutative Noetherian) ring. Let bbe a second ideal of R, and let M be a finitely generated R-module. Then

λba(M) = fba (M).

Proof. This is now immediate from Theorems 9.3.7 and 9.4.16.

The special case of Faltings’ Annihilator Theorem in which a = b is Groth-endieck’s Finiteness Theorem.

9.5.2 Grothendieck’s Finiteness Theorem. (See A. Grothendieck [26, Ex-pose VIII, Corollaire 2.3].) Assume that R is a homomorphic image of a reg-ular (commutative Noetherian) ring, and let M be a finitely generated R-module. Then

λaa(M) = fa(M).


In other words, there exists i ∈ N such that Hia(M) is not finitely generated

if and only if SuppM ⊆ Var(a); moreover, when this is the case, the leasti ∈ N such that Hi

a(M) is not finitely generated is equal to

min {depthMp + ht(a+ p)/p : p ∈ SuppM \Var(a)} .

Proof. Put a = b in Faltings’ Annihilator Theorem 9.5.1, and use 9.1.2 forthe last part.

9.5.3 Example. Let K be a field, and let R = K[X,Y 2, XY, Y 3], a sub-ring of the ring of polynomials K[X,Y ]. Let m denote the maximal ideal(X,Y 2, XY, Y 3) of R.

In the ring of polynomials R[Z], let N = mR[Z] + ZR[Z]. We first cal-culate the invariant λNN(R[Z]). Let P ∈ Spec(R[Z]) \ Var(N): we calculateadjN depthR[Z]P. Two cases arise: if P = 0, then the fact that htN = 3

implies that adjN depthR[Z]0 = 3; if P = 0, then the fact that R[Z]P is alocal domain which is not a field ensures that

adjN depthR[Z]P = depthR[Z]P + ht(N+P)/P ≥ 1 + 1 = 2.

Thus λNN(R[Z]) ≥ 2. To see that λNN(R[Z]) is exactly 2, note that mR[Z] ∈Spec(R[Z]) \ Var(N). Now htN/mR[Z] = 1 since the ideal N/mR[Z] isprincipal. Moreover, it follows from 2.3.6(v) that H1

m(R) = 0; hence, sinceR is a domain, we can deduce from 6.2.7 that gradem = 1. Now X is anon-zerodivisor on R; the fact that m is a maximal ideal of R ensures thatm ∈ assRRX; and it is now easy to deduce that depthR[Z]mR[Z] = 1. Hence

adjN depthR[Z]mR[Z] = depthR[Z]mR[Z] + htN/mR[Z] = 1 + 1 = 2.

Therefore λNN(R[Z]) = 2.Grothendieck’s Finiteness Theorem 9.5.2 thus tells us that the finiteness di-

mension fN(R[Z]) is 2. Let us show this directly.Since Z,X is an R[Z]-sequence contained in N, it follows from 6.2.7 that

H0N(R[Z]) = H1

N(R[Z]) = 0. Further, for each t ∈ N, the exact sequence

0 −→ R[Z]Zt

−→ R[Z] −→ R[Z]/ZtR[Z] −→ 0


0 −→ H1N(R[Z]/ZtR[Z]) −→ H2

N(R[Z])Zt

−→ H2N(R[Z]);

hence (0 :H2N(R[Z]) Z

t) ∼= H1N(R[Z]/ZtR[Z]). But, by 1.1.3 and 4.2.2,

H1N(R[Z]/ZtR[Z]) = H1

mR[Z]+ZtR[Z](R[Z]/ZtR[Z])

∼= H1mR[Z](R[Z]/Z

tR[Z]).


The Independence Theorem 4.2.1 shows that, as R-modules,

H1mR[Z](R[Z]/Z

tR[Z]) ∼= H1m(R[Z]/Z

tR[Z]) ∼= H1m(R

t);

this is isomorphic to the R-module Kt, by 2.3.6(v). Therefore

(0 :H2N(R[Z]) Z

t) ⊂ (0 :H2N(R[Z]) Z

t+1) for all t ∈ N,

and so H2N(R[Z]) is not finitely generated. Thus we have shown directly that

the finiteness dimension fN(R[Z]) is 2.

9.5.4 Exercise. Assume that the local ring (R,m) is a homomorphic imageof a regular local ring, and let M be a finitely generated R-module.

(i) Show that

fm(M) = inf {depthMp + 1 : p ∈ Spec(R) and dimR/p = 1} .

(ii) Suppose that n := dimM > 0. Prove that fm(M) = n if and only ifMp is a Cohen–Macaulay Rp-module of dimension n− dimR/p for allp ∈ SuppM \ {m}.

9.5.5 Exercise. Assume that (R,m) is a local domain which is a homomor-phic image of a regular local ring; assume also that d := dimR > 0, and thatRp is a Cohen–Macaulay ring for all p ∈ Spec(R) \ {m}.

Show that, if Him(R) = 0 for all integers i such that 1 < i < d, then there

exists a Cohen–Macaulay subring of the quotient field of R which contains Rand is a finitely generated R-module. (Here is a hint: think of Dm(R).)

9.5.6 Exercise. Assume that (R,m) is a local domain which is a homomor-phic image of a regular local ring, and that the ideal a is proper. Show thatfa(R) > 1 if and only if ht a = 1.

9.5.7 Exercise. Assume that (R,m) is local, and let M be a non-zero finitelygenerated R-module of dimension n > 0. We say that M is a generalizedCohen–Macaulay R-module precisely when Hi

m(M) is finitely generated forall i = n. (Such modules were called ‘quasi-Cohen–Macaulay modules’ by P.Schenzel in [72, p. 238]; these modules were investigated by Schenzel, N. V.Trung and N. T. Cuong in [76], and, since the publication of that paper, theterminology ‘generalized Cohen–Macaulay module’ seems to have becomemore commonplace than ‘quasi-Cohen–Macaulay module’.)

(i) Show that, if M is a generalized Cohen–Macaulay R-module, then wehave dimR/p = n for all p ∈ AssM \ {m} and Mq is a Cohen–Macaulay Rq-module for all q ∈ SuppM \ {m}.

(ii) Show conversely, that, if


(a) R is a homomorphic image of a regular ring,(b) dimR/p = n whenever p is a minimal member of SuppM , and(c) Mq is a Cohen–Macaulay Rq-module for all q ∈ SuppM \{m},

then M is a generalized Cohen–Macaulay R-module.

9.5.8 Exercise. Assume that (R,m) is local, and that the non-zero finitelygeneratedR-moduleM of dimension n > 0 is a generalized Cohen–MacaulayR-module (see 9.5.7 above). Note that m ∈ Ass(M/Γm(M)) and Mp

∼=(M/Γm(M))p for all p ∈ Spec(R) \ {m}.

(i) Let r ∈ m be such that dimM/rM = n − 1. Show that r is a non-zerodivisor on M/Γm(M) and that, if n > 1, then M/rM is a general-ized Cohen–Macaulay R-module.

(ii) Use part (i) and induction on n to show that every saturated chain ofprime ideals from a minimal member of SuppM (as smallest term) tom (as largest term) has length n.

(iii) Deduce from part (ii) that the ring R/(0 :M) is catenary.

9.5.9 Definitions. Assume that (R,m) is local, and let M be a non-zerofinitely generated R-module of dimension n > 0.

(i) By a system of parameters for M we mean a sequence (ri)ni=1 of n

elements of m such that M/∑n

i=1 riM has finite length. We say thatr1, . . . , rn form a system of parameters for M precisely when (ri)

ni=1

is a system of parameters for M . By a parameter for M , we mean amember of a system of parameters for M .

(ii) Let (ri)ni=1 be a system of parameters forM ; let q :=∑n

i=1 riR. We saythat (ri)ni=1 is a standard system of parameters for M precisely when

qHjm

(M/∑k

i=1riM)= 0 for all j, k ∈ N0 with j + k < n.

(See Stuckrad and Vogel [84, p. 261].)

9.5.10 Exercise. Assume (R,m) is local, and let M be a non-zero finitelygenerated R-module of dimension n > 1. Let (vi)ni=1 be a system of pa-rameters for M , let t ∈ N with t < n, and let M ′ := M/

∑ti=1 viM . Let

r1, . . . , rn−t ∈ m. Show that

(i) dimM ′ = n− t;(ii) v1, . . . , vt, r1, . . . , rn−t form a system of parameters for M if and only

if (ri)n−ti=1 is a system of parameters for M ′;

(iii) if (ri)n−ti=1 is a system of parameters for M ′ and h ∈ N0 is such that

h < n− t, then M ′/∑h

i=1 riM′ ∼=M/(

∑ti=1 viM +

∑hi=1 riM);


(iv) if every system of parameters for M is standard, then every system ofparameters for M ′ is standard.

9.5.11 Exercise. Assume that (R,m) is local, and that M is a generalizedCohen–Macaulay R-module of dimension n > 1. Let r be a parameter for M ,and write M =M/Γm(M). Show that

(i) (0 : Him(M))(0 : Hi+1

m (M)) ⊆ (0 : Him(M/rM)) for all i = 1, . . . ,

n− 2;(ii) (0 : H0

m(M))(0 : H1m(M/(0 :M r))) ⊆ (0 : H0

m(M/rM));(iii) (0 :M r) is m-torsion; and(iv) (0 : Hi

m(M))(0 : Hi+1m (M)) ⊆ (0 : Hi

m(M/rM)) for all i = 0, 1, . . . ,

n− 2.

9.5.12 Exercise. Assume that (R,m) is local, and that M is a generalizedCohen–Macaulay R-module of dimension n > 0. Let

qM :=(0 : H0

m(M))(n−1

0 ) (0 : H1

m(M))(n−1

1 ). . .(0 : Hn−1

m (M))(n−1

n−1) .

Prove that every system of parameters for M composed of elements in qMis standard (see 9.5.9). (Here is a hint: use Exercises 9.5.11 and 9.5.8, togetherwith induction.)

9.5.13 Exercise. Assume (R,m) is local, and let M be a non-zero finitelygenerated R-module of dimension n > 0. Assume that, for every system ofparameters (ri)ni=1 for M ,

mΓrk+1R

(M/∑k

i=1riM)= 0 for all k ∈ N0 with k < n.

(i) Show that (0 :M r) = Γm(M) for each parameter r for M .(ii) Use part (i) and induction to prove that mHj

m(M) = 0 for all j =

0, . . . , n− 1.

9.5.14 Definitions and Exercise. Assume that (R,m) is local, and let M bea non-zero finitely generated R-module of dimension n > 0.

Let a1, . . . , ah ∈ m. We say that a1, . . . , ah is a weakM -sequence preciselywhen(∑i−1

j=1ajM :M ai

)=(∑i−1

j=1ajM :M m)

for all i = 1, . . . , h.

We say that M is a Buchsbaum R-module precisely when every sequencer1, . . . , rn forming a system of parameters for M is a weak M -sequence; wesay that R is a Buchsbaum ring if and only if it is a Buchsbaum module overitself.


Show that M is a Buchsbaum R-module if and only if, for every system ofparameters (ri)ni=1 for M ,

mΓrk+1R

(M/∑k

i=1riM)= 0 for all k ∈ N0 with k < n.

There is an extensive theory of Buchsbaum rings and modules: see Stuckradand Vogel [84]. The following final exercise in this section provides some al-ternative characterizations of Buchsbaum modules.

9.5.15 Exercise. Assume (R,m) is local, and let M be a non-zero finitelygenerated R-module of dimension n > 0.

Prove that the following conditions on M are equivalent:

(i) for every system of parameters (ri)ni=1 for M ,

mHjm

(M/∑k

i=1riM)= 0 for all j, k ∈ N0 with j + k < n;

(ii) every system of parameters for M is standard (see 9.5.9);(iii) M is a Buchsbaum R-module (see 9.5.14).

(It is perhaps appropriate to give some hints. For the implication ‘(ii) ⇒(iii)’, note that m can be generated by parameters for M , and show that, ifr1, . . . , rn form a system of parameters for M , then condition (ii) impliesthat M/

∑ki=1 riM is a generalized Cohen–Macaulay R-module (see Exer-

cise 9.5.7) for all k = 0, . . . , n− 1. For the implication ‘(iii)⇒ (i)’, use 9.5.14and 9.5.13.)

9.6 Extensions

The remaining exercises in this chapter are directed at those readers who arefamiliar with some fairly advanced concepts in commutative algebra, includ-ing formal fibres of local rings, universally catenary rings, and L. J. Ratliff’sTheorem [50, Theorem 31.7] that a local ring (R,m) is universally catenary ifand only if, for every p ∈ Spec(R) and for every minimal prime P of the idealpR of R, we have dim R/P = dimR/p.

So far in this chapter, we have established the main results, that is, the An-nihilator and Finiteness Theorems, under the hypothesis that the underlyingring is a homomorphic image of a regular ring. The remaining exercises inthis chapter, in conjunction with the Local-global Principle 9.6.2 for Finite-ness Dimensions, show that the result of the Finiteness Theorem holds underslightly weaker hypotheses, namely under the assumption that the underlyingring is universally catenary and has the property that all the formal fibres of

9.6 Extensions 189

all its localizations are Cohen–Macaulay rings. We consider the Local-globalPrinciple, which is a consequence of the theorem of Faltings presented in 9.6.1below, to be of considerable independent interest.

9.6.1 Theorem. (See G. Faltings [16, Satz 1].) Let M be a finitely generatedR-module, and let t ∈ N. Then the following statements are equivalent:

(i) Hia(M) is finitely generated for all i < t;

(ii) HiaRp

(Mp) is a finitely generated Rp-module for all i < t and all p ∈Spec(R).

Proof. (i) ⇒ (ii) Since 4.3.3 shows that HiaRp

(Mp) ∼= (Hia(M))p for all

i ∈ N0 and all p ∈ Spec(R), this implication is clear.(ii) ⇒ (i) We use induction on t. When t = 1, there is nothing to prove,

since H0a(M) = Γa(M) is a submodule of M , and so is finitely generated. So

suppose that t > 1 and that the result has been proved for smaller values of t.By this assumption, Hi

a(M) is finitely generated for i = 0, 1, . . . , t− 2, and itonly remains for us to prove that Ht−1

a (M) is finitely generated.Set M :=M/Γa(M). Then, for each p ∈ Spec(R),

Mp∼=Mp/(Γa(M))p =Mp/ΓaRp

(Mp),

and so it follows from 2.1.7(iii) that HiaRp

(Mp) ∼= HiaRp

(Mp) for all i ∈ N

and all p ∈ Spec(R). It follows that M also satisfies condition (ii) in thestatement of the theorem. Moreover, it would be sufficient for us to prove thatHi

a(M) is finitely generated for all i < t, since Hia(M) ∼= Hi

a(M) for alli ∈ N (by 2.1.7(iii)) and it is automatic that H0

a(M) is finitely generated. NowM is an a-torsion-freeR-module, by 2.1.2. Hence we can, and do, assume thatM is an a-torsion-free R-module.

We now use 2.1.1(ii) to deduce that a contains an element r which is a non-zerodivisor on M . Let n ∈ N and let p ∈ Spec(R). The localization of the

exact sequence 0 −→ Mrn−→ M −→ M/rnM −→ 0 at p induces, for each

i ∈ N0, an exact sequence

Hi−1aRp

(Mp) −→ Hi−1aRp

((M/rnM)p) −→ HiaRp

(Mp).

When i < t, the two outer modules in this last exact sequence are finitelygenerated, and therefore so also is the middle one. Thus Hi−1

aRp((M/rnM)p) is

a finitely generated Rp-module for all i < t and all p ∈ Spec(R). Therefore,by the inductive hypothesis, Hi−1

a (M/rnM) is finitely generated for all i < t.

Since the exact sequence 0 −→ Mrn−→ M −→ M/rnM −→ 0 induces an


exact sequence

Ht−2a (M/rnM) −→ Ht−1

a (M)rn−→ Ht−1

a (M),

it therefore follows that (0 :Ht−1a (M) r

n) is finitely generated.Set H := Ht−1

a (M), and Hn := (0 :H rn) for all n ∈ N. Recall that ouraim is to prove that H is finitely generated; we have just proved that Hn isfinitely generated for all n ∈ N. Note that

H1 ⊆ H2 ⊆ · · · ⊆ Hn ⊆ Hn+1 ⊆ · · · ,

and that, since r ∈ a and H is an a-torsion R-module, H =⋃

n∈NHn. Weshall achieve our aim by showing that there exists k ∈ N such thatHk = Hk+i

for all i ∈ N: it will then follow that H = Hk, and the latter module is finitelygenerated.

For each n ∈ N, let Un denote the open subset Spec(R)\Supp(Hn+1/Hn)

of Spec(R). The next two stages in our proof aim to show that Spec(R) =⋃n∈N Un and

U1 ⊆ U2 ⊆ · · · ⊆ Ui ⊆ Ui+1 ⊆ · · · .

Let p ∈ Spec(R). Then Hp = (Ht−1a (M))p ∼= Ht−1

aRp(Mp), and this is a

finitely generated, aRp-torsion, Rp-module. Therefore, there exists n ∈ Nsuch that Hp = (0 :Hp

rn/1). Since

(0 :Hprn/1) ⊆ (0 :Hp

rn+1/1) ⊆ Hp,

it follows that (0 :Hprn/1) = (0 :Hp

rn+1/1). But (0 :Hpri/1) = (Hi)p

for all i ∈ N. Therefore (Hn+1/Hn)p = 0, and so p ∈ Un. It follows thatSpec(R) =

⋃n∈N Un.

Next, let i ∈ N and p ∈ Ui. Then (0 :Hpri/1) = (0 :Hp

ri+1/1), and so

(0 :Hpri+2/1) =

((0 :Hp

ri+1/1) :Hpr/1)

=((0 :Hp

ri/1) :Hpr/1)= (0 :Hp

ri+1/1).

Thus (Hi+2/Hi+1)p = 0, and so p ∈ Ui+1. Hence Ui ⊆ Ui+1 for all i ∈ N.Thus the sets Ui (i ∈ N) form an ascending open cover of the quasi-compact

topological space Spec(R). Therefore there exists k ∈ N such that Uk =

Uk+i = Spec(R) for all i ∈ N. Thus

Supp(Hk+1/Hk) = Supp(Hk+i+1/Hk+i) = ∅ for all i ∈ N,

so that Hk = Hk+i for all i ∈ N. Thus H = Hk, and so is finitely generated.This completes the inductive step.

9.6 Extensions 191

9.6.2 Local-global Principle for Finiteness Dimensions. LetM be a finitelygenerated R-module. Then

fa(M) = inf{faRp

(Mp) : p ∈ Spec(R)}.

Proof. This is now immediate from Theorem 9.6.1 above and the definitionof finiteness dimension in 9.1.3.

9.6.3 Exercise. Assume that (R,m) is local. Let b be a second ideal of R,and let M be a finitely generated R-module. Prove that

fbRaR

(M ⊗R R) = fba (M).

9.6.4 Exercise. Assume that (R,m) is local, that M is a finitely generatedR-module, and that p ∈ Spec(R)\Var(b), where b is a second ideal of R. Leth := ht(a+ p)/p.

(i) Prove that ht(aR+ pR)/pR = h.(ii) Let Q ∈ Spec(R) be a minimal prime of aR + pR with htQ/pR = h,

and let P be a minimal prime of pR such that P ⊆ Q and htQ/P = h.Show that

(a) ht(aR+P)/P ≤ h,(b) P ∩R = p and P ∈ Var(bR),(c) depth(M ⊗R R)P = depthMp, and(d) adjaR depth(M ⊗R R)P ≤ adja depthMp.

(iii) Deduce that λbRaR

(M ⊗R R) ≤ λba(M).

9.6.5 Exercise. Let the situation be as in Exercise 9.6.4 above, and assume inaddition that R is universally catenary and that all its formal fibres are Cohen–Macaulay rings.

(i) Let Q′ ∈ Spec(R) \ Var(bR) and set q′ := Q′ ∩ R. Show that q′ ∈Spec(R) \Var(b) and that

adja depthMq′ ≤ adjaR depth(M ⊗R R)Q′ .

(You might find Ratliff’s Theorem [50, Theorem 31.7] helpful.)(ii) Deduce from part (i) and Exercise 9.6.4(iii) above that

λbRaR

(M ⊗R R) = λba(M).

9.6.6 Exercise. Assume that (R,m) is a universally catenary local ring allof whose formal fibres are Cohen–Macaulay rings. Prove that the conclusionof the Annihilator Theorem holds over R. In other words, prove that, if M isa finitely generated R-module and b is a second ideal of R, then λba(M) =

fba (M).


9.6.7 Exercise. Assume that R is universally catenary and all the formalfibres of all its localizations are Cohen–Macaulay rings. Deduce from Exercise9.6.6 above and the Local-global Principle 9.6.2 that the conclusion of theFiniteness Theorem holds over R. In other words, prove that, if M is a finitelygenerated R-module, then λaa(M) = fa(M).

9.6.8 Exercise. Assume that (R,m) is a universally catenary local ring all ofwhose formal fibres are Cohen–Macaulay rings. Let M be a non-zero finitelygenerated R-module of dimension n > 0 such that

(a) dimR/p = n whenever p is a minimal member of SuppM , and(b) Mq is a Cohen–Macaulay Rq-module for all q ∈ SuppM \ {m}.

Prove that M is a generalized Cohen–Macaulay R-module (see 9.5.7).

9.6.9 Exercise. (This exercise is related to Exercise 9.4.13.) Assume that Ris a universally catenary semi-local integral domain with the property that allthe formal fibres of all its localizations are Cohen–Macaulay rings. Assumethat there exists 0 = c ∈ R such that the ring Rc is Cohen–Macaulay. Provethat there exists n ∈ N such that, for all choices of r ∈ N0 and all choices ofan ideal B′ of the polynomial ring R[X1, . . . , Xr], we have

cnHiB′(R[X1, . . . , Xr]) = 0 for all i < htB′.

10

Matlis duality

Prior to this point in the book, we have not made use of the decompositiontheory (due to E. Matlis [49]) for injective modules over our (Noetherian) ringR. However, our work in the next Chapter 11 on local duality will involve useof the structure of the terms in the minimal injective resolution of a Gorensteinlocal ring, and so we can postpone no longer use of the decomposition theoryfor injective modules. Our discussion of local duality in Chapter 11 will alsoinvolve Matlis duality.

Our purpose in this chapter is to prepare the ground for Chapter 11 by re-viewing, sometimes in detail, those parts of Matlis’s theories that we shall needlater in the book. Sometimes we simply refer to [50, Section 18] for proofs; inother cases, we provide alternative proofs for the reader’s convenience. Anexperienced reader who is familiar with this work of Matlis should omit thischapter and progress straight to the discussion of local duality in Chapter 11:for one thing, there is no local cohomology theory in this chapter! However,graduate students might find this chapter helpful.

10.1 Indecomposable injective modules

10.1.1 Reminders. Let M be a submodule of the R-module L.

(i) We say that L is an essential extension of M precisely when B∩M = 0

for every non-zero submodule B of L.

(ii) We say that L is an injective envelope (or injective hull) of M preciselywhen L is an injective R-module that is an essential extension of M .

(iii) (See [50, Theorem B4, p. 281].) The R-module M is injective if andonly if the only essential extension of M is M itself.

194 Matlis duality

(iv) If L is an injective envelope of M , and g :M −→ K is an R-monomor-phism from M into an injective R-module K, then there is a homomor-phism g′ : L→ K such that the diagram

0 M L�⊆�

gg′

K

��

��

commutes. Since Ker g′ ∩M = Ker g = 0, it follows from the fact thatL is an essential extension of M that Ker g′ = 0 and g′ is a monomor-phism. Hence L is isomorphic to a direct summand of K.

(v) (See [50, p. 281].) Each R-module has an injective envelope which isuniquely determined up to isomorphism. In fact, if E and E′ are bothinjective envelopes of M (so that each of E and E′ is an injective R-module which contains M as a submodule, and each of E and E′ is anessential extension of its submodule M ), then there is an isomorphismf : E → E′ such that the diagram

M E�

f∼=

E′

��

(in which the unnamed homomorphisms are the inclusion maps) com-mutes.

(vi) We denote by E(M) (or ER(M) if it is necessary to specify the under-lying ring) one choice of injective envelope of M .

10.1.2 Example and Warning. The reader should be warned that the iso-morphism f : E → E′ of 10.1.1(v) need not be uniquely determined; for thisreason, we cannot regard E( • ) as a functor from C(R) to itself.

To illustrate this point, consider the submodule

C2(∞) :={α ∈ Q/Z : α =

r

2n+ Z for some r ∈ Z and n ∈ N0

}of the Z-module Q/Z. Since C2(∞) is a divisible Abelian group, it is an injec-tive Z-module, by [71, Theorem 3.24] for example. AlsoC2(∞) is an essentialextension of its Z-submodule C2 := Z( 12 + Z). Now ‘f = IdC2(∞)’ and ‘f =

10.1 Indecomposable injective modules 195

− IdC2(∞)’ are two different choices of automorphism f : C2(∞) → C2(∞)

for which the diagram

C2 C2(∞)�

f∼=

C2(∞)

��

��

(in which the unnamed homomorphisms are the inclusion maps) commutes.

Injective envelopes play an essential role in the very satisfactory decompo-sition theory for injective R-modules: for one thing, the fundamental ‘buildingblocks’ on which the whole theory is based are described using the concept ofinjective envelope.

10.1.3 Reminders. We shall assume that the reader is familiar with the fol-lowing facts about injective R-modules.

(i) Let (Mι)ι∈Λ be a non-empty family of R-modules. It is immediate fromthe definition of injective module that the direct product

∏ι∈ΛMι is an

injective R-module if and only if Mι is injective for all ι ∈ Λ. However,since our ring R is Noetherian, it is also the case that the direct sum⊕

ι∈ΛMι is injective if and only if Mι is injective for all ι ∈ Λ: see [50,Theorem 18.5(i)].

(ii) AnR-module is said to be indecomposable precisely when it is non-zeroand cannot be written as the direct sum of two proper submodules. Foreach p ∈ Spec(R), the injective R-module E(R/p) is indecomposable(see [50, Theorem 18.4(i)]); moreover, each indecomposable injectiveR-module is isomorphic to E(R/q) for some q ∈ Spec(R) (see [50,Theorem 18.4(ii)]).

(iii) Let p ∈ Spec(R) and let r ∈ R\p. Then multiplication by r provides anautomorphism of E(R/p) (see [50, Theorem 18.4(iii)]); moreover, eachelement of E(R/p) is annihilated by some power of p, that is, E(R/p)

is p-torsion (see [50, Theorem 18.4(v)]).(iv) Let p, q ∈ Spec(R). Then E(R/p) ∼= E(R/q) if and only if p = q (see

[50, Theorem 18.4(iv)]).(v) The results recounted in parts (ii) and (iv) above can be reformulated as

follows: there is a set of isomorphism classes of indecomposable injec-tive R-modules, and there is a bijective correspondence between this set

196 Matlis duality

and Spec(R), under which a p ∈ Spec(R) corresponds to the isomor-phism class of E(R/p).

10.1.4 Exercise. Let M1, . . . ,Mn be R-modules. Show that the map

M1 ⊕ · · · ⊕Mn −→ E(M1)⊕ · · · ⊕ E(Mn)

(obtained by taking the direct sum of the inclusion maps) provides an injectiveenvelope of

⊕ni=1Mi.

10.1.5 Exercise. Let M be a non-zero R-module. Show that the followingstatements are equivalent:

(i) E(M) is indecomposable;(ii) E(M) is an injective envelope of every non-zero submodule of itself;

(iii) the zero submodule of M cannot be expressed as the intersection of twonon-zero submodules of M .

10.1.6 Exercise. Let a be a proper ideal of R. Show that E(R/a) is inde-composable if and only if a is irreducible.

Let q be a p-primary ideal of R. Prove that E(R/q) is isomorphic to a directsum of finitely many copies of E(R/p). What can you say about the numberof copies? (Here is a hint: recall that q can be expressed as an intersectionq =⋂n

i=1 ji, where each ji (for 1 ≤ i ≤ n) is irreducible and irredundant inthe intersection. If you still find this exercise difficult, you might like to consult[81, Exercise 8.30].)

10.1.7 Exercise. Let p ∈ Spec(R).

(i) Let 0 = x ∈ E(R/p). Show that (0 : x) is an irreducible ideal of R.(ii) In 10.1.3(iii), we saw that E(R/p) =

⋃n∈N(0 :E(R/p) pn). Show that

E(R/p) =⋃

n∈N(0 :E(R/p) p(n)).

10.1.8 �Exercise. Let I be a non-zero injective R-module. Show that I is adirect sum of indecomposable injective submodules, perhaps by means of thefollowing intermediate steps.

(i) Apply Zorn’s Lemma to the set of all sets of indecomposable injectivesubmodules of I whose sum is direct, in order to find a maximal memberM of this set.

(ii) Let J :=⊕

D∈MD. Suppose that J ⊂ I and seek a contradiction. Usethe injectivity of J to find a submodule K of I such that I = J ⊕K.

(iii) Let q ∈ AssK. Show that K has a submodule isomorphic to ER(R/q).(iv) Use 10.1.3(ii) to find a contradiction to the maximality ofM, and deduce

that J = I .

10.1 Indecomposable injective modules 197

In 10.1.3, we reviewed fundamental facts about indecomposable injectiveR-modules. One of the reasons for the importance of these modules is providedby the following.

10.1.9 Reminder. (See [50, Theorem 18.5(ii)].) By 10.1.8, each injective R-module I is a direct sum of indecomposable injective submodules. Therefore,by 10.1.3(ii), there is a family (pα)α∈Λ of prime ideals of R for which I ∼=⊕

α∈ΛE(R/pα).

10.1.10 Remark. Let the situation be as in 10.1.9. Then p is equal to pα forsome α ∈ Λ if and only if p ∈ Ass I .

10.1.11 Exercise. Let I be an injective R-module, so that, by 10.1.9, there isa family (pα)α∈Λ of prime ideals of R for which I ∼=

⊕α∈ΛE(R/pα). Prove

that

Γa(I) ∼=⊕α∈Λa⊆pα

E(R/pα).

(This exercise provides another proof of the result of Proposition 2.1.4 thatΓa(I) is an injective R-module.)

In fact, the direct decompositions described in 10.1.9 have uniqueness prop-erties. The route taken by Matsumura in [50, §18] to these uniqueness proper-ties involves facts about the behaviour of indecomposable injectiveR-modulesunder localization. We review this behaviour next.

10.1.12 Lemma. Let S be a multiplicatively closed subset of R, and let G bean S−1R-module. Then G is R-injective (that is, injective when viewed as anR-module by means of the natural homomorphism R → S−1R) if and only ifG is S−1R-injective.

Proof. (⇒) Let H be an S−1R-submodule of the S−1R-module J , and leth : H → G be an S−1R-homomorphism. Then H is an R-submodule ofthe R-module J , and h is an R-homomorphism. Since G is R-injective, thereexists an R-homomorphism j : J → G which extends h. It is easy to checkthat j must be an S−1R-homomorphism.

(⇐) Let M be an R-submodule of the R-module N , and let λ : M → G

be an R-homomorphism. As G is an S−1R-module, the natural map ψ : G→S−1G is not only an R-isomorphism but also an S−1R-isomorphism. ThusS−1M is an S−1R-submodule of the S−1R-module S−1N , and the composi-tion ψ−1 ◦ S−1λ : S−1M → G is an S−1R-homomorphism.

SinceG is S−1R-injective, there is an S−1R-homomorphism μ′ : S−1N →

198 Matlis duality

G which extends ψ−1 ◦ S−1λ. Then μ′ ◦ θ : N → G, where θ : N → S−1N

is the natural map, is an R-homomorphism which extends λ.

10.1.13 Lemma. Let S be a multiplicatively closed subset of R, and let p ∈Spec(R) be such that p ∩ S = ∅. By 10.1.3(iii), the indecomposable injectiveR-module ER(R/p) has a natural structure as an S−1R-module.

As S−1R-module, ER(R/p) is isomorphic to ES−1R(S−1R/S−1p).

Furthermore, ES−1R(S−1R/S−1p), when considered as an R-module by

means of the natural homomorphism R→ S−1R, is isomorphic to ER(R/p).

Proof. It follows from Lemma 10.1.12 that the S−1R-module ER(R/p) isS−1R-injective. Since an S−1R-submodule of ER(R/p) is automatically anR-submodule, it is immediate from 10.1.3(ii) thatER(R/p) is indecomposableas S−1R-module; we can therefore deduce from the same result thatER(R/p)

is S−1R-isomorphic to ES−1R(S−1R/P) for some P ∈ Spec(S−1R). By

10.1.3(iii),

P ={α ∈ S−1R : α IdER(R/p) is not an isomorphism

}.

Hence the contraction of P to R is just{r ∈ R : r IdER(R/p) is not an isomorphism

}= p.

Therefore P = S−1p.The final claim is now immediate.

10.1.14 Proposition. Let S be a multiplicatively closed subset of R.

(i) Let p ∈ Spec(R). Then

S−1(ER(R/p))

{= 0 if p ∩ S = ∅,∼= ES−1R(S

−1R/S−1p) if p ∩ S = ∅.

(ii) Let I be an injective R-module. Then S−1I is both S−1R-injective andR-injective.

Proof. (i) Suppose that s ∈ p ∩ S. By 10.1.3(iii), each x ∈ E(R/p) is anni-hilated by some power of s, and so S−1(ER(R/p)) = 0.

If, on the other hand, p ∩ S = ∅, then, by 10.1.3(iii) again, E(R/p) has anatural structure as an S−1R-module, so that S−1(E(R/p)) ∼= E(R/p) bothas R-modules and S−1R-modules. We can now use 10.1.13 to complete theproof of part (i).

(ii) By 10.1.9, there is a family (pα)α∈Λ of prime ideals of R for which I ∼=⊕α∈ΛE(R/pα). Then S−1I ∼=

⊕α∈Λ S

−1(E(R/pα)) (as S−1R-modules);since, by (i), each S−1(E(R/pα)) in the above display is S−1R-injective, it

10.2 Matlis duality 199

follows from 10.1.3(i) that S−1I is S−1R-injective. An appeal to 10.1.12 nowcompletes the proof.

The behaviour of injective R-modules under fraction formation is used inMatsumura’s proof of the following fundamental uniqueness property of thedirect sum decompositions described in 10.1.9.

10.1.15 Reminder. (See [50, Theorem 18.5(iii)].) Let I be an injective R-module. By 10.1.9, there is a family (pα)α∈Λ of prime ideals of R for whichI ∼=⊕

α∈ΛE(R/pα).Let p ∈ Spec(R), and let k(p) = Rp/pRp, the residue field of the local ring

Rp. Then the cardinality of the set {α ∈ Λ : pα = p} depends only on I andp and not on the particular decomposition of I (as a direct sum of indecom-posable injective submodules) chosen; in fact, this cardinality is equal to thevector space dimension dimk(p) HomRp

(k(p), Ip).

10.1.16 Lemma. Suppose that the R-module M is annihilated by the idealb of R. We can regard M and (0 :ER(M) b) as modules over R/b in naturalways: when this is done, (0 :ER(M) b) ∼= ER/b(M).

Proof. Let ι : (0 :ER(M) b)→ ER(M) denote the inclusion map. Let

0 G H� �α

γ

(0 :ER(M) b)�

be a diagram of R/b-modules and R/b-homomorphisms in which the rowis exact. We can regard these modules as R-modules by means of the naturalhomomorphismR→ R/b, and then α, γ and ι◦γ becomeR-homomorphisms.Since ER(M) is an injective R-module, there exists an R-homomorphism β :

H −→ ER(M) such that β ◦ α = ι ◦ γ. However, since bH = 0, it followsthat Imβ ⊆ (0 :ER(M) b). We can regard β as a map from H to (0 :ER(M) b),and, when we do that, β is an R/b-homomorphism such that β ◦ α = γ.

Thus (0 :ER(M) b) is injective as an R/b-module. Obviously (0 :ER(M) b)

is an essential extension of its R/b-submodule M .

10.2 Matlis duality

Although there is a treatment of Matlis duality in [50, Theorem 18.6], weare going to present, for the reader’s convenience, a different approach to this

200 Matlis duality

theory. Our approach makes use of Melkersson’s Theorem [51, Theorem 1.3]:we presented this theorem in 7.1.2, and applied it to local cohomology modulesin Chapter 7.

We begin by specifying some notation which we shall frequently use duringour discussion of Matlis duality.

10.2.1 Notation and Remarks. Suppose (R,m) is local. Set E = E(R/m),the injective envelope of the simple R-module R/m. As usual, we shall use Rto denote the m-adic completion lim

←−n∈N

R/mn ofR. We shall useD to denote the

exact, contravariant, R-linear functor HomR( • , E) from C(R) to itself. Foreach R-module G, we shall refer to D(G) as the Matlis dual of G. Note thatD(R) is naturally isomorphic to E, and that D(E) = HomR(E,E) is just theR-endomorphism ring of E considered as an R-module in the natural way.

For each R-module G, let

μG : G −→ DD(G) = HomR(HomR(G,E), E)

be the natural R-homomorphism for which (μG(x))(f) = f(x) for all x ∈ Gand f ∈ HomR(G,E).

Note that, as G varies through C(R), the μG constitute a natural transforma-tion μ from the identity functor to the functor DD.

If an R-module M has finite length, then we shall denote that length by�(M).

10.2.2 Remarks. Suppose that (R,m) is local, and use the notation of 10.2.1.Let G be an R-module.

(i) The R-homomorphism μG is injective, since if 0 = x ∈ G, there is anR-homomorphism f ′ : Rx → R/m for which f ′(rx) = r + m for allr ∈ R, and the composition of f ′ and the inclusion map R/m→ E canbe extended to an f ∈ HomR(G,E) for which f(x) = 0.

(ii) The annihilators of G and its Matlis dual D(G) are equal, because thefact that D is an R-linear functor ensures that

(0 : G) ⊆ (0 : D(G)) ⊆ (0 : DD(G)),

while the injectivity of μG : G −→ DD(G) (proved in (i) above) en-sures that (0 : DD(G)) ⊆ (0 : G).

First we analyse the case when (R,m) is an Artinian local ring. Recall that,then, an R-module G is Artinian if and only if it is Noetherian, and this is thecase if and only if G is finitely generated.


10.2.3 Proposition. Suppose that (R,m) is local and Artinian, and use thenotation of 10.2.1. Then

(i) D(R/m) ∼= R/m (as R-modules);(ii) for each finitely generated R-module G (that is, for each R-module G

of finite length), the Matlis dual D(G) is also finitely generated and�(D(G)) = �(G);

(iii) E is finitely generated (and so Artinian), and �(E) = �(R);(iv) for each finitely generated R-module G (that is, for each Artinian R-

module G), the map μG : G −→ DD(G) is an isomorphism; and(v) for each f ∈ HomR(E,E), there is a unique rf ∈ R such that f(x) =

rfx for all x ∈ E.

Note. Condition (v) is equivalent to the statement that the homomorphismθ : R −→ HomR(E,E) for which θ(r) = r IdE , for all r ∈ R, is an isomor-phism.

Proof. (i) Let k denote the residue field of R. There is an isomorphism ofk-modules

D(R/m) = HomR(R/m, E) ∼= (0 :E m).

But (0 :E m) ∼= Ek(k) by 10.1.16; since every k-module, that is, every vectorspace over k, is injective, Ek(k) = k. The claim follows from this.

(ii) Use induction on length: remember that D is an exact functor.(iii) This is now immediate from (ii), since E ∼= D(R).(iv) Let G be a finitely generated R-module. Two uses of part (ii) show that

DD(G) is also finitely generated and has �(DD(G)) = �(G); since μG isinjective by 10.2.2(i), it follows that μG must be an isomorphism.

(v) By part (iv), the map μR : R→ HomR(HomR(R,E), E) is an isomor-phism. The claim therefore follows from the fact that the composition

θ : RμR−→ HomR(HomR(R,E), E)

∼=−→ HomR(E,E),

in which the second isomorphism is the obvious natural one, is such that θ(r),for r ∈ R, is the endomorphism of E given by multiplication by r.

10.2.4 Exercise. Suppose that (R,m) is local and Artinian, and set E :=

E(R/m). Let L be a submodule of E. Show that L is faithful if and only ifL = E. (Here is a hint: apply the functor HomR( • , E) to the canonical exactsequence 0 −→ L −→ E −→ E/L −→ 0.)

Proposition 10.2.3 provides the main ingredients of Matlis duality for the(very special!) case of an Artinian local ring. However, as we shall show below,

202 Matlis duality

this special case can be used to build up to the general case quickly. We provedin 10.2.3(iii) that, if (R,m) is local and Artinian, then E(R/m) is Artinian.We now use Melkersson’s Theorem [51, Theorem 1.3] (see 7.1.2) to obtain amuch more general statement.

10.2.5 Theorem. Let m be a maximal ideal of R. Then E(R/m) is an Ar-tinian injective R-module.

Proof. By 10.1.16, we have (0 :ER(R/m) m) ∼= ER/m(R/m). Since R/mis a field, ER/m(R/m) = R/m. Hence (0 :ER(R/m) m) is an Artinian R-module. Now ER(R/m) is m-torsion, by 10.1.3(iii), and so it follows fromMelkersson’s Theorem 7.1.2 that ER(R/m) is Artinian.

10.2.6 Definition. Let M be an R-module. The socle Soc(M) of M is de-fined to be the sum of all simple submodules of M .

10.2.7 Remark. LetA be a non-zero ArtinianR-module. ThenA is an essen-tial extension of its own socle, because each non-zero submodule B of A mustcontain a simple submodule (since a minimal member of the set of non-zerosubmodules of B must be simple).

10.2.8 Corollary. Suppose that (R,m) is local, and set E := E(R/m). LetM be an R-module. Then M is Artinian if and only if M is isomorphic to asubmodule of Et, the direct sum of t copies of E, for some t ∈ N.

Proof. (⇐) It is clear from 10.2.5 that, for every t ∈ N, every submodule ofEt is Artinian.

(⇒) Assume that M is Artinian. Clearly, we can assume that M = 0. NowM is an essential extension of Soc(M), by 10.2.7, and Soc(M) is an Artinianmodule annihilated by m. Thus Soc(M) has a natural structure as a (finite-dimensional) vector space over R/m, and so there exists t ∈ N such thatSoc(M) ∼= (R/m)t. Now compose this isomorphism with the direct sum ofthe inclusion maps to obtain an R-monomorphism f : Soc(M) → Et; sinceEt is injective (by 10.1.3(i)), this f can be extended to an R-homomorphismf ′ : M → Et; and f ′ must also be a monomorphism since M is an essentialextension of Soc(M), by 10.2.7.

Key to the theory of Matlis duality are the facts that, when (R,m) is local,E := E(R/m) has a natural structure as an R-module, and, for each f ∈HomR(E,E), there is a unique rf ∈ R such that f(x) = rfx for all x ∈ E.We approach these facts next, and, in doing so, we touch on ideas mentionedin Exercise 8.2.4.


10.2.9 Remark. Suppose that (R,m) is local and that A is an Artinian R-module.

Let a ∈ A. It is immediate from 10.2.8 and 10.1.3(iii) that there exists t ∈ Nsuch that mta = 0. (In fact, one can prove this in a much more elementarymanner: if a = 0, then, since the R-module Ra is Artinian, R/(0 : a) is anArtinian local ring, and so its maximal ideal is nilpotent.) Let, with an obviousnotation,

r = (rn +mn)n∈N = (r′n +mn)n∈N ∈ lim←−n∈N

R/mn = R,

so that rn − r′n ∈ mn for all n ∈ N and rn+h − rn ∈ mn for all n, h ∈ N.Then rt+ha = rta = r′ta = r′t+ha for all h ∈ N. It is straightforward to checkthat A can be given the structure of an R-module such that, with the abovenotation, ra = rta. Note that, if we regard this R-module as an R-module bymeans of the natural ring homomorphismR→ R, then we recover the originalR-module structure on A; note also that a subset of A is an R-submodule ifand only if it is an R-submodule.

10.2.10 �Exercise. Suppose that (R,m) is local, and set E := E(R/m).By 10.2.5, the R-module E is Artinian; therefore, by 10.2.9, it has a naturalstructure as an R-module. Prove that there is an R-isomorphism E(R/m) ∼=ER(R/m), where m denotes the maximal ideal of R.

We are now able to prove the key result for Matlis duality that we mentionedjust before 10.2.9.

10.2.11 Theorem. Suppose that (R,m) is local, and set E := E(R/m).By 10.2.5, the R-module E is Artinian; therefore, by 10.2.9, it has a naturalstructure as an R-module.

The natural R-homomorphism θ : R −→ HomR(E,E) for which θ(r) =

r IdE for all r ∈ R is an isomorphism. Thus, for each f ∈ HomR(E,E), thereis a unique rf ∈ R such that f(x) = rfx for all x ∈ E.

Proof. For each n ∈ N, set En := (0 :E mn). Let f ∈ HomR(E,E); let t ∈N. Of course, R/m is annihilated by mt. By 10.1.16, there is an isomorphismof R/mt-modules Et

∼= ER/mt((R/mt)/(m/mt)). Now f(Et) ⊆ Et, and,of course, R/mt is an Artinian local ring. Therefore, by 10.2.3(v), there existsrt ∈ R such that f(e) = rte for all e ∈ Et, and, moreover, if r′t is any otherelement of R such that f(e) = r′te for all e ∈ Et, then the uniqueness aspectof 10.2.3(v) ensures that rt +mt = r′t +mt, that is, rt − r′t ∈ mt.

We can proceed as above for each t ∈ N, and so construct a uniquely deter-mined sequence (rn+mn)n∈N ∈

∏n∈NR/m

n with the property that, for every

204 Matlis duality

n ∈ N, we have f(e) = rne for all e ∈ En. Furthermore, for n, h ∈ N, wehaveEn ⊆ En+h, and so, since f(e) = rne = rn+he for all e ∈ En, it followsfrom the immediately preceding paragraph in this proof that rn − rn+h ∈ mn.

We have therefore found a uniquely determined sequence

(rn +mn)n∈N ∈ lim←−n∈N

R/mn = R

such that, for every n ∈ N, we have f(e) = rne for all e ∈ En. Since E =⋃n∈NEn by 10.1.3(iii), it follows that there is exactly one r ∈ R such that

f(x) = rx for all x ∈ E.

We are now able to present the main aspects of Matlis duality over a com-plete local ring.

10.2.12 Matlis Duality Theorem. Suppose that (R,m) is local and com-plete, and use the notation of 10.2.1. (Thus E denotes the injective envelopeE(R/m) of the simple R-module and D := HomR( • , E).)

(i) For each f ∈ HomR(E,E), there is a unique rf ∈ R such that f(x) =rfx for all x ∈ E.

(ii) Whenever N is a finitely generated R-module, the natural homomor-phism μN : N −→ DD(N) is an isomorphism and D(N) is Artinian.

(iii) Whenever A is an Artinian R-module, the natural homomorphism μA :

A −→ DD(A) is an isomorphism and D(A) is Noetherian.

Proof. (i) This is immediate from Theorem 10.2.11.(ii) The composition

θ : RμR−→ HomR(HomR(R,E), E)

∼=−→ HomR(E,E),

in which the second map is the obvious natural isomorphism, is such thatθ(r) = r IdE for all r ∈ R. We have seen in part (i) that θ is an isomorphism;therefore μR is an isomorphism.

The identity functor and DD are both additive, and the result of applicationof an additive functor to a split short exact sequence is again a split short exactsequence; also, μ is a natural transformation of functors. We can thereforededuce, by induction on rank, that μF is an isomorphism whenever F is afinitely generated free R-module.

Let N be an arbitrary finitely generated R-module. Then N can be includedin an exact sequence F1 −→ F0 −→ N −→ 0 in which F1 and F0 are finitelygenerated free R-modules. Since the functor DD is additive and exact, and


μ is a natural transformation of functors, the above exact sequence induces acommutative diagram

F1 F0 N 0 0� � � �

∼= μF1∼= μF0

μN

DD(F1) DD(F0) DD(N) 0 0� � � ��

with exact rows. It therefore follows from the Five Lemma that μN is an iso-morphism.

Finally, application of the contravariant, exact functor D to the exact se-quence F0 −→ N −→ 0 shows that D(N) is isomorphic to a submodule ofD(F0); since D(F0) is isomorphic to a direct sum of finitely many copies ofD(R) and D(R) ∼= E, it follows from 10.2.8 that D(N) is Artinian.

(iii) The composition

E HomR(HomR(E,E), E)μE

HomR(R,E)HomR(θ,IdE)

∼= E,∼=� � �

where θ is the isomorphism used in the proof of part (ii) above, is just theidentity map; therefore μE is an isomorphism. We can now use the additivityof DD, and the natural transformation μ, to deduce, by induction on t, thatμEt is an isomorphism for all t ∈ N.

LetA be an arbitrary ArtinianR-module. Two uses of Corollary 10.2.8 showthat there is an exact sequence 0 −→ A −→ En0 −→ En1 for suitable positiveintegers n0 and n1. We can now use the exactness of the functor DD, togetherwith the natural transformation μ, as we did in the above proof of part (ii), toobtain from this exact sequence a commutative diagram with exact rows, andanother application of the Five Lemma will yield the desired conclusion thatμA is an isomorphism.

Finally, application of the contravariant, exact functor D to the exact se-quence 0 −→ A −→ En0 shows that D(A) is a homomorphic image ofD(En0) ∼= (D(E))n0 ; since θ : R → D(E) is an isomorphism, D(A) isa homomorphic image of a finitely generated free R-module, and so isNoetherian.

10.2.13 �Exercise. Suppose that (R,m) is local and complete, and use thenotation of 10.2.1. LetG be anR-module of finite length, so that, by the MatlisDuality Theorem 10.2.12, the Matlis dual D(G) also has finite length. Provethat �(D(G)) = �(G).

206 Matlis duality

10.2.14 Remark. Suppose that (R,m) is local and complete, and use thenotation of 10.2.1.

The Matlis Duality Theorem 10.2.12 allows statements about NoetherianR-modules to be translated into ‘dual’ statements about Artinian R-modules,and vice versa. For example, it shows that every Noetherian R-module is iso-morphic to the Matlis dual of an Artinian R-module, and that every ArtinianR-module is isomorphic to the Matlis dual of a Noetherian R-module.

To give a sample of this type of ‘translation’, let N be a NoetherianR-module, so that D(N) is an Artinian R-module: we can use the MatlisDuality Theorem 10.2.12 to show quickly that AttD(N) = AssN , as fol-lows. Let p ∈ Spec(R). Then p ∈ AssN if and only if N has a submod-ule with annihilator equal to p. Now D is an exact, contravariant functor andDD(N) ∼= N ; also, by 10.2.2(ii), the annihilators of an R-module M and itsMatlis dual D(M) are equal. Therefore N has a submodule with annihilatorequal to p if and only if D(N) has a homomorphic image with annihilatorequal to p; and, by 7.2.5, this is the case if and only if p ∈ AttD(N).

10.2.15 Exercise. Suppose that (R,m) is local and complete, and use thenotation of 10.2.1.

Let N be a Noetherian R-module and let A be an Artinian R-module; letn ∈ N0 and h ∈ N.

(i) Prove that D(anN/an+hN) ∼= (0 :D(N) an+h)/(0 :D(N) a

n).(ii) Prove that D

((0 :A an+h)/(0 :A an)

) ∼= anD(A)/an+hD(A).(iii) Prove that Att(0 :A ah) = Ass

(D(A)/ahD(A)

).

So far, we have restricted our account of Matlis duality to situations wherethe underlying local ring is complete. However, it is possible to obtain fromTheorem 10.2.11 a satisfactory partial result which is valid over any local ring.We include this because we shall find a corollary of it useful in our applica-tions of the local duality theory developed in Chapter 11. We shall need thefollowing technical lemma.

10.2.16 Lemma. Let M, I, J be R-modules.

(i) There exists a (unique) R-homomorphism

ξM,I,J :M ⊗R HomR(I, J) −→ HomR(HomR(M, I), J)

such that, for m ∈ M , f ∈ HomR(I, J) and g ∈ HomR(M, I), wehave (ξM,I,J (m⊗ f)) (g) = f(g(m)). Furthermore, as M, I, J vary


through the category C(R), the ξM,I,J constitute a natural transforma-tion of functors

ξ • , • , • : ( • )⊗R HomR( • , • ) −→ HomR(HomR( • , • ), • )

(from C(R)× C(R)× C(R) to C(R)).(ii) If J is injective, then ξM,I,J is an isomorphism whenever M is finitely

generated.

Proof. (i) This is straightforward and left to the reader.(ii) It will be convenient to use V to denote ( • )⊗R HomR(I, J), to use W

to denote the functor HomR(HomR( • , I), J), and to use ζ • to denote ξ • ,I,J .

The composition HomR(I, J)∼=−→ V (R)

ζR−→ W (R)∼=−→ HomR(I, J), in

which the first and last isomorphisms are the obvious natural ones, is just theidentity map; hence ζR = ξR,I,J is an isomorphism.

The functors V and W are both additive, and ζ is a natural transformationof functors. We can therefore deduce, by induction on rank, that ζF is an iso-morphism whenever F is a finitely generated free R-module.

LetM be an arbitrary finitely generatedR-module. ThenM can be includedin an exact sequence F1 −→ F0 −→M −→ 0 in which F1 and F0 are finitelygenerated free R-modules. Of course the functor V is right exact; since Jis injective, the functor W is also right exact; therefore, since ζ is a naturaltransformation of functors, the above exact sequence induces a commutativediagram

V (F1) V (F0) V (M) 0 0� � � �

∼= ζF1∼= ζF0

ζM

W (F1) W (F0) W (M) 0 0� � � ��

with exact rows. It thus follows from the Five Lemma that ζM is an isomor-phism.

10.2.17 �Exercise. Supply a proof for part (i) of Theorem 10.2.16.

10.2.18 Remarks. Suppose that (R,m) is local (but not necessarily com-plete), and use the notation of 10.2.1. (Thus E denotes the injective envelopeE(R/m) of the simple R-module and D := HomR( • , E).) Let m denote themaximal ideal of R.

208 Matlis duality

(i) By 10.2.9, an Artinian R-module A can be given a natural structure asan R-module such that

(a) for each a ∈ A and each r ∈ R, there exists r ∈ R with ra = ra,and

(b) if we regard this R-module as an R-module by means of the nat-ural ring homomorphism R → R, then we recover the originalR-module structure on A.

It follows that A is an Artinian R-module.(ii) Let A′ be an Artinian R-module. Let a′ ∈ A′. By 10.2.9, there exists

t ∈ N such that a′ is annihilated by mt = mtR. Now let r ∈ R. Thenthere exists r ∈ R such that r − r ∈ mtR. Therefore ra′ = ra′, sothat, when A′ is regarded as an R-module by means of the natural ringhomomorphism R → R, every R-submodule of A′ is automatically anR-submodule, and therefore A′ is an Artinian R-module. If we then re-gard the Artinian R-module A′ as an R-module using the method of10.2.9, we recover the original R-module structure on A′.

(iii) Thus, by parts (i) and (ii), the category of all Artinian R-modules andall R-homomorphisms between them is equivalent to the category of allArtinian R-modules and all R-homomorphisms between them; in fact,we can regard these two categories as essentially the same.

(iv) Let ν be the natural transformation from the identity functor to the func-tor ( • ) ⊗R HomR(E,E) (from C(R) to itself) given by the formulaνM (m) = m⊗ IdE for each R-module M and each m ∈M .

LetN be a finitely generatedR-module. Then, sinceN is a homomor-phic image of Rh for some h ∈ N, the Matlis dual D(N) is isomorphicto a submodule of D(Rh); since D(Rh) ∼= D(R)h ∼= Eh, we deducefrom 10.2.8 that D(N) is Artinian. Furthermore, it is straightforward tocheck that the diagram

NνN

N ⊗R HomR(E,E)�

ξN,E,EμN

DD(N) ,

��

where ξN,E,E is the isomorphism given by 10.2.16, commutes. Now, by10.2.2(i), the natural map μN is injective; by 10.2.11, HomR(E,E) ∼= R

under an isomorphism which maps IdE to 1R, and so, speaking loosely,


we can regard the duality map μN : N −→ DD(N) as the embeddingof N into its completion.

(v) Let A be an Artinian R-module. Part (i) shows that A can be regardedas an Artinian R-module in a natural way; by 10.2.5, the R-moduleE is Artinian, and so it too can be regarded as an Artinian R-modulein a natural way; by 10.2.10, there is an R-isomorphism E(R/m) ∼=ER(R/m). Also, D(A) = HomR(A,E) = HomR(A,E), by part (i)(and 10.2.9), and soD(A) has a natural (unambiguous) R-module struc-ture. It follows from these observations that, as R-module, D(A) ∼=HomR(A,ER(R/m)); the Matlis Duality Theorem 10.2.12 shows thatthis is a Noetherian R-module.

We now summarize some of the main points of 10.2.18 in a Partial MatlisDuality Theorem.

10.2.19 Partial Matlis Duality Theorem. Suppose (R,m) is local (but notnecessarily complete), and use the notation of 10.2.1. (Thus E := E(R/m)

and D := HomR( • , E).) By 10.2.5 and 10.2.9, the R-module E has a naturalstructure as an R-module. Let m denote the maximal ideal of R.

(i) By 10.2.11, for each f ∈ HomR(E,E), there is a unique rf ∈ R suchthat f(x) = rfx for all x ∈ E.

(ii) Whenever N is a finitely generated R-module, the natural homomor-phism μN : N −→ DD(N) is injective, D(N) is Artinian, and there isa commutative diagram

N N ⊗R R�

∼=μN

DD(N) ,

��

in which the horizontal homomorphism is the canonical one.(iii) Whenever A is an Artinian R-module, it has a natural structure as an

R-module, it is Artinian as such, and D(A) is a Noetherian R-moduleR-isomorphic to the Matlis dual of A over R.

Proof. All the claims were explained in 10.2.18.

We show how the Partial Matlis Duality Theorem 10.2.19 can be used toextend 10.2.14 to the case where the underlying local ring is not necessarilycomplete.

210 Matlis duality

10.2.20 Corollary. Suppose that (R,m) is local (but not necessarily com-plete), and use the notation of 10.2.1.

Let N be a Noetherian R-module, so that D(N) is an Artinian R-moduleby 10.2.19(ii), and we can form the set AttD(N). Then AttD(N) = AssN .

Proof. Let p ∈ AssN , so thatN has a submoduleB with annihilator equal top. NowD is an exact, contravariant functor; also, by 10.2.2(ii), the annihilatorsof anR-moduleM and its Matlis dualD(M) are equal. ThereforeD(N) has ahomomorphic image D(B) with annihilator equal to p, and so p ∈ AttD(N)

by 7.2.5.Conversely, let p ∈ AttRD(N), so that D(N) has an R-homomorphic

image A with annihilator equal to p, by 7.2.5. Application of D and anotheruse of 10.2.2(ii) now show that DD(N) has an R-submodule with annihilatorequal to p. By 10.2.19(ii), there is an R-isomorphism DD(N) ∼= N ⊗R R.Therefore there exists an R-submodule C of N ⊗R R with (0 :R C) = p.

Let CR be the R-submodule of N ⊗R R generated by C. Note that CR is afinitely generated R-module, and that

(0 :R CR) ∩R = (0 :R CR) = (0 :R C) = p.

It follows that there exists P ∈ AssR(CR) such that P ∩ R = p. Of course,P ∈ AssR(N ⊗R R). We can now use [50, Theorem 23.2] to deduce thatp ∈ AssRN .

10.2.21 Exercise. Suppose that (R,m) is local and complete, and use thenotation of 10.2.1. Assume that dimR > 0. Prove that there exists an R-module G which is neither Noetherian nor Artinian but for which μG : G −→DD(G) is an isomorphism.

11

Local duality

Suppose, temporarily, that (R,m) is local, and thatM is a finitely generatedR-module. In Theorem 7.1.3, we showed that Hi

m(M) is Artinian for all i ∈ N0.When R is complete, Matlis duality (see 10.2.12) provides a very satisfac-tory correspondence between the category of Artinian R-modules and the cat-egory of Noetherian R-modules, and so it is natural to ask, in this situation,which Noetherian R-modules correspond to the local cohomology modulesHi

m(M) (i ∈ N0). Local duality provides a really useful answer to this ques-tion.

In fact, the Local Duality Theorem 11.2.6 concerns the situation where Ris a (not necessarily complete) homomorphic image of a Gorenstein local ring(R′,m′) of dimension n′. (Cohen’s Structure Theorem for complete local rings(see [50, Theorem 29.4(ii)], for example) ensures that this hypothesis would besatisfied if R were complete.) The Local Duality Theorem tells us that, if Mis a finitely generated R-module, then, for each i ∈ N0, the local cohomologymodule Hi

m(M) is isomorphic to the Matlis dual of the finitely generated R-module Extn

′−iR′ (M,R′), and, as R′ is Gorenstein, quite a lot is known about

these ‘Ext’ modules. The Local Duality Theorem provides a fundamental toolfor the study of local cohomology modules with respect to the maximal ideal ofa local ring. Although it only applies to local rings which can be expressed ashomomorphic images of Gorenstein local rings, this is not a great restriction,because the class of such local rings includes the local rings of points on affineand quasi-affine varieties, and, as mentioned above, all complete local rings.

At the end of the chapter, some exercises are given to show that a local ringwhich cannot be expressed as a homomorphic image of a Gorenstein local ringneed not have all the good properties flowing from the Local Duality Theorem.

212 Local duality

11.1 Minimal injective resolutions

The work in this chapter will involve knowledge of the structure of a mini-mal injective resolution of a Gorenstein local ring; we begin by reviewing theconcept of minimal injective resolution.

11.1.1 Definition. Let M be an R-module. A minimal injective resolution ofM is an injective resolution

I• : 0 −→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

of M such that In is an essential extension of Ker dn for every n ∈ N0.

11.1.2 �Exercise. Let M be an R-module. Prove that M has a minimal in-jective resolution. (Start with the monomorphism M −→ E(M), and, suc-cessively, take the injective envelope of the cokernel of the last map you haveconstructed.)

Suppose that, with the notation of 11.1.1, the injective resolution I• of M isminimal. There is an augmentation R-homomorphism α : M → I0 such thatthe sequence

0 −→Mα−→ I0

d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

is exact. Since I0 is an essential extension of Ker d0 and Ker d0 = Imα ∼=M ,it follows that I0 ∼= E(M), the injective envelope ofM , and so is uniquely de-termined up to isomorphism by M (see 10.1.1(v)). Actually, all the terms in aminimal injective resolution ofM are uniquely determined up to isomorphism:this is the subject of the next exercise.

11.1.3 �Exercise. Let M be an R-module. Let

I• : 0 −→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

and

J• : 0 −→ J0 δ0−→ J1 −→ · · · −→ J i δi−→ J i+1 −→ · · ·

be minimal injective resolutions of M , so that there are R-homomorphismsα :M → I0 and β :M → J0 such that the sequences

0 −→Mα−→ I0

d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

and

0 −→Mβ−→ J0 δ0−→ J1 −→ · · · −→ J i δi−→ J i+1 −→ · · ·

11.1 Minimal injective resolutions 213

are both exact. Show that there is an isomorphism of complexes

φ• =(φi)i∈N0

: I• −→ J•


Mα

I0�

φ0

Mβ

J0��

commutes.

11.1.4 Notation and Definition. LetM be anR-module. By 11.1.3, for eachi ∈ N0, the i-th term in a minimal injective resolution of M is uniquely deter-mined, up to isomorphism, by M , and is independent of the choice of minimalinjective resolution of M . We denote this i-th term by Ei(M), or by Ei

R(M)

when it is desirable to emphasize the underlying ring.Observe that E0(M) ∼= E(M), the injective envelope of M .For each i ∈ N0 and each p ∈ Spec(R), we define the i-th Bass number

of M with respect to p as follows. By 10.1.9, there is a family (pα)α∈Λ ofprime ideals of R for which Ei(M) ∼=

⊕α∈ΛE(R/pα). By 10.1.15 (see [50,

Theorem 18.5(iii)]), the cardinality of the set {α ∈ Λ : pα = p} depends onlyon Ei(M) and p (and therefore only on M and p) and not on the particulardecomposition of Ei(M) (as a direct sum of indecomposable injective sub-modules) chosen. This cardinality is denoted by μi(p,M), and is referred to asthe i-th Bass number of M with respect to p. Symbolically, we write

Ei(M) ∼=⊕

p∈Spec(R)

μi(p,M)E(R/p),

where ⊕μE denotes the direct sum of μ copies of E.

In the notation of 11.1.4, we can already give a description of μ0(p,M),since it follows from 10.1.15 (and the fact that E0(M) ∼= E(M)) that

μ0(p,M) = dimk(p) HomRp(k(p), E(M)p),

where k(p) = Rp/pRp, the residue field of the local ring Rp. In fact, a re-finement of this, and analogues for the higher Bass numbers, are available: weshall refer to Matsumura [50, Theorem 18.7] for these results, although we firstgive, for the reader’s convenience, a slight refinement of one of Matsumura’slemmas.

214 Local duality

11.1.5 Lemma. (See [50, §18, Lemma 6].) Let f : L → M be a homomor-phism of R-modules such that M is an essential extension of Im f . Let S bea multiplicatively closed subset of R. Then S−1M is an essential extensionof its submodule Im(S−1f) (where S−1f : S−1L → S−1M is the S−1R-homomorphism induced by f ).

Proof. Suppose that x/s, where x ∈ M and s ∈ S, is a non-zero elementof S−1M . Then S−1(Rx) is a non-zero submodule of S−1M , and so thereexists p ∈ Ass(Rx) such that p ∩ S = ∅. Also, there exists r ∈ R such that(0 : rx) = p.

Since M is an essential extension of Im f , there exists r′ ∈ R such that0 = r′rx = f(y) for some y ∈ L. Now (0 : r′rx) = p ⊆ R \ S, and so

0 = r′r

1

x

s=r′rx

s=f(y)

s= (S−1f)

(ys

)∈ Im(S−1f).

11.1.6 Corollary. Let M be an R-module, and let

I• : 0 −→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

be a minimal injective resolution of M . Let S be a multiplicatively closedsubset of R. Then

0 � � � � �S−1(I0) · · · S−1(Ii)S−1(di)

S−1(Ii+1) · · ·

is a minimal injective resolution of the S−1R-module S−1M .Thus, for all i ∈ N0,

(i) S−1(EiR(M)) ∼= Ei

S−1R(S−1M) (as S−1R-modules); and

(ii) μi(S−1p, S−1M) = μi(p,M) for all p ∈ Spec(R) with p ∩ S = ∅.

Proof. All the claims follow easily from 10.1.14 and 11.1.5.

11.1.7 Corollary. Let the situation and notation be as in 11.1.6. Then, for allp ∈ Spec(R) and i ∈ N0, the induced homomorphism HomRp

(Rp/pRp, dip)

is zero.

Proof. In view of 11.1.6, we can, and do, assume that (R,m) is local and thatp = m, and then we have to show that HomR(R/m, d

i) = 0. To achieve this,it is sufficient for us to show that (0 :Ii m) ⊆ Ker di. However, (0 :Ii m) isa direct sum of simple R-modules, and each simple R-submodule of Ii hasnon-trivial intersection with Ker di and so is contained in Ker di.

11.1.8 Theorem. (See [50, Theorem 18.7].) LetM be anR-module. For eachi ∈ N0 and each p ∈ Spec(R), the i-th Bass number μi(p,M) is given by

μi(p,M) = dimk(p) ExtiRp

(k(p),Mp) = dimk(p)

(ExtiR(R/p,M)

)p,

11.1 Minimal injective resolutions 215

where k(p) = Rp/pRp, the residue field of the local ring Rp. �

We have not explained the use of the word ‘minimal’ in the phrase ‘minimalinjective resolution’. We do this next.

11.1.9 Conventions. The injective dimension of an R-module M will be de-noted by inj dimM or, occasionally, by inj dimRM if it is desirable to specifythe underlying ring concerned. We adopt the convention that a zero module hasinjective dimension −∞.

11.1.10 Remark. Let M be an R-module, and let n ∈ N0.

(i) If En(M) = 0, then En+i(M) = 0 for all i ∈ N, and, consequently,inj dimM < n.

(ii) Conversely, if inj dimM < n, then, for all p ∈ Spec(R), we haveExtnR(R/p,M) = 0, so that μn(p,M) = 0 by 11.1.8; it follows thatEn(M) = 0.

11.1.11 Exercise. Let M be an R-module, and let

I• : 0 −→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

be a minimal injective resolution of M and

J• : 0 −→ J0 δ0−→ J1 −→ · · · −→ J i δi−→ J i+1 −→ · · ·

be an arbitrary injective resolution of M . Prove that there is a chain map ofcomplexes

φ• =(φi)i∈N0

: I• −→ J•


Mα

I0�

φ0

Mβ

J0��

(in which the horizontal maps are the appropriate augmentation homomor-phisms) commutes, and which is such that φi is a monomorphism for alli ∈ N0. Deduce that, for each i ∈ N0, the i-th term J i has a direct summandisomorphic to Ei(M).

216 Local duality

11.2 Local Duality Theorems

We shall again refer to Matsumura’s treatment of Gorenstein rings in [50, pp.139–145]. We remind the reader of the following necessary and sufficient con-dition, in terms of the Bass numbers μi(p, R), for R to be a Gorenstein ring.

11.2.1 Reminder. (See [50, Theorems 18.1 and 18.8].) The ring R is aGorenstein ring if and only if, for all i ∈ N0 and all p ∈ Spec(R),

μi(p, R) =

{0 if i = ht p,

1 if i = ht p,

that is, if and only if Ei(R) ∼=⊕

p∈Spec(R)ht p= i

E(R/p) for all i ∈ N0.

11.2.2 Exercise. Suppose that (R,m) is a Gorenstein local ring of dimensionn, and let

� � � � � �I• : 0 I0d0

I1 · · · In−1dn−1

In 0

be a minimal injective resolution of R. (We are using 11.2.1 when we ‘termi-nate’ this resolution after the n-th term.) Set E := In, so that E ∼= E(R/m)

by 11.2.1. Let V := {p ∈ Spec(R) : ht p = n − 1 and p ⊇ a}, and, for eachp ∈ V , let c(p) denote the kernel of the natural ring homomorphismR −→ Rp.

(i) Prove that ImΓa(dn−1) =

∑p∈V(⋃

i∈N(0 :E p(i)

)).

(ii) Assume now that R is complete. Prove that

Hna (R)

∼= HomR

(⋂p∈V c(p), E

).

(The results of this exercise provided the key steps in a short proof of thelocal Lichtenbaum–Hartshorne Vanishing Theorem (see 8.2.1) given by F. W.Call and R. Y. Sharp in [8]: a reader who finds this exercise difficult is referredto that paper for details.)

We now start our approach to the Local Duality Theorem.

11.2.3 Lemma. Let (R,m) be a Gorenstein local ring of dimension n. ThenHn

m(R)∼= E(R/m), the injective envelope of the simple R-module R/m.

Proof. Let

� � � � � �I• : 0 I0d0

I1 · · · In−1dn−1

In 0

11.2 Local Duality Theorems 217

be a minimal injective resolution of R. We calculate Hnm(R) by working out

the n-th cohomology module of the complex Γm(I•).

By 11.2.1,

Ii ∼=⊕

p∈Spec(R)ht p= i

E(R/p) for all i ∈ N0.

Let p ∈ Spec(R) with p = m. Then there exists r ∈ m\p. Now, by 10.1.3(iii),multiplication by r provides an automorphism of E(R/p), and so

Γm(E(R/p)) = 0.

Hence Γm(Ii) = 0 for all i ∈ N0 with i = n.

On the other hand, E(R/m) is m-torsion, again by 10.1.3(iii). Thus all theterms other than the n-th of the complex Γm(I

•) are zero, while its n-th termis isomorphic to E(R/m). Hence Hn

m(R)∼= E(R/m).

11.2.4 �Exercise. This exercise is central to our treatment of local duality.

(i) Let T : C(R) −→ C(R) be a covariant R-linear functor, and let B be anR-module.

Show that there is a natural transformation of functors

φ : T −→ HomR(HomR( • , B), T (B))

(from C(R) to itself) which, for an R-module M , is such that

(φM (y))(f) = T (f)(y) for all y ∈ T (M) and all f ∈ HomR(M,B).

Show also that, if each endomorphism of B can be realized as multi-plication by precisely one element of R, then φB is an isomorphism.

(ii) Now suppose that (R,m) is a Gorenstein local ring of dimension n; setE := E(R/m) and D := HomR( • , E).

Use part (i) and 11.2.3 to show that there is a natural transformationof functors

φ0 : Hnm −→ D(HomR( • , R))

(from C(R) to itself) which is such that φ0R is an isomorphism.Let M be a finitely generated R-module. Use Grothendieck’s Vanish-

ing Theorem 6.1.2, together with the fact that M can be included in anexact sequence F1 −→ F0 −→M −→ 0 in which F1 and F0 are finitelygenerated free R-modules, to show that φ0M is an isomorphism.

We can now present the Local Duality Theorem for a Gorenstein local ring.

218 Local duality

11.2.5 Local duality for a Gorenstein local ring. Let (R,m) be a localGorenstein ring of dimension n; set E := E(R/m) and D := HomR( • , E).By 11.2.4, there is a natural transformation of functors

φ0 : Hnm −→ D(HomR( • , R))

(from C(R) to itself) which is such that φ0M is an isomorphism for everyfinitely generated R-module M .

There is a unique extension of φ0 to a homomorphism

Φ := (φi)i∈N0 :(Hn−i

m

)i∈N0

−→(D(ExtiR( • , R))

)i∈N0

of (positive strongly) connected sequences of covariant functors from C(R) toC(R). Furthermore, φiM is an isomorphism for all i ∈ N0 whenever M is afinitely generated R-module.

In particular, for each finitely generated R-module M ,

Hn−im (M) ∼= D(ExtiR(M,R)) for all i ∈ Z.

Proof. By Grothendieck’s Vanishing Theorem 6.1.2, the functor Hnm is right

exact; it is an easy consequence of this that(Hn−i

m

)i∈N0

is a positive stronglyconnected sequence of covariant functors.

Since depthR = n, it follows from 6.2.8 and 3.4.10 that Hn−im (P ) = 0 for

all i ∈ N and all projective R-modules P ; also D(ExtiR(P,R)) = 0 for alli ∈ N and all projective R-modules P . It follows from the analogue of 1.3.4for positive connected sequences that φ0 can be incorporated into a (uniquelydetermined) homomorphism


m

)i∈N0

−→(D(ExtiR( • , R))

)i∈N0

of connected sequences. Furthermore, it is easy to prove by induction that, foreach i ∈ N, the homomorphism φiM is an isomorphism whenever M is afinitely generated R-module: use the fact that such an M can be included inan exact sequence 0 −→ K −→ F −→ M −→ 0 in which F is a finitelygenerated free R-module.

11.2.6 Local Duality Theorem. Suppose that (R,m) is a local ring whichis a homomorphic image of a Gorenstein local ring (R′,m′) of dimension n′:let f : R′ → R be a surjective ring homomorphism. Set E := E(R/m) andD := HomR( • , E).

An R-module M can be regarded as an R′-module by means of f : thenM , and, for each j ∈ N0, the modules ExtjR′(M,R′) and D(ExtjR′(M,R′)),inherit natural (R,R′)-bimodule structures.


There is a homomorphism

Φ := (φi)i∈N0 :(Hi

m

)i∈N0

−→(D(Extn

′−iR′ ( • , R′))

)i∈N0

of (negative strongly) connected sequences of covariant functors from C(R) toC(R) which is such that φiM is an isomorphism for all i ∈ N0 whenever M isa finitely generated R-module.

Consequently (since inj dimR′ R′ = n′), for each finitely generated R-module M ,

Him(M) ∼= D(Extn

′−iR′ (M,R′)) = HomR(Ext

n′−iR′ (M,R′), E)

for all i ∈ Z.

Proof. Let E′ := ER′(R′/m′), and let D′ := HomR′( • , E′).By 11.2.5, there is a homomorphism

Ψ := (ψi)i∈N0 :(Hn′−i

m′

)i∈N0

−→(D′(ExtiR′( • , R′))

)i∈N0

of (positive strongly) connected sequences of covariant functors from C(R′) toC(R′) which is such that ψiM ′ is an isomorphism for all i ∈ N0 whenever M ′

is a finitely generated R′-module. We can interpret

(ψn′−i)i∈N0 :(Hi

m′)i∈N0

−→(D′(Extn

′−iR′ ( • , R′))

)i∈N0

as a homomorphism of negative connected sequences.Let � R′ : C(R) → C(R′) denote the functor obtained from restriction of

scalars (using f ). If we precede each ψn′−i by � R′ , we can deduce from theIndependence Theorem 4.2.1 that there is a homomorphism

(ψi)i∈N0 :(Hi

m

)i∈N0

−→(D′(Extn

′−iR′ ( • , R′))

)i∈N0

of (negative strongly) connected sequences of covariant functors from C(R) toC(R′) which is such that ψi

M is an isomorphism for all i ∈ N0 whenever M isa finitely generated R-module.

Let c := Ker f . Let M be an arbitrary R-module. Then (0 :E′ c), and, foreach j ∈ N0, the modules ExtjR′(M,R′) and D′(ExtjR′(M,R′)) all inheritnatural R-module structures: for each of these modules, and for Hj

m(M), wehave f(r′)y = r′y for all r′ ∈ R′ and y in the module. It follows that

D′(ExtjR′(M,R′)) = HomR′(ExtjR′(M,R′), E′)

= HomR′(ExtjR′(M,R′), (0 :E′ c))

= HomR(ExtjR′(M,R′), (0 :E′ c)).

220 Local duality

Thus, for each i ∈ N0, we can regard

ψiM : Hi

m(M) −→ HomR(Extn′−iR′ (M,R′), (0 :E′ c))

as an R-homomorphism. All that remains in order to complete the proof is tonote that, by 10.1.16, there are R-isomorphisms

(0 :E′ c) ∼= ER(R′/m′) ∼= ER(R/m) = E.

The following exercise will be useful in applications of the Local DualityTheorem.

11.2.7 �Exercise. Suppose that (R,m) is a local ring which is a homomor-phic image of a local ring (R′,m′): let f : R′ → R be a surjective ring ho-momorphism. Let p ∈ Spec(R), and let p′ = f−1(p),∈ Spec(R′). Of course,R′/p′ ∼= R/p, so that these rings have equal dimensions. Moreover, observethat f induces a surjective ring homomorphism f ′ : R′p′ −→ Rp which is suchthat f ′(r′/s′) = f(r′)/f(s′) for all r′ ∈ R′ and s′ ∈ R′ \ p′.

Let M be a finitely generated R-module, and let j ∈ N0. We can formExtjR′

p′(Mp, R

′p′), which has a natural Rp-module structure. We can also lo-

calize the R-module ExtjR′(M,R′) at p. Show that

ExtjR′p′(Mp, R

′p′) ∼=(ExtjR′(M,R′)

)p

as Rp-modules.

The next exercise illustrates how the Local Duality Theorem can be used toobtain some of the results of 7.3.2 and 7.3.3 in the case where the local ringconcerned is a homomorphic image of a Gorenstein local ring.

11.2.8 Exercise. Suppose that (R,m) is a local ring which is a homomorphicimage of a Gorenstein local ring. Let M be a finitely generated R-module. Letp ∈ Spec(R), and let dimR/p = t.

(i) Let q ∈ Spec(R) be such that q ⊆ p, and let i ∈ Z. By 7.1.3, the Rp-module Hi

pRp(Mp) is Artinian, and Hi+t

m (M) is an Artinian R-module.Show that

qRp ∈ AttRp

(Hi

pRp(Mp))

if and only if q ∈ AttR(Hi+tm (M)).

(ii) Suppose p ∈ AssM . Show that Htm(M) = 0 and p ∈ Att(Ht

m(M)).Show further that, if t > 0, then Ht

m(M) is not finitely generated.(iii) Deduce from (ii) that, ifM is non-zero and of dimension n, thenHn

m(M)

= 0 and {p ∈ AssM : dimR/p = n} ⊆ Att(Hnm(M)); deduce also

that, if n > 0, then Hnm(M) is not finitely generated. Compare 7.3.2 and

7.3.3.


11.2.9 Exercise. Suppose that (R,m) is a local ring which is a homomorphicimage of a Gorenstein local ring. Let M be a finitely generated R-module,and let j be an integer for which Hj

m(M) = 0. Show that, for each p ∈AttR(H

jm(M)), we have dimR/p ≤ j.

Among the hypotheses for 11.2.8 and 11.2.9 was the assumption that theunderlying local ring is a homomorphic image of a Gorenstein local ring. Un-fortunately, the results of those exercises are not all true for general local rings,as can be seen from Exercise 11.2.13 below.

The next two exercises show that one of the implications in 11.2.8(i) is truein general. These two exercises are directed at those readers who are expe-rienced at working with the fibres of flat homomorphisms of commutativeNoetherian rings.

11.2.10 Exercise. Suppose that (R,m) is a local ring, that (R′,m′) is a sec-ond local ring, and that f : R −→ R′ is a flat local ring homomorphismsuch that the extension mR′ of m to R′ under f is m′-primary. Let i ∈ N0,and let M be a non-zero, finitely generated R-module with the property thatHi

m(M) = 0.

(i) Use the Flat Base Change Theorem 4.3.2 to see that there is an R′-isomorphism Hi

m′(M ⊗R R′) ∼= Hi

m(M)⊗R R′.

(ii) Let

Him(M) = S1 + · · ·+ Sh with Sj pj-secondary (1 ≤ j ≤ h)

be a minimal secondary representation of Him(M). For all j = 1, . . . , h,

let uj : Sj → Him(M) be the inclusion map, and let

Tj := (uj ⊗ IdR′)(Sj ⊗R R′).

Show that Him(M)⊗R R

′ = T1 + · · ·+ Th and that{f−1(P) : P ∈ AttR′(Tj)

}= {pj} for all j = 1, . . . , h.

(iii) Show that

AttR(Him(M)) =

{f−1(P) : P ∈ AttR′(Hi

m′(M ⊗R R′))}.

11.2.11 Exercise. Suppose that (R,m) is a local ring. Let M be a finitelygenerated R-module. Let p ∈ Spec(R) and let dimR/p = t. Let i ∈ Z andq ∈ Spec(R) be such that q ⊆ p and qRp ∈ AttRp

(Hi

pRp(Mp))

. Prove that

q ∈ AttR(Hi+tm (M)).

(Here are some hints: use Exercise 11.2.10, and Cohen’s Structure Theorem

222 Local duality

that a complete local ring is a homomorphic image of a regular local ring, inconjunction with 11.2.8(i).)

11.2.12 Exercise. Use 11.2.11 to extend the result of 11.2.8(ii) to all localrings.

Thus, in detail, suppose that (R,m) is a local ring. Let M be a non-zero,finitely generated R-module, and let p ∈ AssM . Suppose that dimR/p = t.Prove that Ht

m(M) = 0 and p ∈ Att(Htm(M)).

11.2.13 Exercise. This exercise shows that, if, in 11.2.9, the hypothesis thatthe local ring (R,m) be a homomorphic image of a Gorenstein local ring isdropped, then the corresponding statement is no longer always true. At thesame time, it shows that the result of 11.2.8(i) is not true for every local ring.

For a counterexample, suppose that (R,m) is a 2-dimensional local domainwhose completion (R, m) possesses an embedded prime ideal P (associatedto its zero ideal). Such an example has been constructed by D. Ferrand and M.Raynaud in [18]. Use 11.2.12 and 11.2.10 to prove that 0 ∈ Att(H1

m(R)), anddeduce that the result of 11.2.9 is not true for every local ring.

Show also that, for each p ∈ Spec(R) having dimR/p = 1,

0Rp ∈ AttRp

(H0

pRp(Rp)).

Deduce that the result of 11.2.8(i) is not true for every local ring.

12

Canonical modules

Suppose that (R,m) is local of dimension n > 0. We have seen earlier inthe book some illustrations of the importance of the so-called ‘top’ local co-homology module Hn

m(R) of R. (It is referred to as the ‘top’ local cohomol-ogy module because Hi

m(R) = 0 for all i > n by Grothendieck’s VanishingTheorem 6.1.2, whereas Hn

m(R) = 0 by the Non-vanishing Theorem 6.1.4.)Now Hn

m(R) is an Artinian R-module (by 7.1.3) and is not finitely generated;commutative algebraists tend to be brought up to work with finitely generatedmodules, and Artinian modules are perhaps a little less familiar. The philoso-phy behind the concept of canonical module is the following: wouldn’t it benice if we could, in some sense, replace Hn

m(R) by some finitely generatedR-module that we could work with effectively to achieve our desired results?

The theory of canonical modules for Cohen–Macaulay local rings is devel-oped by Bruns and Herzog in [7, Chapter 3]; their account is partly based on thelecture notes of Herzog and E. Kunz [33]. However, here we are going to workin the more general setting of an arbitrary local ring (R,m), and define a canon-ical module for R to be a finitely generated R-module K whose Matlis dualHomR(K,ER(R/m)) is isomorphic to Hn

m(R), where n := dimR. In thespecial case whereR is Cohen–Macaulay, this condition turns out to be equiva-lent to Bruns’ and Herzog’s definition, namely that a canonical module forR isa Cohen–Macaulay R-module K of dimension n for which inj dimRK < ∞and μn(m,K) = 1. The more general approach that we shall take is based onwork of Y. Aoyama [1], [2]; the definition we use is that employed by Hochsterand Huneke in [39].

If R is a homomorphic image of a Gorenstein local ring R′ of dimension n′

then the Local Duality Theorem 11.2.6 shows that the R-module

Extn′−n

R′ (R,R′)

is a canonical module for R. Thus the assumption that our local ring is a

224 Canonical modules

homomorphic image of a Gorenstein local ring again appears in hypotheses. Infact, there is a result, proved independently by H.-B. Foxby [19] and I. Reiten[69], to the effect that the Cohen–Macaulay local ring R admits a canonicalmodule if and only if R is a homomorphic image of a Gorenstein local ring.

It turns out that canonical modules are intimately related to Serre’s con-dition S2. Recall that a non-zero finitely generated R-module M (here R isnot assumed to be local) is said to satisfy Serre’s condition S2 if and only ifdepthRp

Mp ≥ min{2, dimRpMp} for all p ∈ SuppM . When R is a nor-

mal domain, R itself satisfies S2. It turns out that, if K is a canonical modulefor the local ring R, then depthK ≥ min{2, dimK}, and this is the key tothe connection with the condition S2. Furthermore, the endomorphism ringHomR(K,K) is a semi-local commutative Noetherian ring that, under mildconditions on R, acts as the so-called ‘S2-ification’ of R. (We did not inventthat name!) There are links between this concept and the generalized idealtransforms we studied in Chapter 2.

The concept of canonical module is of fundamental importance in the studyof Cohen–Macaulay local rings, and therefore we shall reconcile our approachwith that of Bruns and Herzog. In connection with the fact that a canonicalmodule over a Cohen–Macaulay local ring has finite injective dimension, it isperhaps worth noting that the present second author proved in [79, Corollary2.3], that, over a Cohen–Macaulay homomorphic image R of a Gorensteinlocal ring, the finitely generated R-modules of finite injective dimension areprecisely those R-modules M that can be included in an exact sequence

0 −→ Gs −→ Gs−1 −→ · · · −→ G1 −→ G0 −→M −→ 0

in which each of G0, . . . , Gs is a direct sum of finitely many copies of thecanonical R-module K. However, the proof of that result is beyond the scopeof this book.

12.1 Definition and basic properties

Many of the results in this and the next section are due to Y. Aoyama [1], [2].Here, we shall adopt the definition of canonical module used by M. Hochsterand C. Huneke in [39].

12.1.1 Notation. Throughout this section, we shall assume (R,m) is local,and we shall use n to denote dimR and (R, m) to denote the completion of R.

12.1.2 Definition. A canonical module for R is a finitely generated R-mod-ule K such that HomR(K,ER(R/m)) ∼= Hn

m(R).

12.1 Definition and basic properties 225

12.1.3 Remarks. We can make the following observations right away.

(i) Recall that the R-module Hnm(R) is Artinian (by 7.1.3), and that the

Matlis dual of a finitely generated R-module is (by 10.2.19(ii)) alsoArtinian.

(ii) Suppose that R has a canonical module K. There are R-isomorphisms

ER(R/m) ∼= ER(R/m)⊗R R

(by 8.2.4, 10.2.9 and 10.2.10) and Hnm(R)

∼= Hnm(R)⊗R R (by the Flat

Base Change Theorem 4.3.2). Therefore, by [50, Theorem 7.11], thereare R-isomorphisms

Hnm(R)

∼= Hnm(R)⊗R R ∼= HomR(K,ER(R/m))⊗R R

∼= HomR(K ⊗R R, ER(R/m)⊗R R)

∼= HomR(K ⊗R R, ER(R/m)).

Therefore K ⊗R R is a canonical module for R.(iii) It is immediate from the Local Duality Theorem 11.2.6 that, if R is a

homomorphic image of an n′-dimensional Gorenstein local ringR′, thenR has a canonical module, namely Extn

′−nR′ (R,R′).

(iv) In particular, if R itself is Gorenstein, then R is a canonical module foritself.

(v) In particular, when R is complete, so that it is a homomorphic imageof a regular local ring by Cohen’s Structure Theorem, R has a canon-ical module; in fact, in that case, it follows from the Matlis DualityTheorem 10.2.12 that each canonical module for R is isomorphic toHomR(H

nm(R), ER(R/m)); thus, when R is complete, there is, up to

isomorphism, exactly one canonical module for R.(vi) Suppose, in the case where the local ring (R,m) is not necessarily com-

plete, that K ′ is an R-module for which there is a R-isomorphism

K ′ ⊗R R ∼= HomR(Hnm(R), ER(R/m)).

Then, by the Matlis Duality Theorem 10.2.12 and the faithful flatness ofR over R, it follows that K ′ is a finitely generated R-module such thatK ′ ⊗R R is a canonical module for R. One can use [50, Theorem 7.11],8.2.4, 10.2.10 and 4.3.2 again to see that HomR(K

′, ER(R/m))⊗RR ∼=Hn

m(R)⊗R R as R-modules. Since HomR(K′, ER(R/m)) and Hn

m(R)

are ArtinianR-modules (by 10.2.19(ii) and 7.1.6 respectively), it followsfrom 8.2.4 that there is an R-isomorphism HomR(K

′, ER(R/m)) ∼=Hn

m(R). Thus K ′ is a canonical module for R.


This and part (ii) reconcile the above Definition 12.1.2 with that usedby Herzog and Kunz in [33, Definition 5.6].

Our next aim is to extend 12.1.3(v) by showing that, if R has a canonicalmodule, then any two canonical modules for R are isomorphic.

12.1.4 Lemma. Let K and L be two finitely generated R-modules. Thenthe set IsoR(K,L) of all R-isomorphisms from K to L is an open subset ofHomR(K,L) in the m-adic topology. In fact, if α ∈ IsoR(K,L), then

α+mHomR(K,L) ⊆ IsoR(K,L).

Proof. If K and L are not isomorphic, then IsoR(K,L) = ∅, an open subsetof HomR(K,L). So suppose that α : K

∼=−→ L is an isomorphism. We shallshow that α + mHomR(K,L) ⊆ IsoR(K,L), and this will be enough tocomplete the proof.

So let β ∈ α + mHomR(K,L). Then there exist r1, . . . , rt ∈ m andλ1, . . . , λt ∈ HomR(K,L) such that β = α +

∑ti=1 riλi. To show that

β ∈ IsoR(K,L), it is enough for us to show that α−1 ◦ β is an automor-phism of K, and, since K is finitely generated, it is therefore enough for us toshow that α−1 ◦ β is surjective (by [81, Exercise 7.2]). This we do. Now

α−1 ◦ β = α−1 ◦ (α+∑t

i=1 riλi) = IdK +∑t

i=1 riα−1 ◦ λi.

Let g ∈ K. Then

g = IdK(g) =(α−1 ◦ β −

∑ti=1 riα

−1 ◦ λi)(g) ∈ Im(α−1 ◦ β) +mK.

ThereforeK = Im(α−1◦β)+mK, and so it follows from Nakayama’s Lemmathat α−1 ◦ β is surjective, as required.

12.1.5 Lemma. Let R′ and R′′ be two commutative Noetherian rings andlet T : C(R′) −→ C(R′′) be an exact additive functor which is faithful inthe sense that T (M) = 0, for an R′-module M , implies that M = 0. Letf :M −→ N be an R′-homomorphism.

(i) If T is covariant and T (f) is a monomorphism (respectively, an epimor-phism), then f is a monomorphism (respectively, an epimorphism).

(ii) If T is contravariant and T (f) is a monomorphism (respectively, an epi-morphism), then f is an epimorphism (respectively, a monomorphism).

Proof. These are all proved similarly, and so we just prove the first part of(ii). Let C := Coker f . Apply T to the exact sequence

Mf−→ N −→ C −→ 0


to obtain an exact sequence 0 −→ T (C) −→ T (N)T (f)−→ T (M). Since T (f)

is a monomorphism, we must have T (C) = 0. Since T is faithful, C = 0, sothat f is an epimorphism.

12.1.6 Theorem. Any two canonical modules K and K ′ for R are isomor-phic.

Proof. By 12.1.3(ii), both K ⊗R R and K ′ ⊗R R are canonical modules forR, and so, by 12.1.3(v), there is an R-isomorphism α : K⊗R R

∼=−→ K ′⊗R R.SinceK andK ′ are finitely generatedR-modules, HomR(K,K

′) is finitelygenerated, and so the canonical injective map

HomR(K,K′) −→ HomR(K,K

′)⊗R R

provides the completion of HomR(K,K′). Now, by [50, Theorem 7.11], there

is an R-isomorphism

λ : HomR(K,K′)⊗R R

∼=−→ HomR(K ⊗R R,K′ ⊗R R)

which is such that λ(f ⊗ r) = r(f ⊗ IdR) for all f ∈ HomR(K,K′) and

r ∈ R.Since the image of HomR(K,K

′) is dense in HomR(K,K′) ⊗R R, there

exists g ∈ HomR(K,K′) such that

g ⊗ 1 ∈ λ−1(α) + m(HomR(K,K

′)⊗R R).

Apply λ to deduce that g ⊗ IdR ∈ α + mHomR(K ⊗R R,K ′ ⊗R R). But,by 12.1.4, the coset α + mHomR(K ⊗R R,K ′ ⊗R R) consists entirely ofisomorphisms; therefore g⊗IdR is an isomorphism fromK⊗R R toK ′⊗R R,so that g : K −→ K ′ is an isomorphism by 12.1.5, because R is faithfully flatover R.

12.1.7 Remark. Suppose that there exists a canonical module for R. By12.1.6, any two canonical modules for R are isomorphic. We shall denote byωR one choice of canonical module for R. We shall sometimes use the clause‘ωR exists’ as an abbreviation for ‘there exists a canonical module for R’.

Note that, in particular, if R is a homomorphic image of an n′-dimensionalGorenstein local ringR′, and also a homomorphic image of an n′′-dimensionalGorenstein local ring R′′, then, in view of 12.1.3(iii), there is an isomorphismExtn

′−nR′ (R,R′) ∼= Extn

′′−nR′′ (R,R′′) of R-modules.

12.1.8 Exercise. Let M be a non-zero finitely generated R-module, and sup-pose thatR is a homomorphic image of a Gorenstein local ringR′. For i ∈ N0,let Ki(M) denote the R-module ExtdimR′−i

R′ (M,R′).


(i) Show that Ki(M) = 0 for i < depthRM and for i > dimM , and thatKi(M) = 0 for i = depthRM and for i = dimM .

(ii) Show that, if R is complete, then, for a fixed i ∈ N0, the R-moduleKi(M) is, up to R-isomorphism, independent of the choice of the localGorenstein ring R′ (of which R is a homomorphic image).

(iii) Prove the result of (ii) without the assumption that R is complete. Inother words, assume only that R is a homomorphic image of a localGorenstein ring R′ and prove that Ki(M) is, up to R-isomorphism, in-dependent of the choice of R′. (Here is a hint: use 12.1.4 in a mannersimilar to the way it was used in 12.1.6.)

The module Ki(M), for i = dimM , is called the i-th module of deficiencyof M . In a sense, these modules give, if M is not Cohen–Macaulay, an indica-tion of the extent of the failure of M to be Cohen–Macaulay. They have beenstudied by P. Schenzel in [74] and [75].

We shall see during the chapter that the existence of a canonical module fora local ring R imposes some restrictions on R.

12.1.9 Proposition. Suppose that dimR = n > 0 and that ωR exists. Leta1, . . . , an be a system of parameters for R. Then

(i) AssR ωR = {p ∈ Spec(R) : dimR/p = n}; and(ii) a1 is a non-zerodivisor on ωR and, if n ≥ 2, then a1, a2 is an

ωR-sequence.

Proof. Let D denote the Matlis duality functor HomR( • , ER(R/m)).(i) By definition, D(ωR) ∼= Hn

m(R); also,

Att(Hnm(R)) = {p ∈ Spec(R) : dimR/p = n}

by 7.3.2; therefore, by 10.2.20,

AssωR = Att(D(ωR)) = {p ∈ Spec(R) : dimR/p = n} .

(ii) Since dimR/Ra1 = n − 1, it follows from part (i) that a1 does notbelong to any associated prime of ωR. Therefore a1 is an ωR-sequence.

Now suppose that n ≥ 2. Let α : R/(0 :R a1) −→ R be theR-monomorph-ism induced by multiplication by a1. The exact sequence

0 −→ R/(0 :R a1)α−→ R −→ R/Ra1 −→ 0

yields an exact sequence

Hn−1m (R/a1R) � � �Hn

m(R/(0 :R a1)) Hnm(R)

Hnm(α)

0


because dimR/a1R < n. Let π : R −→ R/(0 :R a1) be the natural epimor-phism. The exact sequence

0 −→ (0 :R a1) −→ Rπ−→ R/(0 :R a1) −→ 0

yields the isomorphism Hnm(π) : Hn

m(R)∼=−→ Hn

m(R/(0 :R a1)) becausedim(0 :R a1) < n. It follows that there is an exact sequence

Hn−1m (R/a1R) −→ Hn

m(R)a1−→ Hn

m(R) −→ 0.

But the exact sequence 0 −→ ωRa1−→ ωR −→ ωR/a1ωR −→ 0 yields the

exact sequence

0 −→ D(ωR/a1ωR) −→ D(ωR)a1−→ D(ωR) −→ 0.

Since Hnm(R)

∼= D(ωR), it follows that D(ωR/a1ωR) is a homomorphic im-age of Hn−1

m (R/a1R).The natural images of a2, . . . , an in R/a1R are a system of parameters for

that (n− 1)-dimensional local ring, and, by the Independence Theorem 4.2.1,we have Hn−1

m (R/a1R) ∼= Hn−1m/a1R

(R/a1R). Now it follows from the aboveproof of part (i) that the natural image of a2 is not in any attached prime ideal ofHn−1

m/a1R(R/a1R), and therefore a2Hn−1

m (R/a1R) = Hn−1m (R/a1R). Since

D(ωR/a1ωR) is a homomorphic image of Hn−1m (R/a1R), it follows that

a2D(ωR/a1ωR) = D(ωR/a1ωR).

Therefore, as D is a contravariant faithful exact R-linear additive functor fromC(R) to itself, we can use 12.1.5(ii) to see that a2 is a non-zerodivisor onωR/a1ωR, so that a1, a2 is an ωR-sequence.

12.1.10 �Exercise. Let the situation and notation be as in 12.1.9, and let Mbe a finitely generated R-module for which HomR(M,ωR) = 0. Show thata1 is a non-zerodivisor on HomR(M,ωR) and, if n ≥ 2, then a1, a2 is anHomR(M,ωR)-sequence.

Our next aim is to identify, in the case where ωR exists, the annihilator ofωR. By Matlis Duality, this is the same as the annihilator of HdimR

m (R): see10.2.2(ii).

12.1.11 �Exercise. Let c be the intersection of the (uniquely determined)primary components q of the zero ideal of R for which dimR/q = n, and let

g :=⋂

p∈AssRdimR/p<n

p.


(i) Show that

c = Γg(R)

= {r ∈ R : br = 0 for some ideal b ⊂ R with dimR/b < n}.

(ii) Let J (R) denote the set of all ideals of R. Set

L(R) = {b ∈ J (R) : dim b < n}= {b ∈ J (R) : bRp = 0 ∀ p ∈ Spec(R) with dimR/p = n} .

Recall from 7.3.1 that there is a maximum member d of the set L(R).Show that d = c.

12.1.12 Notation. Bearing in mind 12.1.11, we shall denote the maximummember of the set

{b : b is an ideal of R with dim b < n}

by uR(0). (Note that we really are discussing dim b for an ideal b, as opposedto dimR/b.)

We shall use some of the properties of uR(0) established in 12.1.11.

12.1.13 Lemma. A canonical module for R is annihilated by uR(0). More-over, a finitely generated R-module K that is annihilated by uR(0) (so thatit can be regarded as an R/uR(0)-module in the natural way) is a canonicalmodule for R if and only if it is a canonical module for R/uR(0).

Proof. Since dim uR(0) < n, it follows from Grothendieck’s VanishingTheorem 6.1.2 and the exact sequence

0 −→ uR(0) −→ R −→ R/uR(0) −→ 0

that there is an isomorphism Hnm(R)

∼=−→ Hnm(R/uR(0)). Therefore, if ωR is

a canonical module for R, then

HomR(ωR, ER(R/m)) ∼= Hnm(R)

∼= Hnm(R/uR(0)).

Since uR(0) annihilates Hnm(R/uR(0)), it follows that uR(0) annihilates

HomR(ωR, ER(R/m)).

As an R-module and its Matlis dual have the same annihilator (by 10.2.2(ii)),it follows that uR(0) annihilates ωR.

Now let K be a finitely generated R-module that is annihilated by uR(0),and denote ER(R/m) by E. The second displayed isomorphism in the lastparagraph, together with the Independence Theorem 4.2.1, show that there is


an R-isomorphism Hnm(R)

∼= Hnm/uR(0)(R/uR(0)). Now, in view of 10.1.16,

we have

HomR(K,E) = HomR(K, (0 :E uR(0))) = HomR/uR(0)(K, (0 :E uR(0)))

∼= HomR/uR(0)

(K,ER/uR(0) ((R/uR(0))/(m/uR(0)))

).

Therefore there is an R-isomorphism HomR(K,E) ∼= Hnm(R) if and only if

there is an R/uR(0)-isomorphism

HomR/uR(0)(K,ER/uR(0)((R/uR(0))/(m/uR(0))))∼=Hnm/uR(0)(R/uR(0));

thus K is a canonical module for R if and only if it is a canonical module forR/uR(0).

12.1.14 Remark. Often in this chapter, including in the proof of the nexttheorem, we shall want to work under the hypothesis that R is a homomorphicimage of an n′-dimensional Gorenstein local ring (R′,m′), so that there isa surjective ring homomorphism g : R′ −→ R. We explain here why it ispossible to assume that n′ = n.

Let c = Ker g. Now ht c = n′ − n, and, since a Gorenstein ring is Cohen–Macaulay, there exists an R′-sequence r′1, . . . , r

′n′−n contained in c of length

n′−n. NowR′/(r′1, . . . , r′n′−n) is a Gorenstein local ring of dimension n (see

[50, Exercise 18.1]), and so we can assume that n′ = n.

12.1.15 Theorem. Suppose that ωR exists. Then

(0 :R ωR) = (0 :R Hnm(R)) = uR(0).

Proof. It follows from 10.2.2(ii) that (0 :R ωR) = (0 :R Hnm(R)), and we

proved in 12.1.13 that uR(0) ⊆ (0 :R ωR). It is therefore sufficient for us toshow that (0 :R ωR) ⊆ uR(0).

We first consider the case where R is a homomorphic image of an n′-dimensional Gorenstein local ring (R′,m′), so that there is a surjective ringhomomorphism g : R′ −→ R. In view of 12.1.14, we assume that n′ =

n. By 12.1.3(iii), we have ωR∼= HomR′(R,R′). By 12.1.9(i), AssωR =

{p ∈ Spec(R) : dimR/p = n}. Thus, in order to deal with this case, it isenough for us to show that, if p ∈ Spec(R) has dimR/p = n, then the Rp-module (ωR)p has zero annihilator, for that would show that (0 :R ωR)Rp =

(0 :Rp(ωR)p) = 0. See 12.1.11 and 12.1.12.

To achieve this, let p′ := g−1(p),∈ Spec(R′), and use 11.2.7 to see thatthere are Rp-isomorphisms

(ωR)p ∼= (HomR′(R,R′))p∼= HomR′

p′ (Rp, R′p′).


However, R′p′ is a 0-dimensional Gorenstein local ring, and so

R′p′ ∼= ER′p′ (R

′p′/p′R′p′).

Therefore(0 :R′

p′ HomR′p′ (Rp, R

′p′))= (0 :R′

p′ Rp) by 10.2.2(ii), so that(0 :Rp

(ωR)p) = 0, as required.We have therefore proved the theorem under the additional assumption that

R is a homomorphic image of a Gorenstein local ring. We now deal withthe general case. We use 12.1.3(ii) to see that ωR exists and there is an R-isomorphism ωR ⊗R R ∼= ωR. Now R is a homomorphic image of a regularlocal ring by Cohen’s Structure Theorem, and so it follows from the first twoparagraphs of this proof that (0 :R ωR ⊗R R) = uR(0). Note that

(0 :R ωR) = (0 :R ωR ⊗R R) ∩R = uR(0) ∩R,

and so our proof will be complete if we can show that it is not possible forthere to be a P ∈ assR 0 with dim R/P < n and dimR/(P∩R) = n. So wesuppose that such a P exists and seek a contradiction. Denote P ∩R by p.

The inclusion homomorphism R −→ R induces a flat local homomorphismf : Rp −→ RP. Note that depthRP

RP = 0, so that the depth of the fibrering of f over the maximal ideal of Rp is 0 (by [50, Theorem 23.3]). Sincep ∈ AssωR by 12.1.9(i), we have depthRp

(ωR)p = 0, so that another use of[50, Theorem 23.3] shows that

depthRP

((ωR)p ⊗Rp

RP

)= 0.

But (ωR)p ⊗RpRP is RP-isomorphic to (ωR ⊗R R)P and ωR ⊗R R ∼= ωR.

Since dim R/P < n, there exists a ∈ P that is a parameter for R. By12.1.9(ii), the element a is an (ωR ⊗R R)-sequence, so that

depthRP

((ωR)p ⊗Rp

RP

)= depthRP

(ωR ⊗R R)P ≥ 1.

This is a contradiction, and the proof is complete.

We now remind the reader of Serre’s conditions Si (i ∈ N). Recall our con-vention whereby the depth of the zero module over a local ring is interpretedas ∞, and also our occasional abbreviation of depthRp

Mp by depthMp (fora finitely generated R-module M and p ∈ Spec(R)).

12.1.16 Definition. Let R′ be a commutative Noetherian ring. Let k ∈ Nand let M be a faithful finitely generated R′-module. We say that M sat-isfies Serre’s condition Sk, or, more loosely, that M is Sk, precisely whendepthMp ≥ min{k, ht p} for all p ∈ Spec(R′).


More generally, we say that a non-zero finitely generated R′-module N sat-isfies Serre’s condition Sk precisely when it is Sk as an R′/(0 :R′ N)-module,that is, if and only if depthR′

pNp ≥ min{k, dimR′

pNp} for all p ∈ SuppN ,

or, equivalently (in view of our convention concerning the depth of zero mod-ules), if and only if depthR′

pNp ≥ min{k, dimR′

pNp} for all p ∈ Spec(R′).

12.1.17 �Exercise. Let R′ be a commutative Noetherian ring, and let N be anon-zero finitely generated R′-module. Show that N is S2 if and only if

(i) each associated prime of N is a minimal member of SuppN , and(ii) for every a ∈ R′ which is a non-zerodivisor onN , each associated prime

of N/aN is a minimal member of SuppR′(N/aN).

It is important for us to identify the support of a canonical R-module, if oneexists.

12.1.18 Theorem. Suppose that ωR exists.

(i) We have

SuppωR = {p ∈ Spec(R) : ht p+ dimR/p = dimR};

also the R-modules ωR and HomR(ωR, ωR) satisfy the condition S2.(ii) If R is a homomorphic image of a Gorenstein local ring, then, for each

p ∈ SuppωR, the localization (ωR)p is a canonical module for Rp.

Note. Some readers may be surprised by the additional hypothesis in part (ii).In fact, the statement is true without the hypothesis that R be a homomorphicimage of a Gorenstein local ring, but the proof is beyond the scope of this book.Interested readers are referred to Aoyama [2, Corollary 4.3].

Proof. We deal first with the case where R is a homomorphic image of ann′-dimensional Gorenstein local ring (R′,m′) via a surjective ring homomor-phism g : R′ −→ R. In view of 12.1.14, we assume that n′ = n, so that wecan take ωR = HomR′(R,R′).

Let p ∈ SuppωR, so that there exists q ∈ AssωR with p ⊇ q. By 12.1.9,dimR/q = n. Since R is catenary, ht p/q + dimR/p = dimR/q, so thatht p+ dimR/p = n.

Now let p ∈ Spec(R) be such that ht p+dimR/p = n. Let p′ := g−1(p),∈Spec(R′), and use 11.2.7 to see that there are Rp-isomorphisms

(ωR)p ∼= (HomR′(R,R′))p∼= HomR′

p′ (Rp, R′p′).

We have dimR′/p′ = dimR/p and ht p′ ≥ ht p. Therefore we must haveht p′ + dimR′/p′ = n and ht p′ = ht p. Therefore HomR′

p′ (Rp, R′p′) is a


canonical module for Rp, so that there is an Rp-isomorphism (ωR)p ∼= ωRp.

In particular, p ∈ SuppωR. Also, it follows from 12.1.9 that depthωRp≥

min{2, dimRp}, and from 12.1.10 that

depthHomRp(ωRp

, ωRp) ≥ min{2,dimRp}.

We can therefore conclude that both ωR and HomR(ωR, ωR) are S2.All parts of the theorem have now been proved in the case where R is a

homomorphic image of a Gorenstein local ring. We now deal with the generalcase. We use 12.1.3(ii) to see that ωR exists and there is an R-isomorphismωR ⊗R R ∼= ωR. Now R is a homomorphic image of a regular local ring byCohen’s Structure Theorem, and so we know that the claims of part (i) hold truefor the R-canonical module ωR ⊗R R. In particular, the R-modules ωR ⊗R R

and HomR(ωR⊗RR, ωR⊗RR) are S2. An argument similar to one in the proofof [50, Theorem 23.9] shows that the R-modules ωR and HomR(ωR, ωR) areboth S2.

We now introduce some notation for a general prime ideal p of R. Let P ∈Spec(R) be a minimal prime ideal of pR for which dim R/P = dim R/pR =

dimR/p. Then P ∩ R = p, and the inclusion homomorphism R −→ R

induces a flat local homomorphism f : Rp −→ RP with the property thatits fibre ring over the maximal ideal of Rp is Artinian. Hence htP = ht p, andwe have ht p + dimR/p = htP + dim R/P ≤ n. Note also that there is aRP-isomorphism

(ωR ⊗R R)⊗R RP∼= (ωR ⊗R Rp)⊗Rp

RP.

Since f is faithfully flat, it follows that p ∈ SuppωR if and only if P ∈SuppR(ωR ⊗R R).

Now suppose that p ∈ SuppωR. Then P ∈ SuppR(ωR ⊗R R), so thathtP + dim R/P = n because R is a homomorphic image of a Gorensteinlocal ring. Therefore ht p+ dimR/p = n.

Conversely, suppose that ht p + dimR/p = n. Then htP + dim R/P =

n, so that P ∈ SuppR(ωR ⊗R R) because R is a homomorphic image of aGorenstein local ring; therefore p ∈ SuppωR.

All parts of the theorem have now been proved.

12.1.19 Exercise. Suppose R is a homomorphic image of a Gorenstein localring and that p ∈ SuppωR. Show that uRp

(0) = uR(0)Rp.

The following variant of the Local Duality Theorem will be useful.

12.1.20 Theorem. Suppose that ωR exists. Set E := E(R/m), and let D :=

HomR( • , E).


(i) There is a natural transformation of functors

φ0 : Hnm −→ D(HomR( • , ωR))

which is such that φ0M is an isomorphism whenever M is a finitelygenerated R-module.

(ii) Now suppose that R is Cohen–Macaulay. Then there is a unique exten-sion of φ0 to a homomorphism


m

)i∈N0

−→(D(ExtiR( • , ωR))

)i∈N0

of (positive strongly) connected sequences of covariant functors fromC(R) to C(R). Furthermore, φiM is an isomorphism for all i ∈ N0

whenever M is a finitely generated R-module.In particular, for each finitely generated R-module M ,

Hn−im (M) ∼= D(ExtiR(M,ωR)) for all i ∈ Z.

Proof. (i) By definition, Hnm(R)

∼= D(ωR). By 6.1.10, the functor Hnm is

naturally equivalent to ( • ) ⊗R Hnm(R). Therefore Hn

m is naturally equivalentto ( • )⊗R HomR(ωR, E). We can now make use of 10.2.16 in order to arriveat a natural transformation of functors

φ0 : Hnm −→ D(HomR( • , ωR))

which is such that φ0M is an isomorphism whenever M is a finitely generatedR-module.

(ii) Now suppose that R is Cohen–Macaulay. Here, we reason as in theproof of 11.2.5. Since depthR = n, it follows from 6.2.8 and 3.4.10 thatHn−i

m (P ) = 0 for all i ∈ N and all projective R-modules P ; also

D(ExtiR(P, ωR)) = 0

for all i ∈ N and all projective R-modules P . It follows from the analogueof 1.3.4 for positive connected sequences that φ0 can be incorporated into a(uniquely determined) homomorphism


m

)i∈N0

−→(D(ExtiR( • , ωR))

)i∈N0

of connected sequences, and it is easy to prove by induction that, for eachi ∈ N, the homomorphism φiM is an isomorphism whenever M is a finitelygenerated R-module. For i < 0, we have Hn−i

m (M) = ExtiR(M,ωR) = 0 foreach R-module M .

12.1.21 Corollary. Suppose that the local ring R is Cohen–Macaulay, andthat ωR exists. Then ωR has finite injective dimension equal to n, and is aCohen–Macaulay R-module of dimension n.


Proof. By 12.1.20 and the fact that ER(R/m) is an injective cogeneratorfor R, we have ExtiR(M,ωR) = 0 for all i > n and all finitely generatedR-modules M , so that, in particular, ExtiR(R/p, ωR) = 0 for all i > n

and all p ∈ Spec(R). Thus all the Bass numbers μi(p, ωR) are zero for alli > n and all p ∈ Spec(R) (by 11.1.8). Also ExtnR(R/m, ωR) = 0 becauseD(ExtnR(R/m, ωR)) ∼= Γm(R/m) = R/m = 0. Therefore ωR has finite in-jective dimension equal to n.

It also follows similarly from 12.1.20 that

D(ExtiR(R/m, ωR)) ∼= Hn−im (R/m) = 0 for all i ∈ {0, . . . , n− 1},

so that depthωR = n and ωR is a Cohen–Macaulay R-module.

12.1.22 Corollary. Suppose that the local ring R is Cohen–Macaulay, andthat ωR exists. Then R is Gorenstein if and only if ωR

∼= R, that is, if and onlyif R is a canonical module for itself.

Proof. We observed in 12.1.3(iv) that a Gorenstein local ring is a canonicalmodule for itself. Conversely, suppose that R is a Cohen–Macaulay local ringand that ωR exists and is isomorphic to R. Then inj dimRR <∞, by 12.1.21,and so R is Gorenstein.

12.1.23 �Exercise. Suppose that the local ring R is Cohen–Macaulay andthat ωR exists. Show that μi(m, ωR) = δi,htm (the Kronecker delta) for alli ∈ N0. (Here is a hint: use the method of proof in 12.1.21.)

12.1.24 �Exercise. Suppose that the local ring R is Cohen–Macaulay anda homomorphic image of a Gorenstein local ring, so that R has a canonicalmodule ωR, by 12.1.3(iii). Show that, for all i ∈ N0 and all p ∈ Spec(R),

μi(p, ωR) =

{0 if i = ht p,

1 if i = ht p,

that is, μi(p, ωR) is equal to the Kronecker delta δi,ht p. (Here is a hint: use12.1.18 and 12.1.23.)

12.1.25 �Exercise. Suppose that the local ring R is Cohen–Macaulay. Sup-pose that K is a finitely generated R-module such that μi(m,K) = δi,htmfor all i ∈ N0. The purposes of this exercise are to show that K is actually acanonical module forR, and to prove a result of I. Reiten [69] and H.-B. Foxby[19] that R is a homomorphic image of a Gorenstein local ring. Recall that ndenotes dimR.

(i) Show that there exists an R-sequence r1, . . . , rn which is also a K-sequence.


(ii) Denote the local ring R/(r1, . . . , rn) by (R,m) and use K to denoteK/(r1, . . . , rn)K. Use [7, 3.1.16] to calculate the μj

R(m,K) (j ∈ N0),

and conclude that there is an R-isomorphism K ∼= ER(R/m). Deducethat

(0 :R K/(r1, . . . , rn)K) = (r1, . . . , rn).

(iii) By considering rt1, . . . , rtn for t ∈ N, show that (0 :R K) = 0.

(iv) Let R′ be the trivial extension of R by K, introduced in 6.2.12. It fol-lows from that exercise that R′ is an n-dimensional local ring. Showthat (r1, 0), . . . , (rn, 0) is anR′-sequence, and deduce thatR′ is Cohen–Macaulay.

(v) Show thatR′/ ((r1, 0), . . . , (rn, 0)) is isomorphic to the trivial extensionof R by K, and calculate the socle of the latter ring. Deduce that R′ isGorenstein.

(vi) Conclude that R is a homomorphic image of a Gorenstein local ring, sothat ωR exists, by 12.1.3(iii). Prove that K ∼= ωR. (Here is a hint: recallfrom 12.1.3(iii) that ωR

∼= HomR′(R,R′).)

12.1.26 Remark. Suppose that the local ring R is Cohen–Macaulay.It follows from 12.1.3(iii), 12.1.23 and 12.1.25 that R has a canonical mod-

ule if and only if R is a homomorphic image of a Gorenstein local ring, andthat, when this is the case, an R-module K is a canonical module for R if andonly if it is finitely generated and μi(m,K) = δi,htm for all i ∈ N0.

In their book on Cohen–Macaulay rings [7], Bruns and Herzog (essentially)defined a canonical module for a Cohen–Macaulay local ring (R,m) to be afinitely generated R-module K for which μi(m,K) = δi,htm for all i ∈ N0.Thus 12.1.23 and 12.1.25 reconcile their approach with the one we have takenin this chapter.

12.1.27 Remark. Suppose (R,m) is a Cohen–Macaulay local ring. It followsfrom 12.1.25, 12.1.26 and 12.1.24 that a finitely generated R-module C is acanonical module for R if and only if μi(m, C) = δi,htm (Kronecker delta)for all i ∈ N0, and that, when this is the case, R is a homomorphic image of aGorenstein local ring andCp is a canonical module forRp for all p ∈ Spec(R).

Bruns and Herzog also extended the definition of canonical module to thecase where the underlying ring is not necessarily local (but is Cohen–Macaulay).

12.1.28 Definition. Let R′ be a (not necessarily local) Cohen–Macaulay(commutative Noetherian) ring. Let C be a finitely generated R′-module. We


say that C is a canonical module for R′ precisely when Cm is a canonicalmodule for R′m for all maximal ideals m of R′.

In that case, Cp is a canonical module for R′p for all p ∈ Spec(R′), by12.1.27.

12.1.29 Remark. LetR′ be a (not necessarily local) Cohen–Macaulay (com-mutative Noetherian) ring. Let C be a finitely generated R′-module. It followsfrom 12.1.27 that C is a canonical module for R′ if and only if μi(p, C) =

δi,ht p for all p ∈ Spec(R′) and all i ∈ N0.

12.1.30 Proposition. Let R′ be a (not necessarily local) Cohen–Macaulay(commutative Noetherian) ring, and assume that R′ has a canonical moduleC. Then the trivial extension R′ ∝ C of R′ by C (see 6.2.12) is a Gorensteinring, so that R′ is a homomorphic image of a Gorenstein ring.

Proof. By 6.2.12(ii), a general prime ideal P of R′ ∝ C has the form p× Cfor a prime ideal p of R′; also, 6.2.12(iv) shows that (R′ ∝ C)P ∼= R′p ∝ Cp,which is Gorenstein by 12.1.25. Therefore R′ ∝ C is Gorenstein. The ringhomomorphism φ : R′ ∝ C −→ R′ for which φ((r, c)) = r for all r ∈ R′ andc ∈ C is surjective.

12.2 The endomorphism ring

When (R,m) is local and ωR exists, it turns out that HomR(ωR, ωR) has somevery interesting and useful properties, some of which are relevant to the theoryof S2-ifications that we shall develop in the final section of this chapter. In thissection, we shall concentrate on the R-module structure of HomR(ωR, ωR);we shall consider its ring structure later in the chapter.

12.2.1 Notation. Throughout this section also, we shall assume that (R,m)

is local, we shall use n to denote dimR, and we shall denote the completionof R by (R, m).

In order to present the nice properties of HomR(ωR, ωR), we are going toconcentrate first on the case where (R,m) is a Cohen–Macaulay local ring(and ωR exists): by 12.1.15, we have (0 :R ωR) = uR(0), and this is zerowhen R is Cohen–Macaulay; in fact, we shall show, in that case, that eachR-endomorphism of ωR is given by multiplication by a uniquely determinedelement of R.

12.2.2 Proposition. Suppose the local ring R is Cohen–Macaulay, and ωR

exists. Let a1, . . . , aj ∈ m. Then a1, . . . , aj is an R-sequence if and only if it

12.2 The endomorphism ring 239

is an ωR-sequence; moreover, when this is the case, ωR/(a1,...,aj) exists andωR/(a1, . . . , aj)ωR

∼= ωR/(a1,...,aj).

Proof. Let D denote the Matlis duality functor HomR( • , ER(R/m)). Notethat there is nothing to prove if n = 0, and so we suppose that n > 0.

We first deal with the case in which j = 1. Since R is Cohen–Macaulay,AssR = {p ∈ Spec(R) : dimR/p = n}, and this is equal to AssωR, by12.1.9(i). Therefore a1 is a non-zerodivisor on R if and only if it is a non-zerodivisor on ωR. Suppose that this is the case. Since gradem = n, we haveHn−1

m (R) = 0, by 6.2.7. Therefore the exact sequence

0 −→ Ra1−→ R −→ R/a1R −→ 0

yields an exact sequence

0 −→ Hn−1m (R/a1R) −→ Hn

m(R)a1−→ Hn

m(R) −→ 0.

Also, the exact sequence 0 −→ ωRa1−→ ωR −→ ωR/a1ωR −→ 0 induces

an exact sequence 0 −→ D(ωR/a1ωR) −→ D(ωR)a1−→ D(ωR) −→ 0. By

the definition of canonical module, we have D(ωR) ∼= Hnm(R). It follows that

there is an R-isomorphism D(ωR/a1ωR) ∼= Hn−1m (R/a1R).

Now Hn−1m (R/a1R) ∼= Hn−1

m/a1R(R/a1R) by the Independence Theorem

4.2.1. Also

D(ωR/a1ωR) = HomR(ωR/a1ωR, ER(R/m))

= HomR(ωR/a1ωR, (0 :ER(R/m) a1)),

and 10.1.16 shows that there is an R/a1R-isomorphism

(0 :ER(R/m) a1) ∼= ER/a1R((R/a1R)/(m/a1R)).

We can therefore conclude that there is an R/a1R-isomorphism

HomR/a1R(ωR/a1ωR, ER/a1R((R/a1R)/(m/a1R)))∼= Hn−1

m/a1R(R/a1R).

Since the local ring R/a1R has dimension n− 1, this shows that ωR/a1ωR isa canonical module for it.

To complete the proof, proceed by induction on j.

12.2.3 Lemma. Suppose that the local ring R is Cohen–Macaulay, and thatωR exists. Let a ∈ m be a non-zerodivisor on R. Then the R-homomorphism

ξ : HomR(ωR, ωR) −→ HomR(ωR/aωR, ωR/aωR)

for which ξ(g)(x+ aωR) = g(x) + aωR, for each endomorphism g of ωR andeach x ∈ ωR, is surjective, and Ker ξ = aHomR(ωR, ωR).


Proof. Denote ωR/aωR by ωR, and let π : ωR −→ ωR denote the naturalepimorphism.

Proposition 12.1.9(ii) shows that a is an ωR-sequence, and 12.2.2 shows thatdepthR ωR = n. Therefore Hn−1

m (ωR) = 0, and it follows from 12.1.20 thatExt1R(ωR, ωR) = 0.

Therefore the exact sequence 0 −→ ωRa−→ ωR −→ ωR −→ 0 yields the

exact sequence

0 −→ HomR(ωR, ωR)a−→ HomR(ωR, ωR) −→ HomR(ωR, ωR) −→ 0.

It also yields the exact sequence

0 −→ HomR(ωR, ωR) −→ HomR(ωR, ωR)a−→ HomR(ωR, ωR).

But a annihilates HomR(ωR, ωR), and so the latter exact sequence shows thatHom(π, ωR) : HomR(ωR, ωR) −→ HomR(ωR, ωR) is an isomorphism. It isstraightforward to check that the diagram

HomR(ωR, ωR) HomR(ωR, ωR)�ξ

Hom(π,ωR)∼=Hom(ωR,π)

HomR(ωR, ωR)

��

��

commutes. The sequence

0 −→ HomR(ωR, ωR)a−→ HomR(ωR, ωR)

ξ−→ HomR(ωR, ωR) −→ 0

is therefore exact, and this proves all the claims of the lemma.

12.2.4 Notation. Suppose that ωR exists. We shall use

hR : R −→ HomR(ωR, ωR)

to denote the natural homomorphism for which hR(r) = r IdωR for all r ∈ R.Note that hR is both an R-module homomorphism and a ring homomor-

phism, and that ImhR is contained in the centre of the ring HomR(ωR, ωR).We are going to be interested in the kernel and cokernel of hR, and, in partic-ular, we shall show that h is an isomorphism when R is Cohen–Macaulay.

12.2.5 Remark. Suppose that ωR exists and that n > 0. By 12.1.11 and12.1.15,

Kerhr = uR(0)

= {r ∈ R : br = 0 for some ideal b ⊂ R with dimR/b < n}.


12.2.6 Theorem. Suppose that the local ringR is Cohen–Macaulay, and thatωR exists. Then hR : R −→ HomR(ωR, ωR) is an isomorphism.

Proof. We argue by induction on n. When n = 0, the ring R is an Artinianlocal ring, and therefore complete; also Γm(R) = R; thus 12.1.3(v) shows that

ωR∼= HomR(H

0m(R), ER(R/m)) ∼= HomR(R,ER(R/m)) ∼= ER(R/m).

By 10.2.3(v), for each endomorphism f ofER(R/m), there is a unique rf ∈ Rsuch that f(x) = rfx for all x ∈ ER(R/m). It follows easily that, for ourendomorphism g of ωR, there is a unique rg ∈ R such that g(x) = rgx for allx ∈ ωR.

Now suppose that n > 0 and assume, inductively, that the result has beenproved for all Cohen–Macaulay local rings of dimension n − 1 that possesscanonical modules. We already know that hR is injective, because KerhR =

(0 :R ωR) = uR(0) by 12.1.15, and this ideal is zero because R is a Cohen–Macaulay local ring. So let C := CokerφR.

Let a ∈ m be a non-zerodivisor on R, and let R denote the (n − 1)-dimensional Cohen–Macaulay local ring R/Ra. Now R has a canonical mod-ule and we can identify ωR/Ra = ωR/aωR (by 12.2.2). The exact sequence

0 −→ RhR−→ HomR(ωR, ωR) −→ C −→ 0 induces the exact sequence

R⊗R R� � �HomR(ωR, ωR)⊗R R C ⊗R R

hR⊗R0.

But it follows from 12.2.3 that there is an R-isomorphism

β : HomR(ωR, ωR)⊗R R∼=−→ HomR(ωR/aωR, ωR/aωR)

such that β ◦ (hR ⊗R) maps 1⊗ (r +Ra), for r ∈ R, to hR(r +Ra). SincehR is an isomorphism by the inductive hypothesis, it follows that hR ⊗ R isan isomorphism, so that C/aC = 0. Since C is finitely generated because ωR

is, we can use Nakayama’s Lemma to deduce that C = 0. Therefore hR issurjective, and so is an isomorphism.


12.2.7 Theorem. Suppose that ωR exists. Then


has Supp(CokerhR) ⊆ {p ∈ Spec(R) : ht p ≥ 2} . Thus each element ofCokerhR has annihilator of height at least 2.

Proof. Let C := CokerhR. Let p ∈ Spec(R) have ht p ≤ 1; it is enough forus to show thatCp = 0. Since SuppC ⊆ Supp(HomR(ωR, ωR)) ⊆ SuppωR,


we can, and do, assume that p ∈ SuppωR. Therefore ht p+dimR/p = n, by12.1.18.

Now argue as in that part of the proof of 12.1.18 where the general case istreated to find P ∈ Spec(R) such that P ∩ R = p, dim R/P = dimR/p

and htP = ht p. Then the inclusion homomorphism R −→ R induces a flatlocal homomorphism f : Rp −→ RP, and so it is enough for us to show thatCp⊗Rp

RP = 0. Hence it is enough for us to show that (C⊗R R)⊗R RP = 0.Now, by 12.1.3(ii), we have ωR ⊗R R ∼= ωR. In view of the natural R-

isomorphism HomR(ωR, ωR) ⊗R R∼=−→ HomR(ωR ⊗R R, ωR ⊗R R) (see

[50, Theorem 7.11]), we can therefore assume for the remainder of this proofthat R is complete. Recall that p ∈ SuppωR, ht p ≤ 1 and our aim is to showthat Cp = 0.

By 12.1.15 and 12.1.13, we have (0 :R ωR) = uR(0) and ωR is a canonicalmodule for R/uR(0). Note that p ⊇ uR(0) and ht p = ht p/uR(0), in view of12.1.15 and 12.1.18(i). Therefore, we can, and do, assume henceforth in thisproof not only that R is complete, but also that uR(0) = 0. Since ht p ≤ 1 anduR(0) = 0, the localization Rp is Cohen–Macaulay. By 12.1.18(ii), we have(ωR)p = ωRp

. It therefore follows from 12.2.6 that hRpis an isomorphism.

As the standard Rp-isomorphism

ψ : (HomR(ωR, ωR))p∼=−→ HomRp

((ωR)p, (ωR)p)

is such that ψ ◦ (hR)p = hRp, it follows that Cp = 0. This completes the

proof.

The following proposition, due to Hartshorne and Grothendieck [28, Propo-sition 2.1 and Remark 2.4.1] will be helpful in the determination of a neces-sary and sufficient condition (in the case where ωR exists) for hR : R −→HomR(ωR, ωR) to be an isomorphism.

12.2.8 Proposition. Suppose (R,m) is local and S2, and let 0 =⋂t+u

i=1 qi bea minimal primary decomposition of the zero ideal of R, where both t and uare positive integers. Then

(i) every minimal prime ideal of⋂t

i=1 qi +⋂t+u

j=t+1 qj has height 1;(ii) if, in addition, R is catenary, then dimR/p′ = dimR for all p′ ∈ ass 0.

Proof. Write a :=⋂t

i=1 qi and b =⋂t+u

i=t+1 qi, and let√qi = pi for all

i = 1, . . . , t+ u. Observe that a+ b is a proper ideal.(i) Let p be a minimal prime of a+ b. There exists i ∈ {1, . . . , t} such that

p ⊇ pi, and there exists j ∈ {t+1, . . . , t+u} such that p ⊇ pj . Localize at p:


we obtain, in Rp, that

aRp =

t⋂i=1pi⊆p

qiRp, bRp =

t+u⋂j=t+1pj⊆p

qjRp and 0 =

t+u⋂i=1pi⊆p

qiRp

are all minimal primary decompositions, that 0 = aRp ∩ bRp and that pRp isthe only associated prime ideal of aRp + bRp, and it is enough, in order for usto complete the proof of this part, to show that ht pRp = 1. It is thus enoughfor us to prove the claim under the additional assumption that p = m, and wenow assume this.

As R is S2, all associated primes of 0 are minimal; since there is an i ∈{1, . . . , t} such that m ⊇ pi, and there is a j ∈ {t + 1, . . . , t + u} such thatm ⊇ pj , we have htm ≥ 1. Thus m ∈ ass 0, m ∈ ass a and m ∈ ass b; onthe other hand, m ∈ ass(a + b). Thus Γm(R) = Γm(R/a) = Γm(R/b) = 0,whereas Γm(R/(a+ b)) = 0. By 3.2.1, there is an exact sequence

0 −→ R −→ (R/a)⊕ (R/b) −→ R/(a+ b) −→ 0.

Application of local cohomology (with respect to m) to this sequence thereforeyields that H1

m(R) = 0, so that depthR = 1. Since R is S2, we must havedimR = htm = 1.

(ii) Now suppose, in addition, that R is catenary. Suppose that there ex-ists p′ ∈ ass 0 such that dimR/p′ < n, and seek a contradiction. By thisassumption, we may suppose that the numbering, used above, of the primarycomponents of the zero ideal is such that dimR/pi = n for all i = 1, . . . , t

and dimR/pj < n for all j = t+1, . . . , t+ u. Then, with the above notation,a+ b will have a minimal prime ideal p, and by part (i), ht p = 1. Now p ⊇ pifor some i ∈ {1, . . . , t} and p ⊇ pj for some j ∈ {t+ 1, . . . , t+ u}. Then, bythe catenarity, we have

n = dimR/pi = dimR/p+ ht p/pi = dimR/p+ 1

= dimR/p+ ht p/pj = dimR/pj ,

and this is a contradiction.

12.2.9 �Exercise. Suppose R is a homomorphic image of a Gorenstein localring, so that ωR exists. Show that hR : R −→ HomR(ωR, ωR) is an isomor-phism if and only if R satisfies the condition S2.

Here are some hints for the implication ‘(⇐)’.

(a) Argue by induction on dimR. Use 12.2.6 to establish the claim in thecases where dimR ≤ 2.


(b) When dimR > 2, use 12.2.8(ii) and 12.2.5 to show that hR is monomo-rphic, and consider the exact sequence

0 −→ RhR−→ HomR(ωR, ωR) −→ CokerhR −→ 0.

12.2.10 Lemma. Suppose that R is a homomorphic image of a Gorensteinlocal ring, and that uR(0) = 0. Then {p ∈ Spec(R) : Rp is S2}, which werefer to as the S2-locus of R, is an open subset of Spec(R) in the Zariskitopology.

Proof. Consider hR : R −→ HomR(ωR, ωR), and let C := CokerhR, afinitely generatedR-module. Since uR(0) = 0, we see that hR is monomorphicand SuppωR = Spec(R), by 12.1.15. Let p ∈ Spec(R). On use of the natu-ralRp-isomorphism between (HomR(ωR, ωR))p and HomRp

((ωR)p, (ωR)p),and the fact that (ωR)p ∼= ωRp

(by 12.1.18(ii)), we see from 12.2.9 that Rp isnot S2 if and only if p ∈ SuppC.

The next exercise, which can be solved by use of 12.2.9, establishes a resultdue to Y. Aoyama.

12.2.11 Exercise (Y. Aoyama [1, Proposition 2]). Suppose that ωR exists.Show that hR : R −→ HomR(ωR, ωR) is an isomorphism if and only if thecompletion R satisfies the condition S2.

12.2.12 Exercise. Suppose that ωR exists, thatR is S2 and that all the formalfibres of R are Cohen–Macaulay. Show that hR is an isomorphism.

12.2.13 Exercise. Suppose that R is a homomorphic image of a Gorensteinlocal ring and that R is S2. Show that, for all p ∈ Spec(R), the localization(ωR)p is a canonical module for Rp.

The following exercise has important significance for algebraic geometry.

12.2.14 Exercise. Suppose that (R,m) is a Cohen–Macaulay local domainof dimension n that is not Gorenstein but admits a canonical module ω := ωR.

(i) Show that ω is isomorphic to an ideal of R. (Here is a hint: considerω ⊗R Q, where Q is the quotient field of R.)

(ii) In the light of (i), regard ω as an ideal of R. Show that

(a) htω = 1 and ω is unmixed;(b) R/ω is a Cohen–Macaulay ring of dimension n− 1; and(c) R/ω is Gorenstein.

12.3 S2-ifications 245

(Here are some more hints: for (ii)(a) and (ii)(b), apply local cohomology tothe exact sequence 0 −→ ω −→ R −→ R/ω −→ 0; for (c), use 12.1.20 tosee that the R/ω-module Ext1R(R/ω, ω) is a canonical module for R/ω, andthen apply the functor HomR( • , ω) to the above-mentioned exact sequencein order to deduce that R/ω is a canonical module for itself. If you still findthe exercise difficult, you might like to consult [7, Proposition 3.3.18].)

12.2.15 Exercise. Prove the following result of M. P. Murthy [55]: a Cohen–Macaulay UFD that is a homomorphic image of a Gorenstein local ring mustitself be Gorenstein.

12.3 S2-ifications

The purpose of this section is to relate the theory of canonical modules to theconcept of S2-ification of a local ring discussed by Hochster and Huneke in[39, Discussion (2.3)]. Our starting point is, however, a little more general, andconcerns the theory of generalized ideal transforms, discussed in §2.2.

12.3.1 Notation. In this section, we shall only assume that R is local whenthat is explicitly stated. Throughout this section, (Λ,≤) will denote a (non-empty) directed partially ordered set, and B = (bα)α∈Λ will denote a systemof ideals of R over Λ in the sense of 2.1.10.

Later in the section, we shall take B to be a particular system of idealsrelevant to the condition S2.

By a subsystem of B we shall mean a system of ideals C of R such that eachideal in the family C is a member of B and C can be written as (bα)α∈Θ forsome directed subset Θ of the indexing set Λ.

12.3.2 Proposition. Let M be a finitely generated R-module whose supportis equal to the whole of Spec(R). Then H0

B(M) = H1B(M) = 0 if and only

if H0bα(M) = H1

bα(M) = 0 for all α ∈ Λ.

Proof. Since

Γbα(M) ⊆ ΓB(M) ⊆⋃β∈Λ

Γbβ(M),

we see that H0B(M) = 0 if and only if H0

bα(M) = 0 for all α ∈ Λ. Suppose

that this is so.Let S be the set of non-zerodivisors on M . Then, by 2.1.1(ii), S ∩ bα = ∅

for all α ∈ Λ. It follows from 2.2.6(i) that H1B(M) = 0 if and only if ηB,M is

an isomorphism, and H1bα(M) = 0 (for a given α ∈ Λ) if and only if ηbα,M


is an isomorphism. Moreover, by 2.2.18, the map ηB,M is an isomorphism

if and only if the inclusion monomorphism M⊆−→⋃

α∈Λ(M :S−1M bα)

is an isomorphism, and, for an α ∈ Λ, the map ηbα,M is an isomorphism if

and only if the inclusion monomorphism M⊆−→⋃

n∈N(M :S−1M bnα) is anisomorphism. The proof can now be completed easily.

12.3.3 Reminders. We remind the reader of some properties of the naturalring homomorphism ηB,R : R −→ DB(R) that were established in 2.2.6(i),2.2.15, 2.2.16 and 2.2.17.

(i) Both the kernel and cokernel of ηB,R are B-torsion.(ii) Suppose that R′ is a ring (with identity, but not necessarily commuta-

tive), and let e : R −→ R′ be a ring homomorphism such that Im e iscontained in the centre ofR′ and, whenR′ is regarded as a leftR-moduleby means of e, both Ker e and Coker e are B-torsion. If ΓB(R′) = 0,then the ring R′ is commutative.

(iii) If now R′ is a commutative ring (with identity) and e : R −→ R′ isa ring homomorphism such that the R-modules Ker e and Coker e areB-torsion, then the unique R-homomorphism ψ′ : R′ −→ DB(R) suchthat the diagram

Re

R′�

ηRψ′

DB(R)

��

commutes is actually a ring homomorphism. The fact that this diagramcommutes can simply be recorded by the statement that ψ′ is an R-algebra homomorphism.

(iv) In the situation of part (iii), it follows from the formula for ψ′ in 2.2.15that ψ′ is injective if and only if ηB,R′ is injective, and this is the case ifand only if ΓB(R′) = 0. Furthermore, ψ′ is an R-algebra isomorphismif and only if ηB,R′ is an isomorphism, and this is the case if and only ifΓB(R′) = H1

B(R′) = 0.

12.3.4 Remark. Let C be a subsystem of B, and suppose that both H0B(R)

andH1B(R) are C-torsion. Then it follows from 12.3.3 that there is a uniqueR-

algebra homomorphism DB(R) −→ DC(R); also the facts that Ker ηC,R ∼=H0

C(R) and Coker ηC,R ∼= H1C(R) are C-torsion, and therefore B-torsion,

show that there is a unique R-algebra homomorphism e′ : DC(R)→ DB(R).


The uniqueness aspects of these statements therefore mean that e′ is an R-algebra isomorphism.

Similarly, if d is a member of B andH0B(R) andH1

B(R) are d-torsion, thenthere is a uniquely determined R-algebra isomorphism Dd(R)

∼=−→ DB(R).

12.3.5 Remark. Suppose DB(R) is a finitely generated R-module. Then itfollows from 2.2.6(i) that H1

B(R) is a finitely generated R-module; of course,H0

B(R) is finitely generated. Since B is a system of ideals, there exists α ∈ Λ

such that bα annihilates both H0B(R) and H1

B(R). Let C be a subsystem ofB such that bα belongs to the family C. Then it is immediate from 12.3.4 thatthere are uniquely determined R-algebra isomorphisms

Dbα(R)∼=−→ DC(R)

∼=−→ DB(R).

12.3.6 Definition. A B-closure of R is a commutative R-algebra θ : R −→A (with identity) such that

(i) the structural ring homomorphism θ makes A into a finitely generatedR-module;

(ii) both Ker θ and Coker θ are B-torsion; and(iii) whenever e : R −→ R′ is a commutativeR-algebra such that both Ker e

and Coker e are B-torsion, there is a unique R-algebra homomorphismψ′ : R′ −→ A.

We are using the terminology ‘B-closure’ in 12.3.6 because the concept hassimilarities with ‘la Z-cloture’ studied in Grothendieck [24, §5.9, §5.10].

12.3.7 Remarks. We can make the following comments about existence of aB-closure of R.

(i) It is immediate from the definition that a B-closure of R, if it exists, isuniquely determined up to R-algebra isomorphism.

(ii) Conditions (ii) and (iii) in Definition 12.3.6 require that a B-closure ofR has to be a solution to a certain universal problem. As ηB,R : R −→DB(R) is a solution to that universal problem (by 12.3.3(iii)), it followsthat there is a B-closure ofR if and only ifDB(R) is a finitely generatedR-module, and then ηB,R : R −→ DB(R) is the B-closure of R.

(iii) If there exists a B-closure of R, then, by part (ii), the R-module DB(R)

is finitely generated, and so it is immediate from 12.3.5 that there is anideal a in the family B for which H0

B(R) and H1B(R) are a-torsion, and

then, for any such a, and any subsystem C of B such that a belongs toC, there are uniquely determined R-algebra isomorphisms

Da(R)∼=−→ DC(R)

∼=−→ DB(R).


12.3.8 Notation. Throughout this section, H will denote the system of idealsof R formed by the set of all ideals of R of height at least 2 (indexed by itself,partially ordered by reverse inclusion). (Recall that we interpret the height ofthe improper ideal R of R as∞.)

We define the non-S2 locus of R to be {p ∈ Spec(R) : Rp is not S2}. Alsothroughout this section, S will denote the system of ideals of R formed by theset of all ideals s such that Var(s) is contained in the non-S2 locus of R. Notethat an R-module M is S-torsion if and only if (0 :R m) ∈ S for all m ∈M .

12.3.9 Definition. We define an S-closure of R, in the sense of 12.3.6, to bean S2-ification ofR. It follows from 12.3.7(ii) that there is an S2-ification ofRif and only if DS(R) is a finitely generated R-module, and then ηS,R : R −→DS(R) provides the S2-ification of R.

We shall reconcile this definition of S2-ification with that made by Hochsterand Huneke in [39, Discussion (2.3)] in the case where R is local, uR(0) = 0

and ωR exists.

12.3.10 Theorem. Suppose that (R,m) is local, that uR(0) = 0 and that ωR

exists. Then

(i) S is a subsystem of H, and the ring homomorphism


of 12.2.4 has (kernel and) cokernel that are H-torsion;(ii) the ring HomR(ωR, ωR) is commutative;

(iii) there is a unique R-algebra isomorphism ψ′ : HomR(ωR, ωR)∼=−→

DH(R);(iv) DH(R) andH1

H(R) are finitely generatedR-modules, and the ideal a :=

(0 :R H1H(R)) belongs to the family H; also Var(a) is the non-S2 locus

of R;(v) there are unique R-algebra isomorphisms

HomR(ωR, ωR)∼=−→ Da(R)

∼=−→ DS(R)∼=−→ DH(R),

and each of these R-algebras provides the S2-ification of R.

Proof. (i) Let a be an ideal in S, and suppose that a ⊆ p, where p is aprime ideal of height less than 2. Then Rp is not S2, and this is a contradic-tion because uR(0) = 0. Now hR is injective (since its kernel is uR(0)), andCokerhR is H-torsion by 12.2.7.

(ii) Note that ImhR is contained in the centre of H := HomR(ωR, ωR);also ΓH(H) = 0, since otherwise there would be an associated prime of the


R-module H having height at least 2, and this is not possible because H isa faithful R-module satisfying the condition S2 (by 12.1.18). It now followsfrom 2.2.16 that H is a commutative ring.

(iii) It is now immediate from 2.2.17 that there is a unique R-algebra homo-morphism ψ′ : HomR(ωR, ωR) =: H −→ DH(R). Since H is (faithful and)S2 by 12.1.18, we see that H0

a(H) = H1a(H) = 0 for all ideals a of R with

ht a ≥ 2. Hence H0H(H) = H1

H(H) = 0 by 12.3.2. It therefore follows from2.2.15 that ψ′ is an isomorphism.

(iv) Since H is a finitely generated R-module, DH(R) must also be finitelygenerated, by part (iii). Therefore its epimorphic image H1

H(R) is finitely gen-erated, and, since this module is H-torsion, its annihilator must be in H.

By 12.3.5, there is a uniquely determined R-algebra isomorphism

Da(R)∼=−→ DH(R).

We show next that Var(a) is the non-S2 locus of R. Let p ∈ Spec(R). Ifp ∈ Var(a), then there is an Rp-isomorphism Rp

∼= Hp, so that Rp is S2

because H is. On the other hand, if p is a minimal member of Var(a), then

0 = (H1a(R))p

∼= H1aRp

(Rp)

and depthRp ≤ 1; since ht p ≥ 2, this means that Rp is not S2.(v) By part (iv), the ideal a lies in the family S. Also, S is a subsystem of

H, by part (i). We can now use part (iii) and 12.3.5 to complete the proof.

12.3.11 Corollary. Suppose (R,m) is local, ωR exists and uR(0) = 0. LetS denote the set of all non-zerodivisors of R, so that R can be considered asa subring of S−1R, the total quotient ring of R. Then the S2-ification of R(which exists by 12.3.10) is given by the R-algebra A :=

⋃b∈H(R :S−1R b).

Furthermore,

(i) A is finitely generated and S2 as an R-module,(ii) for all a ∈ A, we have ht(R :R a) ≥ 2, and

(iii) there are unique R-algebra isomorphisms⋃b∈H(R :S−1R b)

∼=−→ HomR(ωR, ωR)∼=−→ Da(R)

∼=−→ DS(R)∼=−→ DH(R),

where a := (0 :R H1H(R)).

Proof. Set H := HomR(ωR, ωR). Since uR(0) = 0, it follows from 12.1.10that S consists entirely of non-zerodivisors on H , and that S ∩ b = ∅ forall b ∈ H. We can now use 2.2.18 to see that there is a unique R-algebra


isomorphism A∼=−→ DH(R). It therefore follows from 12.3.10 that R

⊆−→ A

provides the S2-ification of R. All the other claims are now immediate from12.3.10.

12.3.12 Remark. Suppose that R is local and that uR(0) = 0. Hochster andHuneke [39, Discussion (2.3)] defined an S2-ification of R to be a subring A′

of the total quotient ring of R such that R ⊆ A′, such that A′, as R-module, isfinitely generated and S2, and such that, for all a ∈ A′, ht(R :R a) ≥ 2. If ωR

exists, then the S2-ification A of R found in 12.3.11 is an S2-ification in thesense of Hochster and Huneke.

12.3.13 Remark. Suppose (R,m) is local and ωR exists. By 12.1.13, theideal uR(0) annihilates ωR and ωR is a canonical module for R/uR(0). Ittherefore follows from 12.3.11 that the endomorphism ring HomR(ωR, ωR) ofωR is a commutative Noetherian semi-local ring.

12.3.14 Exercise (Goto [2, Example 3.3]). Let

R := K[[X,Y, Z,W ]]/(X,Y ) ∩ (Z,W ),

where K is a field and X,Y, Z,W are independent indeterminates. Show thatthe R-homomorphism hR : R −→ HomR(ωR, ωR) of 12.2.4 is not an iso-morphism, and that the (commutative) ring HomR(ωR, ωR) is not local.

12.3.15 Exercise. Suppose that (R,m) is local and ωR exists. Assume thatRis analytically irreducible; that is, assume that the completion R is an integraldomain. Show that HomR(ωR, ωR) is also an analytically irreducible localring. (Here is a hint: you might find [50, Theorem 8.15] helpful.)

Note. In [39, Theorem (3.6)], Hochster and Huneke provide, in the casewhere (R,m) is local, complete and equidimensional, several conditions equi-valent to the statement that the semi-local ring HomR(ωR, ωR) is local.

13

Foundations in the graded case

If our Noetherian ring R is (Z-)graded, and our ideal a is graded, then it isnatural to ask whether the local cohomology modules Hi

a(R) (i ∈ N0), andHi

a(M) (i ∈ N0) for a graded R-module M , also carry structures as gradedR-modules. Some of the realizations of these local cohomology modules thatwe have obtained earlier in the book suggest that they should. For instance, ifa1, . . . , an (where n > 0) denote n homogeneous elements which generate a,then the Cech complex C•(M) of M with respect to a1, . . . , an is composedof graded R-modules and homogeneous homomorphisms, and so Hi

a(M) (fori ∈ N0), which, by Theorem 5.1.20, is isomorphic to Hi(C(M)•), inherits agrading. But is this grading independent of the choice of homogeneous gener-ators for a?

Additional hopeful evidence is provided by the isomorphism

Hia(M) ∼= lim

−→n∈N

ExtiR(R/an,M)

of 1.3.8. For each n ∈ N, since R/an is a finitely generated graded R-module,ExtiR(R/a

n,M) is actually the graded R-module

*ExtiR(R/an,M)

(see [7, pp. 32–33]) with its grading forgotten, and, for n,m ∈ N with n ≥ m,the natural homomorphism hnm : R/an → R/am is homogeneous, so that theinduced homomorphism

*ExtiR(hnm,M) : *ExtiR(R/a

m,M) −→ *ExtiR(R/an,M)

is homogeneous; hence lim−→n∈N

ExtiR(R/an,M) is graded, and Hi

a(M) inherits

a grading by virtue of the above isomorphism. But is this grading the same asthat which comes from the approach using the Cech complex described in thepreceding paragraph?

252 Foundations in the graded case

One could take another approach to local cohomology in this graded situa-tion, an approach which, at first sight, seems substantially different from thosedescribed in the preceding two paragraphs. Again suppose that our Noetherianring R is (Z-)graded, and our ideal a is graded. The category *C(R) of allgraded R-modules and homogeneous R-homomorphisms is an Abelian cate-gory which has enough projective objects [7, p. 32] and enough injective ob-jects [7, 3.6.2]: we can therefore carry out standard techniques of homologicalalgebra in this category. In particular, the a-torsion functor Γa can be viewedas a (left exact, additive) functor from *C(R) to itself, and so we can form itsright derived functors *Hi

a (i ∈ N0) on that category. This will produce, for agraded R-module M , graded local cohomology modules *Hi

a(M) (i ∈ N0),which are constructed by the following procedure. Since *C(R) has enoughinjectives, we can construct an injective resolution of M in this category, thatis, we can construct an exact sequence

0 −→Mα−→ E0 d0

−→ E1 −→ · · · −→ Ei di

−→ Ei+1 −→ · · ·

in *C(R) in which the Ei (i ∈ N0) are injective objects in that category, thatis, they are *injective graded R-modules in the terminology of [7, §3.6]; thenwe apply the functor Γa to the complex

0 −→ E0 d0

−→ E1 −→ · · · −→ Ei di

−→ Ei+1 −→ · · · ;

the i-th cohomology module of the resulting complex is *Hia(M) (for each

i ∈ N0). But if we forget the grading on *Hia(M), is the resulting R-module

isomorphic to Hia(M), and, if so, is the grading on Hi

a(M) induced by thisisomorphism the same as that which comes from the approaches in the firsttwo paragraphs of this chapter?

Our main purpose in this chapter is to reconcile these various approaches.We think it is desirable that it should be established that they all give (up toisomorphism in *C(R)) the same object of *C(R). Thus we are going to showthat the questions posed at the ends of the first three paragraphs of this chapterall have affirmative answers.

In fact, because many modern applications of local cohomology are to multi-graded rings and modules, we are going to deal in this chapter with a Zn-graded commutative Noetherian ring and Zn-graded modules over it (note thatn will always denote a positive integer in this context).

13.1 Basic multi-graded commutative algebra 253

13.1 Basic multi-graded commutative algebra

13.1.1 Notation and Terminology. Throughout this chapter, G will denotea finitely generated, torsion-free Abelian group, written additively. (Thus G iseither 0 or isomorphic to Zn for some n ∈ N.)

When we write thatR =⊕

g∈GRg is aG-graded ring, it is to be understoodthat the direct decomposition is as Z-module, that RgRg′ ⊆ Rg+g′ for allg, g′ ∈ G, and that all other ‘graded’ objects related to R (such as R-modules,ideals of R) are graded by G. Thus, when R is G-graded as above and wewrite that M is a graded R-module, it is to be understood that M has a directdecomposition M =

⊕g∈GMg as Z-module and that RgMg′ ⊆ Mg+g′ for

all g, g′ ∈ G. The elements of⋃

g∈GMg are called the homogeneous elementsof M , and an element of Mg′ \ {0}, where g′ ∈ G, is said to have degree g′.We shall use ‘deg’ as an abbreviation for degree.

Suppose that R =⊕

g∈GRg is G-graded. An R-homomorphism f : M =⊕g∈GMg −→ N =

⊕g∈GNg between graded R-modules will be said to

be homogeneous precisely when f(Mg) ⊆ Ng for all g ∈ G; we shall denoteby *C(R) (or by *CG(R) when it is desirable to specify the grading group G)the category of all graded R-modules and homogeneous R-homomorphismsbetween them.

For g0 ∈ G, we shall denote the g0-th shift functor by ( • )(g0) : *C(R) −→*C(R): thus, for a graded R-module M =

⊕g∈GMg , we have (M(g0))g =

Mg+g0 for all g ∈ G; also, f(g0)� (M(g0))g = f� Mg+g0for each morphism f

in *C(R) and all g ∈ G.If S is a multiplicatively closed subset of R consisting of non-zero homoge-

neous elements, then it is routine to check that the ring S−1R is alsoG-graded,with g-th component, for g ∈ G, equal to the set of all elements of S−1R thatcan be expressed in the form r/s with s ∈ S, r a homogeneous element of Rand (r = 0 or) r = 0 and deg r−deg s = g. Similarly, for a graded R-moduleM , the S−1R-module S−1M is also graded.

13.1.2 Remark. A total order ≺ on G is said to be compatible with additionif, whenever g, g′, h ∈ G with g ≺ g′, we have g + h ≺ g′ + h. Note that itis always possible to put a total order compatible with addition on G: this istrivial if G = 0, and otherwise G ∼= Zn for some n ∈ N, and, for example, thelexicographical order on Zn is compatible with addition. This lexicographicalorder ≺ is defined as follows: for a = (a1, . . . , an) and b = (b1, . . . , bn) ∈Zn, we set a ≺ b if and only if a = b and, if i is the least integer in {1, . . . , n}for which ai = bi, then ai < bi.


13.1.3 Elementary Reminders. Assume that R is G-graded, and let M be agraded R-module.

(i) We have 1R ∈ R0; furthermore, R0 is a Noetherian subring of R andRg is a finitely generated R0-module, for each g ∈ G. (See [61, §2.11,Theorem 21, and §4.3, Theorem 13].)

(ii) We shall say that a submodule of M is graded precisely when it can begenerated by homogeneous elements: see [61, §2.11, Proposition 28].

(iii) Recall from [61, §2.11, Proposition 29] that the sum and intersection ofthe members of an arbitrary family of graded submodules ofM are againgraded.

(iv) Let N be a graded submodule of M and assume that a is graded. Then(N :R M), aN and (N :M a) are all graded: see [61, §2.11, Propositions30 and 31].

(v) Assume that a is graded. Then a is prime if and only if it is proper and,whenever r and r′ are homogeneous elements of R \ a, then rr′ ∈ R \ atoo: see [61, §2.13, Lemma 13].

(vi) For an arbitrary ideal b of R, we denote by b* the (necessarily graded)ideal generated by all homogeneous elements of b. Thus b* is the largestgraded ideal of R contained in b. By [61, §2.13, Proposition 33], if b isprime, then so too is b*.

(vii) We denote by * Spec(R) (or * SpecG(R)) the set of all graded primeideals of R.

(viii) The radical of a graded ideal ofR is again graded: see [61, §2.13, Propo-sition 32].

13.1.4 Example. Let R0 be a commutative Noetherian ring, let n ∈ N, andlet R := R0[X1, . . . , Xn], the ring of polynomials over R0.

(i) We shall often consider R as Zn-graded, with grading given by

R(i1,...,in) = R0Xi11 . . . Xin

n for all (i1, . . . , in) ∈ Zn.

Thus R(i1,...,in) = 0 unless ij ≥ 0 for all j = 1, . . . , n. Observe that Xi

is homogeneous of degree (0, . . . , 0, 1, 0, . . . , 0) (where the ‘1’ is in thei-th spot), for all i = 1, . . . , n.

(ii) Consider the special case in which R0 is a field. Then the graded idealsofR are just the ideals that can be generated by monomialsXi1

1 . . . Xinn ,

where (i1, . . . , in) ∈ N0n. The graded prime ideals of R are just the

ideals that can be generated by a subset of the set {X1, . . . , Xn} of thevariables. It follows that there are only finitely many graded prime idealsof R.

13.1 Basic multi-graded commutative algebra 255

(iii) Now consider the case where R0 = S is a G-graded ring, with gradinggiven by S =

⊕g∈G Sg . ThenR = S[X1, . . . , Xn] is (G⊕Zn)-graded,

with (g, (i1, . . . , in))-th component equal to SgXi11 . . . Xin

n for all g ∈G and (i1, . . . , in) ∈ Zn.

13.1.5 Example. Let n ∈ N. A simplicial complex on the vertex set V :=

{1, . . . , n} is a set Δ of subsets of V which is closed under passage to subsets,that is, whenever F ∈ Δ and F ′ ⊆ F , then F ′ ∈ Δ too.

LetR0 be a commutative Noetherian ring and letR := R0[X1, . . . , Xn], thering of polynomials over R0 in indeterminates X1, . . . , Xn, considered to beZn-graded as in 13.1.4. Let Δ be a simplicial complex on {1, . . . , n}. Let aΔbe the ideal of R generated by all square-free monomials Xi1 . . . Xit such that{i1, . . . , it} ∈ Δ. Observe that, if F ′ is a subset of V that does not belong toΔ, then no subset of V that contains F ′ can belong to Δ. Observe also that aΔis a graded ideal of R, as it is generated by homogeneous elements. Thereforethe ring R0[Δ] := R/aΔ inherits a Zn-grading from R. The ring R0[Δ] iscalled the Stanley–Reisner ring of Δ with respect to R0.

13.1.6 Lemma. Assume that R is G-graded, let M be a graded R-module,and let p be a prime ideal of R.

(i) If p ∈ SuppM , then p* ∈ SuppM .(ii) If p ∈ AssM , then p is graded; in addition, p = (0 :R m) for some

homogeneous element m ∈M .(iii) In particular, if a is a graded ideal of R, then ass a consists of graded

prime ideals.

Proof. The special case of this lemma in which G = Z is proved in [7,Lemma 1.5.6(b)]. The same proof with minor modifications works in thisG-graded case provided one puts a total order ≺ on G compatible with ad-dition (see 13.1.2), and uses ≺ instead of the usual order < on Z.

13.1.7 Categorical Reminders. Assume that R is G-graded.

(i) The category *C(R) of all gradedR-modules and homogeneousR-hom-omorphisms is Abelian.

(ii) If f : M −→ N is a morphism in *C(R) (that is, if M and N aregraded R-modules and f is a homogeneous R-homomorphism), thenthe ordinary kernel and image of f are graded submodules of M and Nrespectively, and act as Ker f and Im f in *C(R).

(iii) A sequence M −→ N −→ P of objects and morphisms in *C(R) isexact in that category if and only if it is exact in C(R).


(iv) Projective (respectively injective) objects in the category *C(R) will bereferred to as *projective (respectively *injective) graded R-modules.The category *C(R) ‘has enough projectives’, for if M is a graded R-module, then M is a homogeneous homomorphic image of a gradedR-module

⊕i∈I R(gi), where (gi)i∈I is a family of elements of G, and

it is easy to see that⊕

i∈I R(gi) is *projective. A graded R-moduleof the form

⊕i∈I R(gi) will be called *free. That *C(R) ‘has enough

injectives’ is proved in §13.2 below.

13.1.8 �Exercise and Definitions. Assume that R is G-graded, and let M =⊕g∈GMg and N =

⊕g∈GNg be graded R-modules.

(i) Let g0 ∈ G. We say that an R-homomorphism f : M −→ N ishomogeneous of degree g0 precisely when f(Mg) ⊆ Ng+g0 for allg ∈ G. We denote by *HomR(M,N)g0 the set of all homogeneousR-homomorphisms from M to N of degree g0. Show that

(a) *HomR(M,N)g0 is an R0-submodule of HomR(M,N), and(b) the sum

∑g′∈G *HomR(M,N)g′ is direct.

(ii) We set

*HomR(M,N) :=∑g′∈G

*HomR(M,N)g′ =⊕g′∈G

*HomR(M,N)g′ .

Show that this is an R-submodule of HomR(M,N), and that the abovedirect decomposition turns *HomR(M,N) into a graded R-module.Deduce that *HomR( • , • ) : *C(R) × *C(R) −→ *C(R) is a leftexact, additive functor.

(iii) Show that, if M is finitely generated, then HomR(M,N) is actuallyequal to *HomR(M,N) with its grading forgotten.

(iv) Let i ∈ N0. In order to avoid lengthy excursions into the homologicalalgebra of the category *C(R), we shall define *ExtiR( • , N) to be thei-th right derived functor in *C(R) of *HomR( • , N).

Show that, if M is finitely generated, then ExtiR(M,N) is actuallyequal to *ExtiR(M,N) with its grading forgotten.

13.1.9 �Exercise. SupposeR isG-graded, and letM =⊕

g∈GMg andN =⊕g∈GNg be graded R-modules. Let (M ⊗R N)g , for a g ∈ G, be the Z-

submodule ofM⊗RN generated by all elementsmg1⊗ng2 , where g1, g2 ∈ Gare such that g1 + g2 = g, and mg1 ∈ Mg1 , ng1 ∈ Ng1 . It is clear thatM ⊗R N =

∑g∈G(M ⊗R N)g; the aim of this exercise is to show that this

sum is direct, and provides M ⊗R N with a structure as a graded R-module.

13.2 *Injective modules 257

(i) Let F be a *free R-module; thus F =⊕

i∈I R(gi), where (gi)i∈I is afamily of elements ofG. Show that the sumF⊗RN =

∑g∈G(F⊗RN)g

is direct, and that the decomposition

F ⊗R N =⊕g∈G

(F ⊗R N)g

provides a grading for F ⊗R N .(ii) Consider an exact sequence F1 −→ F0 −→ M −→ 0 in the category

*C(R), where F1 and F0 are *free R-modules. Use part (i) to show thatthe sum M ⊗R N =

∑g∈G(M ⊗R N)g is direct; deduce that the de-

composition M ⊗R N =⊕

g∈G(M ⊗R N)g provides a grading forM ⊗R N .

(iii) Deduce that ⊗R� can be considered as a functor

⊗R� : *C(R)× *C(R) −→ *C(R).

13.2 *Injective modules

One of the main aims of this section is to show that, when R is G-graded,the category *C(R) ‘has enough injectives’. Our route to this will involve theconcept of *injective envelope.

13.2.1 Definition. Assume thatR isG-graded. LetM be a graded submoduleof the graded R-module L.

(i) We say that L is a *essential extension of M precisely when B∩M = 0

for every non-zero graded submodule B of L. Such a *essential exten-sion of M is said to be proper if and only if it is not equal to M .

(ii) We say that L is a *injective envelope (or *injective hull) of M preciselywhen L is a *injective R-module and also a *essential extension of M .

13.2.2 Lemma. Assume that R is G-graded. Let M be a graded submoduleof the graded R-module L such that L is a *essential extension of M . Then,with the gradings forgotten, L is an essential extension of M .

Proof. Let 0 = x ∈ L. Write x as a sum of homogeneous elements x =

xg1 + · · · + xgr , where g1, . . . , gr are r different members of G, and 0 = xgifor all i = 1, . . . , r. We show by induction on r that there exists a homogeneouselement a ∈ R such that 0 = ax ∈ M ; this will suffice to prove the lemma.This is clear when r = 1, since then x is itself homogeneous, and soRx∩M =0 because L is a *essential extension of M .


Thus we suppose that r > 1 and make the obvious inductive assumption.There exists a homogeneous element b ∈ R such that 0 = bxgr ∈ M ; letx′ := x− xgr = xg1 + · · ·+ xgr−1 . If bx′ = 0, then bx = bxgr is a non-zeroelement of M , as wanted; otherwise, 0 = bx′, and bx′ has fewer than r non-zero homogeneous components; therefore, by the inductive hypothesis, thereis a homogeneous element c ∈ R such that 0 = cbx′ ∈ M . Then, since b andc are both homogeneous, 0 = cbx = cbx′ + cbxgr ∈M .

13.2.3 Proposition. Assume that R is G-graded, and let M be a gradedR-module. Then M is *injective if and only if M has no proper *essentialextension.

Proof. (⇒) Assume that M is *injective, and let ι : M −→ N be the in-clusion homomorphism from M into a *essential extension N of M . (ThusN is graded and ι is homogeneous.) Since M is *injective, there exists a ho-mogeneous R-homomorphism ϕ : N −→ M such that ϕ ◦ ι = IdM , andN = M

⊕Kerϕ. Since N is a *essential extension of M , we must have

Kerϕ = 0, so that N =M .(⇐) This can be proved just as in the ungraded case: see [7, Proposition

3.2.2].

In the next Theorem 13.2.4, we shall show that each graded R-module M(where R is G-graded) has a *injective envelope; in particular, this will showthat M can be embedded, by means of a homogeneous R-homomorphism,into a *injective R-module, and so prove that the category *C(R) has enoughinjectives. The proof of 13.2.4 is modelled on Bruns’ and Herzog’s proof in [7,Theorem 3.6.2] of the particular case in which G = Z.

13.2.4 Theorem. Assume that R is G-graded, and let M be a graded R-module.

(i) The gradedR-moduleM has a *injective envelope, which, with its grad-ing forgotten, is an R-submodule of E(M), the ordinary injective enve-lope of M .

(ii) Between any two *injective envelopes of M there is a homogeneousisomorphism which restricts to the identity map on M . We denote by*E(M) or *ER(M) one choice of *injective envelope of M .

(iii) Considered as (ungraded) R-module, *ER(M) is an essential extensionof M .

Proof. (i) Consider M as an ungraded R-module, and embed M into an un-graded injective R-module I; for example, we could take I to be E(M). Let Sdenote the set of all R-submodules N of I that carry a grading with respect to

13.2 *Injective modules 259

which M is a graded submodule of N and N is a *essential extension of M .There is a partial order ≤ on S defined as follows: for N1, N2 ∈ S, we declarethat N1 ≤ N2 if and only if N1 is a graded submodule of N2 (with the spec-ified gradings). By Zorn’s Lemma, S has a maximal member E; we note thatE is a graded R-module, a *essential extension of M , and an R-submoduleof I .

Suppose that E is not *injective; it then follows from 13.2.3 that E has aproper *essential extension E ⊂ E′. By 13.2.2, when the gradings are for-gotten, E′ is an essential extension of E. As I is injective, there exists anR-homomorphism (possibly not homogeneous) ψ : E′ −→ I that extends theinclusion homomorphism E

⊆−→ I . Since Kerψ ∩ E = 0, and E′ is an es-sential extension of E, we see that Kerψ = 0 and ψ is a monomorphism. Wecan now use the grading on E′ and the monomorphism ψ to put a grading onImψ with respect to which E is a homogeneous submodule of Imψ. SinceE ⊂ E′ was a proper *essential extension, it follows that Imψ is a proper*essential extension of E, and so a proper *essential extension of M , and wehave a contradiction to the maximality of E in S . This contradiction showsthat E is *injective.

(ii) Suppose that M ⊆ E1 and M ⊆ E2 are two *injective envelopes of M .Since E2 is *injective, there is a homogeneous R-homomorphism β : E1 −→E2 whose restriction toM is just the identity map onM . Since Kerβ∩M = 0,we can deduce that Kerβ = 0 because M ⊆ E1 is a *essential extension;therefore β is monomorphic, and Imβ is a *injective graded submodule ofE2, and so has no proper *essential extension, by 13.2.3. ButE2 is a *essentialextension of M , and therefore of Imβ; hence Imβ = E2, and β : E1 −→ E2

is a homogeneous isomorphism.(iii) This is immediate from 13.2.2, because *ER(M) is a *essential exten-

sion of M .

Now that we know that the category *C(R) has enough injectives, we canmake progress with multi-graded local cohomology. We shall build on theabove introduction of the concept of *injective envelope, but not until §14.2.

13.2.5 Lemma. Assume that R is G-graded, and let I,M be graded R-modules with I *injective.

(i) We have *ExtiR(M, I) = 0 for all i ∈ N.(ii) If M is finitely generated, then ExtiR(M, I) = 0 for all i ∈ N.

Proof. (i) There is a *projective graded R-module F and a homogeneousR-epimorphism ψ : F −→ M ; let K := Kerψ. Since F is *projective,


*ExtiR(F, I) = 0 for all i ∈ N. The exact sequence

0 −→ K −→ Fψ−→M −→ 0

therefore induces an exact sequence

0 −→ *HomR(M, I) −→ *HomR(F, I) −→ *HomR(K, I)

−→ *Ext1R(M, I) −→ 0

(in the category *C(R)) and homogeneous isomorphisms

*ExtiR(K, I) ∼= *Exti+1R (M, I) for all i ∈ N.

The induced homomorphism *HomR(F, I) −→ *HomR(K, I) is surjec-tive because I is *injective. Hence *Ext1R(M, I) = 0. ButM was an arbitrarygraded R-module; therefore *Ext1R(K, I) = 0, and so *Ext2R(M, I) = 0.Use induction to complete the proof of (i).

(ii) This is now immediate, since if, for finitely generated M , we forget thegrading on *ExtiR(M, I), then, by 13.1.8(iv), we obtain the ordinary ‘Ext’module ExtiR(M, I).

Lemma 13.2.5 already enables us to prove the following result, which willplay a key role in this chapter.

13.2.6 Proposition. Assume thatR isG-graded. Let I be a *injective gradedR-module. Let (Λ,≤) be a (non-empty) directed partially ordered set, and letB = (bα)α∈Λ be a system of ideals of R over Λ (as in 2.1.10), with theproperty that all ideals in the system are graded. Then I is ΓB-acyclic.

In particular, if the ideal a is graded, then I is Γa-acyclic.

Proof. For each α ∈ Λ, the gradedR-moduleR/bα is finitely generated, andso ExtiR(R/bα, I) = 0 for all i ∈ N, by 13.2.5. Hence, by 1.3.7,

HiB(I) ∼= lim

−→α∈Λ

ExtiR(R/bα, I) = 0 for all i ∈ N.

The following exercise establishes a G-graded analogue of the well-knownBaer Criterion (see [71, Theorem 3.20]) for a module to be injective.

13.2.7 �Exercise. Assume that R is G-graded, and let I be a graded R-module. Show that I is *injective if and only if, for each graded ideal b ofR, each g0 ∈ G and each homogeneous homomorphism f : b −→ I of degreeg0, there exists a homogeneous homomorphism f ′ : R −→ I of degree g0such that f ′� b = f .

13.3 The *restriction property 261

We made use of the (ungraded) Baer Criterion in our proof of Proposition2.1.4, and the G-graded version in the above exercise can be used in a similarway to establish a G-graded version of that proposition. This is addressed inthe next exercise.

13.2.8 �Exercise. Assume that R is G-graded and that the ideal a is graded.Let I be a *injective graded R-module. Show that Γa(I) is *injective.

Now let (Λ,≤) be a (non-empty) directed partially ordered set, and letB = (bα)α∈Λ be a system of graded ideals of R over Λ. Show that ΓB(I)

is *injective.

13.2.9 �Exercise. Assume that R is G-graded; let I be a graded R-module.Show that the following conditions are equivalent:

(i) I is *injective;(ii) *Ext1R(M, I) = 0 for all graded R-modules M ;

(iii) Ext1R(L, I) = 0 for all finitely generated graded R-modules L.

13.3 The *restriction property

13.3.1 Hypotheses for the section. We shall assume, throughout §13.3, thatR =⊕

g∈GRg is G-graded. (Recall that we always assume that R is Noethe-rian.) Also throughout this section, we shall let G′ be another finitely gener-ated torsion-free Abelian group, and we shall let R′ =

⊕g′∈G′ R′g′ be a G′-

graded commutative ring. It should be noted that we are not assuming that R′

is Noetherian; we shall assume that the reader is familiar with the elementaryproperties of graded R′-modules expounded in [61, §2.11 and §2.13].

13.3.2 Definition. A sequence (T i)i∈N0 of covariant functors from *CG(R)to *CG′

(R′) is said to be a negative connected sequence (respectively, a nega-tive strongly connected sequence) if the following conditions are satisfied.

(i) Whenever

0 −→ Lf−→M

g−→ N −→ 0

is an exact sequence in *CG(R), there are defined connecting homoge-neous R′-homomorphisms

T i(N) −→ T i+1(L) for all i ∈ N0


(in *CG′(R′)) such that the long sequence

0 T 0(L)T 0(f)

T 0(M)T 0(g)

T 0(N)

T 1(L)T 1(f)

T 1(M)T 1(g)

T 1(N)

· · · · · ·

T i(L)T i(f)

T i(M)T i(g)

T i(N)

T i+1(L) · · ·

� � �

� � �

�

� � �

� �

is a complex (respectively, is exact).(ii) Whenever

0 L M N 0� � � �

λ μ ν

0 L′ M ′ N ′ 0� � � ��

is a commutative diagram of graded R-modules and homogeneous R-homomorphisms with exact rows, then there is induced, by λ, μ and ν, achain map of the long complex of (i) for the top row into the correspond-ing long complex for the bottom row.

13.3.3 Examples. Let M be a graded R-module.

(i) Let T : *CG(R) −→ *CG′(R′) be an additive covariant functor. Since

the category *CG(R) has enough injectives (by 13.2.4), we can carry outa standard procedure of homological algebra in that category and formthe right derived functors RiT (i ∈ N0) of T . In more detail, for thegraded R-module M , we can construct an exact sequence

0 −→Mα−→ E0 d0

−→ E1 −→ · · · −→ Ei di

−→ Ei+1 −→ · · ·

in *CG(R) in which the Ei (i ∈ N0) are *injective graded R-modules;then we apply the functor T to the complex

0 −→ E0 d0

−→ E1 −→ · · · −→ Ei di

−→ Ei+1 −→ · · · ;

the i-th cohomology module of the resulting complex is the graded R′-moduleRiT (M) (for each i ∈ N0).

It should be noted that (RiT )i∈N0 is a negative strongly connectedsequence of covariant functors from *CG(R) to *CG′

(R′); furthermore,if T is left exact, thenR0T is naturally equivalent to T .


(ii) When a is graded, the a-torsion functor Γa can be viewed as a (left ex-act, additive) functor from *CG(R) to itself, and so we can form theconnected sequence (*Hi

a)i∈N0 of its right derived functors on that cat-egory. For each i ∈ N0, this leads to a graded local cohomology module*Hi

a(M): one of the aims of this chapter is to show that, if we forgetthe grading on *Hi

a(M), then the resulting R-module is isomorphic toHi

a(M).(iii) Another example is provided by a system of graded ideals of R. Let

(Λ,≤) be a (non-empty) directed partially ordered set, and let B =

(bα)α∈Λ be a system of graded ideals of R over Λ. The B-torsion func-tor ΓB can be viewed as a (left exact, additive) functor from *CG(R)to itself, and so we can form the connected sequence (*Hi

B)i∈N0 of itsright derived functors on that category. For each i ∈ N0, this leads toa graded generalized local cohomology module *Hi

B(M): in this chap-ter, we shall show that, if we forget the grading on *Hi

B(M), then theresulting R-module is isomorphic to Hi

B(M).

13.3.4 Definition. Let (T i)i∈N0 and (U i)i∈N0 be two negative connected se-quences of covariant functors from *CG(R) to *CG′

(R′). A homomorphismΨ : (T i)i∈N0 −→ (U i)i∈N0 of connected sequences is a family (ψi)i∈N0

where, for each i ∈ N0, ψi : T i −→ U i is a natural transformation offunctors, and which is such that the following condition is satisfied: when-ever 0 −→ L −→M −→ N −→ 0 is an exact sequence of gradedR-modulesand homogeneous R-homomorphisms, then, for each i ∈ N0, the diagram

T i(N) T i+1(L)�

ψiN ψi+1

L

U i(N) U i+1(L)��

(in which the horizontal maps are the appropriate connecting homomorphismsarising from the connected sequences) commutes.

Such a homomorphism Ψ = (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 of con-nected sequences is said to be an isomorphism (of connected sequences) pre-cisely when ψi : T i −→ U i is a natural equivalence of functors for eachi ∈ N0.

13.3.5 �Exercise. Let(T i)i∈N0

and(U i)i∈N0

be negative connected sequen-

ces of covariant functors from *CG(R) to *CG′(R′).


(i) Let ψ0 : T 0 −→ U0 be a natural transformation of functors. Assumethat

(a) the sequence (T i)i∈N0 is strongly connected, and(b) T i(I) = 0 for all i ∈ N and *injective graded R-modules I .

Show that there are uniquely determined natural transformations ψi :

T i −→ U i (i ∈ N) such that (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 is ahomomorphism of connected sequences.

(ii) Now let ψ0 : T 0 −→ U0 be a natural equivalence of functors. Assumethat

(a) the sequence (T i)i∈N0 is strongly connected,(b) the sequence (U i)i∈N0 is strongly connected, and(c) for all i ∈ N and *injective graded R-modules I , we have

T i(I) = U i(I) = 0.

Show that there exist uniquely determined natural equivalences ψi :

T i −→ U i (i ∈ N) such that (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 is anisomorphism of connected sequences.

13.3.6 Definition. Let T : C(R) −→ C(R′) be a covariant functor. If

(i) wheneverM is a gradedR-module, theR′-module T (M) is graded, and(ii) the gradings in (i) are such that, whenever f : M −→ N is a homo-

geneous homomorphism of graded R-modules, then T (f) : T (M) −→T (N) is homogeneous,

then we say that T has the *restriction property (with respect to the gradingsspecified in (i)).

Of course, the restriction of T to *CG(R) is a functor T �: *CG(R) −→C(R′). (We should, strictly speaking, denote T � by T � *CG(R), but we shalluse the shorter notation in the interests of simplicity.) Note that T has the*restriction property if and only if T � can be viewed as a functor from *CG(R)to *CG′

(R′).

13.3.7 Definition. Let T, U : C(R) −→ C(R′) be covariant functors. Letα : T −→ U be a natural transformation of functors (from C(R) to C(R′)).Of course, the restriction of α to *CG(R) provides a natural transformationα�: T �−→ U� of functors from *CG(R) to C(R′).

Now assume that both T and U have the *restriction property. We say that αhas the *restriction property precisely when, for every graded R-module M ,the map αM : T (M) −→ U(M) is homogeneous.


Note that this is the case if and only if α� can be viewed as a natural trans-formation α�: T �−→ U� of functors from *CG(R) to *CG′

(R′).

13.3.8 Remark. Observe that, if, in the notation of 13.3.7, α : T −→ U isa natural equivalence of functors which has the *restriction property, then theinverse natural equivalence α−1 : U −→ T also has the *restriction property.

Note also that, if V : C(R) −→ C(R′) is a third covariant functor havingthe *restriction property, and both α and a natural transformation β : U −→ V

have the *restriction property, then the composition β ◦α : T −→ V again hasthe *restriction property.

13.3.9 Definition. Let (T i)i∈N0 be a negative strongly connected sequence ofcovariant functors from C(R) to C(R′). We can regard (T i�)i∈N0 as a negativestrongly connected sequence of covariant functors from *CG(R) to C(R′).

We shall say that (T i)i∈N0 has the *restriction property precisely when

(i) T i has the *restriction property for all i ∈ N0; and(ii) whenever 0 −→ L −→ M −→ N −→ 0 is an exact sequence in the

category *CG(R), the connecting homomorphisms T i(N) −→ T i+1(L)

(i ∈ N0) (which exist by virtue of the fact that (T i)i∈N0 is a connectedsequence) are all homogeneous.

Note that this is the case if and only if (T i�)i∈N0 can be viewed as a negativestrongly connected sequence of covariant functors from *CG(R) to *CG′

(R′).

13.3.10 Definition. Let (T i)i∈N0 and (U i)i∈N0 be negative strongly connec-ted sequences of covariant functors from C(R) to C(R′); assume that boththese sequences have the *restriction property of 13.3.9. Let Ψ := (ψi)i∈N0 :

(T i)i∈N0 −→ (U i)i∈N0 be a homomorphism of connected sequences. We saythat Ψ has the *restriction property if and only if, for all i ∈ N0, the naturaltransformation ψi has the *restriction property of 13.3.7.

13.3.11 Proposition. Let V,M be graded R-modules with V finitely gen-erated. By 13.1.8(iv), for each i ∈ N0, ExtiR(V,M) is actually the gradedR-module *ExtiR(V,M) with its grading forgotten; hence each ExtiR(V,M)

has a natural structure as a graded R-module, and these structures are suchthat, if V ′ is a second finitely generated graded R-module and h : V −→ V ′

is a homogeneous homomorphism, then ExtiR(h,M) : ExtiR(V′,M) −→

ExtiR(V,M) is also homogeneous.Moreover, with respect to these graded R-module structures, the negative

strongly connected sequence(ExtiR(V, • )

)i∈N0

of covariant functors (fromC(R) to itself) has the *restriction property (see 13.3.9).


Proof. Only the claim in the second paragraph requires proof. Let F• be a*free resolution of V in *C(R) by *free graded R-modules of finite rank.

An exact sequence 0 −→ L′ −→ M ′ −→ N ′ −→ 0 in C(R) induces asequence

0 −→ HomR(F•, L′) −→ HomR(F•,M

′) −→ HomR(F•, N′) −→ 0

of complexes of R-modules and chain maps of such complexes such that, foreach i ∈ N0, the sequence

0 −→ HomR(Fi, L′) −→ HomR(Fi,M

′) −→ HomR(Fi, N′) −→ 0

(where Fi denotes the i-th term of F•) is exact. It is straightforward to usethe long exact sequences of cohomology modules induced by such sequencesof complexes to turn

(Hi(HomR(F•, • ))

)i∈N0

into a negative strongly con-nected sequence of covariant functors from C(R) to itself, and a standard ap-plication of Exercise 1.3.4(ii) then shows that this negative connected sequenceis isomorphic to

(ExtiR(V, • )

)i∈N0

.In particular, when we apply this to a morphism f : M −→ M ′ in *C(R),

and to an exact sequence 0 −→ L −→ M −→ N −→ 0 of graded R-modules and homogeneous R-homomorphisms, there result, for each i ∈ N0,commutative diagrams

Hi(HomR(F•,M)) Hi(HomR(F•,M′))�

∼= ∼=

ExtiR(V,M) ExtiR(V,M′)

ExtiR(V,f)��

and

Hi(HomR(F•, N)) Hi+1(HomR(F•, L))�

∼= ∼=

ExtiR(V,N) Exti+1R (V, L) .�

� �

In these circumstances, HomR(F•,M), HomR(F•,M′), HomR(F•, N) and

HomR(F•, L) are complexes of gradedR-modules and homogeneousR-hom-omorphisms, so that all their cohomology modules are graded, and it is easyto check that the top horizontal homomorphisms in the above two commu-tative diagrams are homogeneous. Furthermore, the graded module structures


on ExtiR(V,M), ExtiR(V,M′), ExtiR(V,N) and Exti+1

R (V, L) induced by thevertical isomorphisms in the diagrams are precisely the graded module struc-tures referred to in the first paragraph of the statement of the proposition. Allthe claims follow easily from these observations.

13.3.12 Remark. Let (Ω,≤) be a (non-empty) directed partially ordered set,and let (Wω)ω∈Ω be a direct system of graded R-modules and homogeneousR-homomorphisms over Ω, with constituent R-homomorphisms hων : Wν →Wω (for each (ω, ν) ∈ Ω × Ω with ω ≥ ν). We can forget the gradings andcalculate the ordinary direct limit

W∞ := lim−→ω∈Ω

Wω,

in C(R): let hω : Wω −→ W∞ be the canonical map (for each ω ∈ Ω). Itshould be noted that W∞ inherits a natural grading, for which all the hω (ω ∈Ω) are homogeneous, and thatW∞ actually acts as the direct limit of the directsystem (Wω)ω∈Ω in the category *C(R).

13.3.13 Examples. The following examples will be important for us.

(i) Let (Λ,≤) be a (non-empty) directed partially ordered set, and let B =

(bα)α∈Λ be a system of graded ideals of R over Λ. For α, β ∈ Λ withα ≥ β, the natural homomorphism hαβ : R/bα −→ R/bβ is homo-geneous. It follows from 13.3.11 and 13.3.12 that the negative stronglyconnected sequence of covariant functors(

lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

from C(R) to itself has the *restriction property.(ii) In particular, when the ideal a is graded, the negative strongly connected

sequence of covariant functors(lim−→n∈N

ExtiR(R/an, • )

)i∈N0

from C(R) to itself has the *restriction property.(iii) Assume that a is graded, and let a1, . . . , an (where n > 0) denote n

homogeneous elements which generate a. Let C• denote the Cech com-plex of R with respect to a1, . . . , an, as in 5.1.5. Then it is straightfor-ward to check that the negative strongly connected sequence of covariantfunctors (Hi(( • ) ⊗R C

•))i∈N0 from C(R) to itself has the *restrictionproperty.


13.3.14 �Exercise. Let (Λ,≤) be a (non-empty) directed partially orderedset, and let B = (bα)α∈Λ be a system of graded ideals of R over Λ. Showthat the functor DB of 2.2.3 has the *restriction property, and that the naturaltransformation ηB of 2.2.6(i) has the *restriction property.

Deduce that, when the ideal a is graded, the functor Da of 2.2.1 has the*restriction property, and the natural transformation ηa of 2.2.6(i) has the *rest-riction property.

13.3.15 Theorem. Let (T i)i∈N0 be a negative strongly connected sequence ofcovariant additive functors from C(R) to C(R′). Suppose that, for each gradedR-module M , a grading is given on T 0(M), and that, with respect to thesegradings, T 0 has the *restriction property. Suppose further that T i(I) = 0 forall i ∈ N whenever I is a *injective graded R-module.

Then there is exactly one choice of gradings on the T i(M) (i ∈ N, M agraded R-module) with respect to which (T i)i∈N0 has the *restriction prop-erty.

Furthermore, if we denote by *T 0 : *CG(R) −→ *CG′(R′) the (neces-

sarily left exact) functor induced by T 0� (see 13.3.6), then there is a uniqueisomorphism (ψi)i∈N0 : (T i�)i∈N0

∼=−→ (Ri(*T 0))i∈N0 of negative connectedsequences of covariant functors from *CG(R) to *CG′

(R′) for which ψ0 :

T 0� = *T 0 −→ R0(*T 0) is the canonical natural equivalence.

Proof. We first show, by induction on t ∈ N0, that, if (T i)i∈N0 has the*restriction property, then the gradings on theR′-modules T t(M) (M a gradedR-module) are uniquely determined by the gradings on the R′-modules T 0(L)

(L a gradedR-module). This statement is certainly true when t = 0, and so wesuppose now that t > 0 and the statement is true for smaller values of t.

Let M be an arbitrary graded R-module. Since the category *CG(R) hasenough injective objects, there is an exact sequence

0 −→M −→ Ig−→ N −→ 0

in *CG(R) with I a *injective graded R-module. The hypotheses and the as-sumption that (T i)i∈N0 has the *restriction property therefore lead to an exactsequence

T t−1(g)T t−1(I) T t−1(N) T t(M) 0� � �

of graded R′-modules and homogeneous homomorphisms. Since, by the in-ductive hypothesis, the grading on T t−1(N) is uniquely determined by ourassumptions, it follows that the grading on T t(M) is similarly uniquely deter-mined.


Thus there is at most one choice of gradings on the T i(M) (i ∈ N, M agradedR-module) with respect to which (T i)i∈N0 has the *restriction property.We still have to show that there is one such choice.

Since (T i)i∈N0is a negative strongly connected sequence of covariant func-

tors from C(R) to C(R′), it is automatic that T 0 is left exact. Therefore *T 0 :

*CG(R) −→ *CG′(R′) is left exact (and additive), (Ri(*T 0))i∈N0 is a nega-

tive strongly connected sequence of functors from *CG(R) to *CG′(R′), and

there is a canonical natural equivalence ψ0 : T 0� = *T 0 −→ R0(*T 0). Notethat ψ0

M is homogeneous for each graded R-module M .Now forget, just temporarily, the grading on R′: then (Ri(*T 0))i∈N0 and

(T i�)i∈N0 are negative strongly connected sequences of covariant functorsfrom *CG(R) to C(R′) and ψ0 : T 0� −→ R0(*T 0) is a natural equivalence(of functors from *CG(R) to C(R′)). Moreover, T i(I) = Ri(*T 0)(I) = 0 forall i ∈ N and all *injective graded R-modules I .

At this point, regard, again temporarily,R′ as trivially graded, that is, gradedby 0: we can use 13.3.5(ii) to see that there exist uniquely determined nat-ural equivalences ψi : T i� −→ Ri(*T 0) (i ∈ N) such that (ψi)i∈N0 :

(T i� )i∈N0 −→ (Ri(*T 0))i∈N0 is an isomorphism of connected sequences(of functors from *CG(R) to C(R′) = *C0(R′)).

Now remember the original G′-gradings on R′ and its graded modules: foreach graded R-module M and each i ∈ N, the R′-module Ri(*T 0)(M) isgraded. For such i and M , we can define a grading on T i(M) in such a waythat the R′-isomorphism

ψiM : T i(M)

∼=−→ Ri(*T 0)(M)

is homogeneous. With these gradings in place, all the remaining claims of thetheorem follow routinely.

13.3.16 Corollary. Let T be a left exact additive covariant functor from C(R)to C(R′). Suppose that, for each graded R-module M , a grading is given onT (M), and that, with respect to these gradings, T has the *restriction property.Suppose further that each *injective graded R-module I is T -acyclic, that is,RiT (I) = 0 for all i ∈ N.

Then there is exactly one choice of gradings on the RiT (M) (i ∈ N, M agraded R-module) with respect to which (RiT )i∈N0 has the *restriction prop-erty.

Furthermore, if we denote by *T : *CG(R) −→ *CG′(R′) the functor in-

duced by T �, then there is a unique isomorphism

(ψi)i∈N0 : ((RiT )�)i∈N0

∼=−→ (Ri(*T ))i∈N0


of negative connected sequences of functors from *CG(R) to *CG′(R′) for

which ψ0 : T � = *T −→ *T is the identity. �

13.3.17 Theorem. Let

(T i)i∈N0 and (U i)i∈N0

be negative strongly connected sequences of covariant additive functors fromC(R) to C(R′) which have the *restriction property of 13.3.9, and supposethat T i(I) = 0 for all i ∈ N whenever I is a *injective graded R-module.

Let Ψ := (ψi)i∈N0 : (T i)i∈N0 −→ (U i)i∈N0 be a homomorphism of con-nected sequences. Suppose that ψ0 has the *restriction property of 13.3.7.Then it is automatic that ψi has the *restriction property for all i ∈ N0, so thatΨ has the *restriction property of 13.3.10.

Furthermore, (ψi�)i∈N0 : (T i�)i∈N0 −→ (U i�)i∈N0 is the unique extensionof the natural transformation ψ0�: T 0�−→ U0� (of functors from *CG(R)to *CG′

(R′)) to a homomorphism of connected sequences of functors from*CG(R) to *CG′

(R′).

Proof. We prove by induction on t that ψt has the *restriction property forall t ∈ N0. We know that ψ0 has the *restriction property, and so we supposenow that t > 0 and that ψi has the *restriction property for all i < t.

Let M be an arbitrary graded R-module. Since the category *CG(R) hasenough injective objects, there is an exact sequence

0 −→M −→ Ig−→ N −→ 0

in *CG(R) with I a *injective graded R-module. The hypotheses thereforelead to a commutative diagram

T t−1(g)T t−1(I) T t−1(N) T t(M) 0� � �

Ut−1(g)U t−1(I) U t−1(N) U t(M) U t(I)� � �

� � �

ψt−1I ψt−1

Nψt

M

of graded R′-modules and homogeneous homomorphisms with exact rows. Itfollows from the inductive hypothesis, and the fact that (T i)i∈N0 and (U i)i∈N0

have the *restriction property, that all the homomorphisms in the above dia-gram, with the possible exception of ψt

M , are homogeneous. It is immediatefrom the fact that the top row is exact that ψt

M must be homogeneous too.The claim in the final paragraph is immediate from 13.3.5, 13.3.7 and 13.3.9.

13.4 The reconciliation 271

13.4 The reconciliation

13.4.1 Hypotheses for the section. We shall assume, throughout §13.4, thatR =⊕

g∈GRg is G-graded, and that the ideal a is graded.Also, we shall assume that (Λ,≤) is a (non-empty) directed partially ordered

set, and that B = (bα)α∈Λ is a system of graded ideals of R over Λ.

13.4.2 Theorem. There is a unique choice of gradings on the HiB(M) (i ∈

N,M a gradedR-module) with respect to which (HiB)i∈N0 has the *restriction

property; furthermore, when these gradings are imposed, there is a uniqueisomorphism

(φi)i∈N0 : (HiB�)i∈N0

∼=−→ (*HiB)i∈N0

of connected sequences of covariant functors from *C(R) to itself for which φ0

is the identity.

Proof. By 13.2.6, each *injective graded R-module I is ΓB-acyclic; also, itis clear that ΓB : C(R) −→ C(R) has the *restriction property. The resulttherefore follows from 13.3.16.

We record separately a very important special case of 13.4.2.

13.4.3 Corollary. There is a unique choice of gradings on theHia(M) (i ∈ N,

M a graded R-module) with respect to which (Hia)i∈N0

has the *restrictionproperty; furthermore, when these gradings are imposed, there is a uniqueisomorphism

(ψi)i∈N0 : (Hia�)i∈N0

∼=−→ (*Hia)i∈N0

of connected sequences of covariant functors from *C(R) to itself for whichψ0 is the identity. �

13.4.4 Remark. For a graded R-module M , we do, of course, grade theHi

B(M) (i ∈ N) and the Hia(M) (i ∈ N) using the unique gradings for which

the conclusions of 13.4.2 and 13.4.3 are satisfied.Thus, during the rest of the book, whenever our current assumptions about

gradings (that R is G-graded, and a and all the ideals in B are graded) are inforce, we shall, without further ado, utilize the gradings on the Hi

a(M) andHi

B(M) whenever M is a graded R-module; for each g ∈ G, we shall denotethe g-th component of the graded R-module Hi

a(M) (respectively HiB(M))

by Hia(M)g (respectively Hi

B(M)g). (In many of the examples that we shallconsider, G will actually be the additive group Z of integers.) We shall alsomake much use of the fact that, whenever 0 −→ L −→M −→ N −→ 0 is anexact sequence of graded R-modules and homogeneous homomorphisms, not


only are all the terms in the induced long exact sequence of local cohomologymodules with respect to a (respectively generalized local cohomology mod-ules with respect to B) graded, but all the homomorphisms in this sequence,including the connecting homomorphisms, are homogeneous. These are reallycrucial facts about graded local cohomology.

In the introduction to this chapter, we posed three questions about variouspossible approaches to the construction of gradings on local cohomology mod-ules. The next theorem will enable us to answer those questions.

13.4.5 Theorem. Let (T i)i∈N0 be a negative strongly connected sequence ofcovariant additive functors from C(R) to itself. Suppose that

Ω = (ωi)i∈N0 : (T i)i∈N0

∼=−→ (HiB)i∈N0

is an isomorphism of connected sequences (from C(R) to itself). Suppose that,for each graded R-module M , a grading is given on T 0(M), and that, withrespect to these gradings, T 0 has the *restriction property. Suppose also thatω0 has the *restriction property.

Then there is exactly one choice of gradings on the T i(M) (i ∈ N, M agraded R-module) with respect to which (T i)i∈N0 has the *restriction prop-erty; with these gradings (and those of 13.4.4) imposed, Ω has the *restrictionproperty of 13.3.10.

Proof. By 13.2.6, we have T i(I) = HiB(I) = 0 for all i ∈ N whenever I

is a *injective graded R-module. Therefore, by 13.3.15, there is exactly onechoice of gradings on the T i(M) (i ∈ N, M a graded R-module) with respectto which (T i)i∈N0 has the *restriction property.

By 13.4.4, the connected sequence (HiB)i∈N0 has the *restriction property.

The final claim therefore follows from 13.3.17.

13.4.6 Remarks. Here, we finally answer the three questions posed in theintroduction to this chapter.

(i) We apply Theorem 13.4.5 to the isomorphism

ΦB =(φiB)i∈N0

:

(lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

∼=−→(Hi

B

)i∈N0

of negative strongly connected sequences of functors (from C(R) to it-self) of 2.2.2. It is clear from the definition of φ0B in 1.2.11(ii) that it hasthe *restriction property. Hence Theorem 13.4.5 can be applied: there isexactly one choice of gradings (and that was described in 13.3.13(i)) on

13.4 The reconciliation 273

the lim−→α∈Λ

ExtiR(R/bα,M) (i ∈ N, M a graded R-module) with respect

to which (lim−→α∈Λ

ExtiR(R/bα, • )

)i∈N0

has the *restriction property; with respect to these gradings, ΦB has the*restriction property.

(ii) The following special case of part (i) is worthy of separate mention. Wecan apply Theorem 13.4.5 to the isomorphism

Φa =(φia

)i∈N0

:

(lim−→n∈N

ExtiR(R/an, • )

)i∈N0

∼=−→(Hi

a

)i∈N0

of negative strongly connected sequences of functors (from C(R) to it-self) of 1.3.8. It is clear from the definition of φ0a in 1.2.11(iii) that it hasthe *restriction property. Hence Theorem 13.4.5 can be applied: there isexactly one choice of gradings (and that was described in 13.3.13(ii)) onthe lim

−→n∈N

ExtiR(R/an,M) (i ∈ N,M a graded R-module) with respect to

which (lim−→n∈N

ExtiR(R/an, • )

)i∈N0

has the *restriction property; with these gradings, Φa has the *restrictionproperty.

(iii) Let a1, . . . , an (where n > 0) denote n homogeneous elements whichgenerate a. Let C• denote the Cech complex of R with respect to theelements a1, . . . , an, as in 5.1.5. It is easy to check that Theorem 13.4.5can be applied to the isomorphism

((γi)−1)i∈N0 : (Hi(( • )⊗R C•))i∈N0

∼=−→ (Hia)i∈N0

of negative strongly connected sequences of functors (from C(R) to it-self) of 5.1.20. We conclude that there is exactly one choice of gradings(and that was described in 13.3.13(iii)) on the Hi(M ⊗R C•) (i ∈ N,M a graded R-module) with respect to which (Hi(( • ) ⊗R C•))i∈N0

has the *restriction property; with these gradings, ((γi)−1)i∈N0 has the*restriction property.

(iv) In short, if we use the isomorphism of connected sequences

Φa =(φia

)i∈N0

:

(lim−→n∈N

ExtiR(R/an, • )

)i∈N0

∼=−→(Hi

a

)i∈N0


of 1.3.8 to define gradings on the Hia(M) (i ∈ N) (for a graded R-

module M ), or if we use the isomorphism of connected sequences

((γi)−1)i∈N0 : (Hi(( • )⊗R C•))i∈N0

∼=−→ (Hia)i∈N0

of 5.1.20, with any choice of homogeneous generators for a, to definegradings on the Hi

a(M) (i ∈ N), the resulting gradings are alwaysthe same, and are precisely those with respect to which (Hi

a)i∈N0 hasthe *restriction property: see 13.4.3. (Note also that it would make nodifference if we used C(M)•, the Cech complex of M with respect toa1, . . . , an, instead ofM⊗RC

•: the constituent isomorphisms in the iso-morphism of complexes of 5.1.11 are all homogeneous.) Furthermore,as, when these gradings are imposed, there is a unique isomorphism(ψi)i∈N0 : (Hi

a�)i∈N0

∼=−→ (*Hia)i∈N0 of connected sequences of co-

variant functors from *C(R) to itself for which ψ0 is the identity (seeagain 13.4.3), all the questions posed in the introduction to this chapterhave been answered affirmatively!

13.5 Some examples and applications

In our discussion of examples, we shall sometimes wish to change gradingsand use a ‘less fine’ or ‘coarser’ grading than one that occurs naturally. In thisconnection, the notation and terminology introduced in the following definitionwill be helpful.

13.5.1 Definition. Let φ : G −→ H be a homomorphism of finitely gen-erated torsion-free Abelian groups, and suppose that R =

⊕g∈GRg is G-

graded.For each h ∈ H , let Rφ

h :=⊕

g∈φ−1({h})Rg. Then

Rφ :=⊕h∈H

Rφh =⊕h∈H

⎛⎝ ⊕g∈φ−1({h})

Rg

⎞⎠provides an H-grading on R, and we denote R by Rφ when considering it asan H-graded ring in this manner.

Furthermore, for each G-graded R-module M =⊕

g∈GMg, let Mφh :=⊕

g∈φ−1({h})Mg and Mφ :=⊕

h∈H Mφh ; then Mφ is an H-graded Rφ-

module. Also, if f : M −→ N is a G-homogeneous homomorphism of G-graded R-modules, then the same map f becomes an H-homogeneous homo-morphism of H-graded Rφ-modules fφ :Mφ −→ Nφ.

13.5 Some examples and applications 275

In this way, ( • )φ becomes an exact additive covariant functor from *CG(R)to *CH(R). We refer to it as the φ-coarsening functor, and we shall consider theH-gradings constructed in this way as a coarsening of the given G-gradings.

The following lemma will help us to exploit coarsenings of graded localcohomology modules.

13.5.2 Lemma. Let the situation be as in 13.5.1, so that R is G-graded andφ : G −→ H is a homomorphism of finitely generated torsion-free Abeliangroups. Assume that a is graded. Then(

Hia( • )φ)i∈N0

and(Hi

aφ(( • )φ))i∈N0

are isomorphic connected sequences of functors from *CG(R) to *CH(R).Consequently, for each G-graded R-module M and each i ∈ N0, there is

an H-homogeneous isomorphism Hia(M)φ ∼= Hi

aφ(Mφ).

Proof. Because ( • )φ is an exact additive covariant functor from *CG(R) to*CH(R), one easily sees that

(Hi

a( • )φ)i∈N0

and(Hi

aφ(( • )φ))i∈N0

are both

negative strongly connected sequences of covariant functors from *CG(R) to*CH(R). Now Γa( • )φ and Γaφ(( • )φ) are the same functor. Also, whenever Iis a *injective G-graded R-module, we can see that Hi

a(I)φ = 0 = Hi

aφ(Iφ)

for all i > 0: the first of these claims is immediate, whereas the second followsfrom 13.4.3 because, without the H-grading, Hi

aφ(Iφ) ∼= Hi

a(I), and I is Γa-acyclic (by 13.2.6). The result therefore follows from 13.3.5.

An illustration of the idea of coarsening occurs with our first example inthis section; in this example, we show that the Cech complex approach canquickly lead to important information about graded local cohomology over apolynomial ring.

13.5.3 Example. Let S =⊕

g∈G Sg be aG-graded commutative Noetherianring, let n ∈ N, and let R := S[X1, . . . , Xn], the ring of polynomials over S,graded by G⊕ Zn as in 13.1.4(iii). The reader should note the special case inwhich S = S0 is trivially graded, in which case the grading on R is (essen-tially) as described in 13.1.4(i), with S0 playing the role of R0.

We propose to use the Cech complex of R with respect to the homogeneouselements X1, . . . , Xn,

C• : 0 −→ C0 d0

−→ C1 −→ · · · −→ Ci di

−→ Ci+1 −→ · · · dn−1

−→ Cn −→ 0,

in conjunction with 13.4.6 above, to calculate the graded R-module

Hn(X1,...,Xn)

(S[X1, . . . , Xn]) = Hn(X1,...,Xn)

(R).


There is a homogeneous R-isomorphism between this local cohomology mod-ule and

RX1...Xn/(∑n

t=1dn−1(RX1...Xt−1Xt+1...Xn

)).

Now RX1...Xn is a free S-module with(Xi1

1 . . . Xinn

)(i1,...,in)∈Zn as a base;

for each t = 1, . . . , n, its R-submodule dn−1(RX1...Xt−1Xt+1...Xn

)is again

free as S-module, with base(Xi1

1 . . . Xinn

)(i1,...,in)∈Zn, it≥0

. Use −N to de-note the set {n ∈ Z : n < 0}. It follows that the graded R-module

RX1...Xn/(∑n

t=1dn−1(RX1...Xt−1Xt+1...Xn

))can be considered as a free S-module with base(

Xi11 . . . Xin

n

)(i1,...,in)∈(−N)n

and R-module structure such that, for (i1, . . . , in) ∈ (−N)n and t ∈ N with1 ≤ t ≤ n,

Xt(Xi11 . . . Xin

n ) =

{Xi1

1 . . . Xit−1

t−1 Xit+1t X

it+1

t+1 . . . Xinn if it < −1,

0 if it = −1.

The (G⊕ Zn)-grading is such that, for g ∈ G and sg ∈ Sg \ {0},

deg(sgXi11 . . . Xin

n ) = (g, (i1, . . . , in)) for all (i1, . . . , in) ∈ (−N)n.

We refer to this graded R-module as the module of inverse polynomials inX1, . . . , Xn over S, and denote it by S[X−1 , . . . , X

−n ]. To summarize, the

Cech complex approach to the calculation of graded local cohomology mod-ules quickly yields a (G⊕ Zn)-homogeneous S[X1, . . . , Xn]-isomorphism

Hn(X1,...,Xn)

(S[X1, . . . , Xn]) ∼= S[X−1 , . . . , X−n ].

We could also regard R = S[X1, . . . , Xn] as (G⊕ Z)-graded, with

R(g,m) =⊕

(i1,...,in)∈N0n

i1+···+in=m

R(g,(i1,...,in)) for all (g,m) ∈ G⊕ Z.

However, with this grading, our polynomial ring is just the result Rφ of apply-ing the φ-coarsening functor of 13.5.1 to R, where φ : G ⊕ Zn −→ G ⊕ Z isthe Abelian group homomorphism for which

φ((g, (i1, . . . , in))) = (g, i1 + · · ·+ in) for all (g, (i1, . . . , in)) ∈ G⊕ Zn.


Thus, in view of 13.5.2, there are (G⊕ Z)-homogeneous R-isomorphisms

Hn(X1,...,Xn)φ

(S[X1, . . . , Xn]φ) ∼=(Hn

(X1,...,Xn)(S[X1, . . . , Xn])

)φ∼=(S[X−1 , . . . , X

−n ])φ,

where, in the right-hand module, for (i1, . . . , in) ∈ (−N)n and sg ∈ Sg \ {0}(where g ∈ G), the element sgXi1

1 . . . Xinn has degree (g, i1 + · · ·+ in).

The reader should note the special case of the above in which S is triviallygraded, so that S = S0 = R0: when R is considered to be Zn-graded asdescribed in 13.1.4(i), the isomorphism

Hn(X1,...,Xn)

(R0[X1, . . . , Xn]) ∼= R0[X−1 , . . . , X

−n ]

is Zn-homogeneous and, in the module of inverse polynomials,

deg(Xi11 . . . Xin

n ) = (i1, . . . , in) for all (i1, . . . , in) ∈ (−N)n;

when R is considered to be Z-graded, with degXi = 1 for all i = 1, . . . , n,then the isomorphism

Hn(X1,...,Xn)

(R0[X1, . . . , Xn]) ∼= R0[X−1 , . . . , X

−n ]

is Z-homogeneous and, in the module of inverse polynomials,Xi11 . . . Xin

n hasdegree i1 + · · ·+ in for all (i1, . . . , in) ∈ (−N)n.

13.5.4 �Exercise. Assume that R is G-graded and that the ideal a is graded.Let (Λ,≤) be a (non-empty) directed partially ordered set, and B = (bα)α∈Λbe a system of graded ideals of R over Λ. By 13.3.14, the functor DB of 2.2.3has the *restriction property, and the natural transformation ηB of 2.2.6(i) hasthe *restriction property. In particular, Da has the *restriction property and ηahas the *restriction property.

(i) Show that the natural transformation ζ0B of 2.2.6 also has the *restrictionproperty, so that, for each graded R-module M , all the homomorphismsin the exact sequence

0 � � � � �ξBM ηBM

ΓB(M) M DB(M)ζ0BM

H1B(M) 0

are homogeneous. Note that, as a special case, this exercise shows thatζ0a (see 2.2.6(i)) has the *restriction property.

(ii) Let e : M −→ M ′ be a homogeneous homomorphism of graded R-modules such that Ker e and Coker e are both B-torsion. Show that the


unique R-homomorphism ψ′ :M ′ → DB(M) for which the diagram

Me

M ′�

ηBMψ′

DB(M)

��

commutes (see 2.2.15) is homogeneous.

The special case of this result in which B is taken to be the system formedby the powers of a should be noted.

13.5.5 �Exercise. Assume that R is G-graded and let a be a homogeneouselement of R. Show that the natural equivalence of functors

ω′ : DRa = lim−→n∈N

HomR(Ran, • ) −→ ( • )a

of 2.2.19 has the *restriction property, and deduce that, for a graded R-moduleM , the isomorphism

H1Ra(M) ∼=Ma/(M/ΓRa(M))

of 2.2.21(i) is homogeneous.

13.5.6 Exercise. Assume that R is G-graded and that the ideal a is graded.Let a1, . . . , an (where n > 0) denote n homogeneous elements which generatea. Let u ∈ N and let K(au)• denote the Koszul complex of R with respect toau1 , . . . , a

un, as in 5.2.1.

(i) Show that K(au)• has the structure of a complex of graded R-modulesand homogeneous R-homomorphisms in which e1 ∧ . . .∧ en has degree0, the element 1 ∈ K(au)0 has degree −u

∑ni=1 deg ai, and, for k ∈

{1, . . . , n} with k < n and i ∈ I(k, n) (the notation is as in 5.1.4), thedegree of ei(1)∧. . .∧ei(k) is−u

∑n−kh=1 deg aj(h), where j ∈ I(n−k, n)

is the n-complement of i (see 5.1.4).(ii) Let

(δi)i∈N0 :

(lim−→u∈N

Hn−i(K(au, • )•)

)i∈N0

∼=−→(Hi

a

)i∈N0

be the isomorphism of connected sequences of functors (from C(R) to

itself) of Theorem 5.2.9. Show that

(lim−→u∈N

Hn−i(K(au, • )•)

)i∈N0

has


the *restriction property, and that (δi)i∈N0has the *restriction property,

so that, for each graded R-module M , the isomorphism

δiM : lim−→u∈N

Hn−i(K(au,M)•)∼=−→ Hi

a(M)

is homogeneous.

13.5.7 �Exercise. Assume that R is G-graded and that the ideal a is graded.It follows from 13.3.14 that Da has the *restriction property.

(i) Use 13.3.11 and 13.3.12 to show that the negative strongly connectedsequence of covariant functors(

lim−→n∈N

ExtiR(an, • )

)i∈N0

from C(R) to itself has the *restriction property.(ii) Recall from 2.2.4 that there is a unique isomorphism of connected se-

quences (of functors from C(R) to itself)

Ψa =(ψia

)i∈N0

:(RiDa

)i∈N0

∼=−→(lim−→n∈N

ExtiR(an, • )

)i∈N0

which extends the identity natural equivalence from Da to itself.Let M be a graded R-module. We define structures as graded R-

modules on the RiDa(M) (i ∈ N) so that the R-isomorphism ψiaM is

homogeneous for all i ∈ N. A consequence of this definition is that theconnected sequence (RiDa)i∈N0 from C(R) to itself has the *restrictionproperty.

Denote by *Da : *C(R) −→ *C(R) the functor induced by Da�.Show that the three strongly connected sequences of covariant functors(from *C(R) to itself)

(Ri(*Da)

)i∈N0

,((RiDa)�

)i∈N0

and

(lim−→n∈N

ExtiR(an, • )�)

i∈N0

are isomorphic.

(iii) Let i ∈ N. Show that the natural equivalence γi : RiDa

∼=−→ Hi+1a of

2.2.6(ii) has the *restriction property, so that, for each graded R-moduleM , there is a homogeneous isomorphism RiDa(M) ∼= Hi+1

a (M) ofgraded R-modules.


13.5.8 Exercise. Generalize 13.5.7 to systems of graded ideals.In detail, assume that R is G-graded, let (Λ,≤) be a (non-empty) directed

partially ordered set, and let B = (bα)α∈Λ be a system of graded ideals of Rover Λ. By 13.3.14, the functor DB of 2.2.3 has the *restriction property.

(i) Recall from 2.2.4 that (lim−→α∈Λ

ExtiR(bα, • )

)i∈N0

is a negative strongly connected sequence of covariant functors fromC(R) to itself; show that it has the *restriction property.

(ii) Recall from 2.2.4 that there is a unique isomorphism of connected se-quences (of functors from C(R) to itself)

ΨB =(ψiB

)i∈N0

:(RiDB

)i∈N0

∼=−→(lim−→α∈Λ

ExtiR(bα, • )

)i∈N0

which extends the identity natural equivalence from DB to itself. De-duce that the first connected sequence here has the *restriction property.

Denote by *DB : *C(R) −→ *C(R) the functor induced by DB�.Show that the three strongly connected sequences of covariant functors(from *C(R) to itself)

(Ri(*DB)

)i∈N0

,((RiDB)�

)i∈N0

and

(lim−→α∈Λ

ExtiR(bα, • )�)

i∈N0

are isomorphic.(iii) Let i ∈ N. Show that there is a natural equivalence

γiB : RiDB

∼=−→ Hi+1B

that has the *restriction property.

Up to this point, this chapter has been fairly technical. In contrast, we endthe chapter with some concrete illustrations of some of the ideas developed sofar: the next exercise introduces the important concept of Veronesean subring,and this idea is involved in Example 13.5.12 and Exercise 13.5.13.

13.5.9 �Exercise: Veronesean subrings and functors. Assume that R =⊕n∈ZRn is Z-graded and that the ideal a is graded. Let r ∈ N and s ∈ Z be

fixed. DefineR(r) :=⊕

n∈ZRrn: thenR(r) is a subring ofR, and is itself a Z-graded (commutative Noetherian) ring with grading given by (R(r))n = Rrn


for all n ∈ Z. We refer to R(r), with this grading, as the r-th Veroneseansubring of R.

Let M =⊕

n∈ZMn and L =⊕

n∈Z Ln be general graded R-modules, andlet f : M −→ L be a homogeneous homomorphism, with n-th componentfn : Mn −→ Ln for all n ∈ Z. We define M (r,s) :=

⊕n∈ZMrn+s, an R(r)-

submodule of M� R(r) ; in fact, M (r,s) is a graded R(r)-module with gradinggiven by (M (r,s))n = Mrn+s for all n ∈ Z. We refer to M (r,s) as the (r, s)-th Veronesean submodule of M� R(r) . Note that M (r,s) = (M(s))

(r,0). Also,we denote by f (r,s) : M (r,s) −→ L(r,s) the homogeneous homomorphism ofgradedR(r)-modules for which (f (r,s))n = frn+s :Mrn+s −→ Lrn+s for alln ∈ Z. With these assignments, ( • )(r,s) : *C(R) −→ *C(R(r)) becomes anexact additive covariant functor, which we refer to as the (r, s)-th Veroneseanfunctor.

Note that, since a is graded, a(r) := a(r,0) is a graded ideal of R(r).

(i) Show that there is an isomorphism of R(r)-modules

r−1⊕i=0

M (r,s+i) ∼=−→M� R(r) .

(ii) Show that√a(r)R =

√a.

(iii) Show that Γa(r)(M (r,s)) = (Γa(M))(r,s).(iv) Let b be a graded ideal of R(r), and let I be a *injective graded R-

module. Let j ∈ N. Show that there is an isomorphism of R(r)-modules

r−1⊕i=0

Hjb(I

(r,s+i))∼=−→ Hj

bR(I),

and deduce that the R(r)-module I(r,s) is Γb-acyclic.(v) Show that there is a unique isomorphism

Φ =(φi)i∈N0

:(Hi

a(r)(( • )(r,s)))i∈N0

−→((Hi

a( • ))(r,s))i∈N0

of negative connected sequences of covariant functors from *C(R) to*C(R(r)) for which φ0 is the identity natural equivalence.

(vi) Use 2.2.15 to show that there is a natural equivalence of functors

(Da( • ))(r,s)∼=−→ Da(r)(( • )(r,s))

from *C(R) to *C(R(r)) .

It is natural to ask whether one can generalize the concept of Veroneseansubring to multi-graded situations. This is the subject of the next exercise.


13.5.10 Exercise. Let n ∈ N and assume that G = Zn and R is G-graded;let a be a graded ideal of R. Let G′ be a subgroup of finite index t in G, sothat rankG′ = n. Let g0 := 0G, g1, . . . , gt−1 be representatives of the distinctcosets of G′ in G. Let M =

⊕g∈GMg be a (G-)graded R-module.

(i) Define RG′:=⊕

g′∈G′ Rg′ , and show that this is a Noetherian subringof R; the decomposition of its definition provides RG′

with a grading byG′. (Here is a hint to help you show that RG′

is Noetherian: note thatRG′

is a direct summand of R as RG′-module; consider an ascending

chain of ideals of RG′, extend the ideals to R and then contract back to

RG′.)

(ii) Let k ∈ {0, 1, . . . , t−1}. SetMG′,gk :=⊕

g′∈G′ Mg′+gk , and show thatthis is an RG′

-submodule of M which is G′-graded (by the decomposi-tion given in its definition). Show further that there is an isomorphism ofRG′

-modules⊕t−1

j=0MG′,gj

∼=−→M� RG′ .

(iii) Let aG′:= aG

′,0G . Show that√aG′R =

√a.

(iv) Let k ∈ {0, 1, . . . , t− 1}. Generalize arguments from 13.5.9 to producean isomorphism

Θ =(θi)i∈N0

:(Hi

aG′ (( • )G′,gk))j∈N0

−→((Hi

a( • ))G′,gk)i∈N0

of negative connected sequences of covariant functors from *CG(R) to*CG′

(RG′) for which θ0 is the identity natural equivalence.

(v) Again for k ∈ {0, 1, . . . , t − 1}, generalize arguments from 13.5.9 toproduce a natural equivalence of functors

(Da( • ))G′,gk ∼=−→ DaG′ (( • )G

′,gk)

from *CG(R) to *CG′(RG′

).

13.5.11 Definition. When R = ⊕n∈ZRn is Z-graded, we shall say that R ispositively graded precisely when Rn = 0 for all n < 0.

More generally, whenR = ⊕g∈ZnRg is Zn-graded for some positive integern, we shall say that R is positively graded if and only if Rg = 0 for all g =

(g1, . . . , gn) ∈ Zn \ N0n.

13.5.12 Example. Let K be a field, and consider the ring K[X,Y ] of poly-nomials over K in two indeterminates X and Y to be Z-graded so that K isthe component of degree 0 and degX = deg Y = 1. (See 13.5.3.) Let d ∈ Nwith d ≥ 3, and let A(d) be the subring of K[X,Y ] given by

A(d) := K[Xd, Xd−1Y,XY d−1, Y d].


This is a subring of the d-th Veronesean subring K[X,Y ](d) of K[X,Y ], de-scribed in 13.5.9; in fact, A(d) inherits a Z-grading from K[X,Y ](d).

In this example, it will be convenient, when considering a positively gradedcommutative Noetherian ring R′ =

⊕n∈N0

R′n, to denote the graded ideal⊕n∈NR

′n by R′+.

Note that A(d)+ is the unique graded maximal ideal of A(d). We shall nowillustrate some of the ideas of this chapter by showing that the ideal transformDA(d)+

(A(d)) can be naturally identified with K[X,Y ](d), and then exploitingthis fact to obtain information about H1

A(d)+(A(d)).

Let φ : A(d) −→ K[X,Y ](d) denote the inclusion homomorphism. If m :=

XiY j with i, j ∈ N0, i+j ≡ 0 (mod d) and i+j > 0, then we can writem =

XduY dvXrY d−r for some u, v, r ∈ N0 with 0 ≤ r ≤ d. If r = 0, 1, d− 1 ord, then m ∈ A(d). Now suppose that 2 ≤ r ≤ d− 2. Then

Y d(r−1)m = XduY dv(XY d−1)r ∈ A(d),

Xd(d−r−1)m = XduY dv(Xd−1Y )d−r ∈ A(d),

(Xd−1Y )rm = XduY dvXdrY d ∈ A(d),

(XY d−1)d−rm = XduY dvXdY d(d−r) ∈ A(d).

Hence Kerφ and Cokerφ are both A(d)+-torsion. Note that√A(d)+K[X,Y ](d) =

√(Xd, Xd−1Y,XY d−1, Y d)K[X,Y ](d)

= (K[X,Y ](d))+ = (K[X,Y ]+)(d).

Therefore, by 13.5.9 and the Independence Theorem 4.2.1, we have, for eachi ∈ N0,

HiA(d)+

(K[X,Y ](d)) ∼= HiA(d)+K[X,Y ](d)(K[X,Y ](d))

= Hi(K[X,Y ]+)(d)(K[X,Y ](d))

∼= (Hi(X,Y )(K[X,Y ]))(d,0),

and this is zero for i = 0, 1.We can now use 2.2.15 to see that there is a unique A(d)-isomorphism φ′ :

K[X,Y ](d) −→ DA(d)+(A(d)) such that the diagram

A(d)

φK[X,Y ](d)�

ηA(d)φ′∼=

DA(d)+(A(d))

��

��


commutes, and we can use 13.5.4(ii) to see that φ′ is homogeneous. So, by13.5.4(i), there is a homogeneous isomorphism Cokerφ ∼= H1

A(d)+(A(d)) of

A(d)-modules. For each n ∈ N0, letAn (respectivelyBn) denote the n-th com-ponent of the graded A(d)-module A(d) (respectively K[X,Y ](d)). We nowcompare An and Bn.

First, when 1 ≤ n < d − 2 the monomial Xd−2Y nd−d+2 cannot be ex-pressed as a product of n factors taken from {Xd, Xd−1Y,XY d−1, Y d}, andso An ⊂ Bn.

Next, we consider the case where n ≥ d − 2. Of course, Xdn ∈ An; wetherefore consider a monomial of the form Xrd+sY (n−r)d−s, where 0 ≤ r ≤n−1 and 0 ≤ s ≤ d−1. We claim that either (a) r+s ≤ n or (b) r+s ≥ d−1:if this were not the case, then we should have

n ≤ r + s− 1 and r + s ≤ d− 2,

which would imply that n ≤ d− 3, a contradiction. In case (a), we have

Xrd+sY (n−r)d−s = (Xd)r(XY d−1)s(Y d)n−r−s ∈ An,

while in case (b) we have

Xrd+sY (n−r)d−s = (Xd)r+s+1−d(Xd−1Y )d−s(Y d)n−r−1 ∈ An.

Thus An = Bn in this case.It follows that the n-th component of H1

A(d)+(A(d)) is non-zero if and only

if 1 ≤ n < d− 2.

13.5.13 �Exercise. Let K be a field and let R := K[X1, . . . , Xn], the ring ofpolynomials over K in n indeterminates (where n ∈ N), Z-graded so that Xi

has degree 1, for all i = 1, . . . , n. Let r ∈ N, and consider the r-th Veroneseansubring R(r) of R, as in 13.5.9. Thus R(r) is the K-subspace of R generatedby

{Xv11 . . . Xvn

n : v1, . . . , vn ∈ N0, v1 + · · ·+ vn ≡ 0 (mod r)} .

Let R(r)+ denote the unique graded maximal ideal of R(r). Show that

Hi

R(r)+

(R(r)) = 0 for all i ∈ N0 \ {n},

and deduce from [7, 1.5.8 and 1.5.9] that R(r) is Cohen–Macaulay.

14

Graded versions of basic theorems

We have now laid the foundations of multi-graded local cohomology theoryin Chapter 13, where the gradings on R and R-modules M are by a finitelygenerated, torsion-free Abelian group G. Indeed, in the case where R is G-graded and the ideal a is G-graded, we now know that, for a G-graded R-module M , there is a natural way in which to define G-gradings on the localcohomology modules Hi

a(M) (i ∈ N0); furthermore, whenever f :M −→ N

is a morphism in *CG(R), then Hia(f) is a homogeneous homomorphism for

all i ∈ N0; also, whenever 0 −→ L −→M −→ N −→ 0 is an exact sequencein the ‘G-graded’ category *CG(R), then all the homomorphisms, includingthe connecting homomorphisms, in the induced long exact sequence of localcohomology modules (with respect to a) are G-homogeneous.

This chapter is concerned with refinements available in this multi-gradedcase of such fundamental results as the Independence Theorem 4.2.1, the FlatBase Change Theorem 4.3.2, Faltings’ Annihilator Theorem 9.5.1, Grothendi-eck’s Finiteness Theorem 9.5.2 and the Local Duality Theorem 11.2.6, and ofthe theory of canonical modules developed in Chapter 12. However, althoughit is true that part of this chapter is a retracing of steps through earlier chapters,revisiting many of the highlights in order to ‘add graded frills’, we have feltit necessary to include quite a bit of the underlying algebra of multi-gradedcommutative Noetherian rings.

For example, there is a multi-graded analogue of Matlis’s decompositiontheory for injective modules over a commutative Noetherian ring, and thismulti-graded analogue has both strong similarities to, and fascinating linkswith, the ungraded theory. We present the multi-graded version in §14.2, andmake much use of it in our treatment of graded local duality in §14.4 and*canonical modules in §14.5.

In §14.3, we present some results due to S. Goto and K.-i. Watanabe [22,§1.2] that also concern the algebra of a commutative Noetherian ring R that is

286 Graded versions of basic theorems

graded by a finitely generated torsion-free Abelian group. Let M be a finitelygenerated graded R-module, and let p ∈ SuppM . We showed in 13.1.6(i)that p* ∈ SuppM . Our principal aims in §14.3 are to present Goto’s andWatanabe’s results that

dimMp = dimMp∗ + ht p/p* and depthMp = depthMp∗ + ht p/p*,

and to apply them to prove graded versions of Faltings’ Annihilator Theorem9.5.1 and Grothendieck’s Finiteness Theorem 9.5.2. However, we also use theGoto–Watanabe results later in the chapter.

Limitations on space mean that almost all of our treatment in §14.5 of *cano-nical modules is concerned with the case whereR is a Cohen–Macaulay multi-graded ring with a unique maximal graded proper ideal m. We define a *can-onical module for R to be a finitely generated graded R-module C for whichthere is a homogeneous isomorphism *HomR(C, *ER(R/m)) ∼= Hn

m(R).

This definition is the obvious graded analogue of our definition of canonicalmodule in the ungraded local case. However, in the special case in which Ris Z-graded, Bruns and Herzog give an alternative definition in [7, Definition3.6.8] which is not obviously equivalent to ours. The work involved in our rec-onciliation of these two approaches (see 14.5.12) has contributed significantlyto the length of this chapter.

Readers who are only interested in the case of Z-graded rings rather than thefull multi-graded case should be able to pass quickly over some parts of thischapter. For example, the special cases of the above-mentioned results of Gotoand Watanabe in which the grading group is Z are essentially covered by [7,Theorems 1.5.8 and 1.5.9].

14.1 Fundamental theorems

14.1.1 Notation and Terminology. Throughout this chapter, G will denote afinitely generated, torsion-free Abelian group, written additively, and we shallassume that our commutative Noetherian ring R is G-graded, with gradinggiven by R = ⊕g∈GRg . Occasionally, we shall consider particular cases inwhich R is Z-graded, that is, in which G is taken to be Z.

We shall employ the notation, conventions and terminology concerning G-graded rings and modules described in 13.1.1. In addition, when the ideal ais graded, and M is a graded R-module, we use Hi

a(M)g to denote the g-thcomponent of the graded R-module Hi

a(M) (for i ∈ N0 and g ∈ G).A maximal member of the set of proper graded ideals of R is referred to as

a *maximal graded ideal of R. We shall say that R is *local precisely when it

14.1 Fundamental theorems 287

has exactly one *maximal graded ideal. The statement ‘(R,m) is *local’ is tobe interpreted as meaning that m is the unique *maximal graded ideal of theG-graded ring R. For an ideal b of R, we shall denote by b* the graded idealgenerated by all homogeneous elements of b.

Let S be a multiplicatively closed subset of R consisting of non-zero ho-mogeneous elements. We pointed out in 13.1.1 that the ring S−1R is also G-graded, and that, if M is a graded R-module, then the S−1R-module S−1M

is also graded.If p is a prime ideal of R and we take S to be the set of all homogeneous

elements of R that lie outside p, then the resulting G-graded ring S−1R (re-spectively module S−1M ) is called the homogeneous localization of R (re-spectively M ) at p, and is denoted by R(p) (respectively M(p)). This conceptshould not be confused with a different concept for which Hartshorne, in [30,p. 18], uses the notation ‘(p)’. Note that R(p) is a *local ring, with pR(p) as itsunique *maximal graded ideal.

Some of our work will be particularly concerned with the case whereG = Z.Suppose that R is Z-graded (that is, that G = Z), and let M =

⊕n∈ZMn be

a graded R-module. We define the end of M by

end(M) := sup {n ∈ Z :Mn = 0}

if this supremum exists, and ∞ otherwise. (We adopt the convention that thesupremum of the empty set of integers is to be taken as −∞, and we interpret−∞+t as−∞ for all t ∈ Z.) With analogous conventions, we similarly definethe beginning of M , denoted by beg(M), to be inf {n ∈ Z :Mn = 0} if thisinfimum exists, and −∞ otherwise.

14.1.2 Remark. Suppose that M is a graded R-module.

(i) Recall from 13.1.6(i) that, if p ∈ SuppM , then the graded prime p*also belongs to SuppM . Hence if M has no graded prime ideal in itssupport, then M = 0.

(ii) Consequently, if (R,m) is *local (see 14.1.1) andMm = 0, thenM = 0.

14.1.3 Definition and Remark. Let R′ =⊕

g∈GR′g be a commutative G-

graded ring, and let f : R −→ R′ be a ring homomorphism.We say that f is homogeneous precisely when f(Rg) ⊆ R′g for all g ∈ G.

Assume that this is the case. Let M ′ =⊕

g∈GM′g be a graded R′-module.

Then the same direct sum decomposition provides the R-module M ′� R witha structure as a graded R-module. In fact, in the terminology of 13.3.6, thefunctor � R : CG(R′) −→ CG(R) has the *restriction property.

Whenever we regard, in such circumstances, a graded R′-module M ′ as a


graded R-module, it is to be understood that the same grading is used for thetwo structures. Thus we can write (M ′� R)g = (M ′

g)� R0 for all g ∈ G.

14.1.4 Proposition. Assume that a is graded, and let R′ =⊕

g∈GR′g be a

second commutative Noetherian G-graded ring. Let f : R −→ R′ be a ringhomomorphism which is homogeneous (see 14.1.3).

The natural equivalence of functors ε : DaR′(•)�R −→ Da(•�R) of 2.2.24(from C(R′) to C(R)) has the *restriction property.

Proof. Since aR′ is a graded ideal of R′, it follows from 13.3.14 and 14.1.3that both DaR′( • )� R and Da( • � R) have the *restriction property.

Let M ′ be a graded R′-module. The homomorphism

ηaR′,M ′� R :M ′� R −→ DaR′(M ′)� R

is homogeneous by 13.3.14, and has kernel and cokernel which are a-torsionby 2.2.6(i)(c). The result therefore follows from 13.5.4(ii).

14.1.5 �Exercise: Graded Mayer–Vietoris sequence. Assume that a isgraded; let b be a second graded ideal of R. Show that, for a graded R-moduleM , all the homomorphisms in the Mayer–Vietoris sequence (see 3.2.3)

0 H0a+b(M) H0

a(M)⊕H0b(M) H0

a∩b(M)

H1a+b(M) H1

a(M)⊕H1b(M) H1

a∩b(M)

· · · · · ·

Hia+b(M) Hi

a(M)⊕Hib(M) Hi

a∩b(M)

Hi+1a+b(M) · · ·

� � �

� � �

�

� � �

� �

are homogeneous.

14.1.6 �Exercise. Assume that a is graded; let i ∈ N0.

(i) Let (Λ,≤) be a (non-empty) directed partially ordered set; let (Wα)α∈Λbe a direct system of gradedR-modules and homogeneousR-homomor-phisms over Λ, with constituent R-homomorphisms hαβ : Wβ → Wα

(for each (α, β) ∈ Λ× Λ with α ≥ β). (See 13.3.12.)Show that the isomorphism

lim−→α∈Λ

Hia(Wα)

∼=−→ Hia

(lim−→α∈Λ

Wα

)given by Theorem 3.4.10 (see also 3.4.1) is homogeneous.


(ii) Let (Lθ)θ∈Ω be a non-empty family of gradedR-modules. Show that theisomorphism

Hia

(⊕θ∈Ω Lθ

) ∼=−→⊕

θ∈ΩHia(Lθ)

given by (Theorem 3.4.10 and) Exercise 3.4.5 is homogeneous.

Next we use Theorem 13.3.15 to establish quickly G-graded versions of theIndependence Theorem 4.2.1 and the Flat Base Change Theorem 4.3.2.

14.1.7 Graded Independence Theorem. Assume that a is graded, and letR′ =⊕

g∈GR′g be a second commutative Noetherian G-graded ring.

Let f : R −→ R′ be a ring homomorphism which is homogeneous (see14.1.3).

(i) Both the negative (strongly) connected sequences of covariant functors

(HiaR′( • )� R)i∈N0 and (Hi

a( • � R))i∈N0

from C(R′) to C(R) have the *restriction property.(ii) The isomorphism of connected sequences

Λ = (λi)i∈N0 : (HiaR′( • )� R)i∈N0

∼=−→ (Hia( • � R))i∈N0

of 4.2.1 has the *restriction property. Consequently, for each i ∈ N0 andeach graded R′-module M ′, there is a homogeneous R-isomorphism

λiM ′ : HiaR′(M ′)

∼=−→ Hia(M

′).

Proof. (i) Since f is homogeneous, the restriction functor � R : C(R′) −→C(R) has the *restriction property of 13.3.6: see 14.1.3. Since, by 13.4.3 and13.4.4, the connected sequence (Hi

a)i∈N0 (from C(R) to C(R)) has the *restric-tion property, it is immediate that (Hi

a(•�R))i∈N0 has the *restriction property.Since the extension aR′ of a to R′ under f is a graded ideal, it is just as easyto see that (Hi

aR′( • )� R)i∈N0 has the *restriction property.(ii) Since λ0 is the identity natural equivalence from ΓaR′(•)�R = Γa(•�R)

to itself, λ0 has the *restriction property.For each i ∈ N0 and each graded R′-module M ′, we can use the grad-

ing on Hia(M

′� R) of (i) to define a grading on HiaR′(M ′)� R in such a

way that the isomorphism λiM ′ : HiaR′(M ′)� R

∼=−→ Hia(M

′� R) is homo-geneous: we recover the grading of part (i) on the ΓaR′(M ′)� R. With respectto these gradings, (Hi

aR′( • )� R)i∈N0 also has the *restriction property. SinceHi

aR′(I ′)� R = 0 for all i ∈ N whenever I ′ is a *injective graded R′-module(by 13.2.6), it follows from 13.3.15 that these gradings coincide with the natu-ral ones used in part (i). All the remaining claims follow from this.


Below, we shall use a similar argument to establish a graded version of theFlat Base Change Theorem 4.3.2. However, the reader might find the followingpreparatory remark helpful.

14.1.8 Remark. Let M =⊕

g∈GMg be a graded R-module. Let R′ =⊕g∈GR

′g be a second commutative Noetherian G-graded ring, and let f :

R −→ R′ be a ring homomorphism which is homogeneous (see 14.1.3). ThenR′�R is a gradedR-module (as is explained in 14.1.3), and thereforeM⊗RR

′

has a structure as a graded R-module. In fact, the direct decomposition

M ⊗R R′ =⊕g∈G

(M ⊗R R′)g

described in 13.1.9 actually provides M ⊗R R′ with a structure as a gradedR′-module, and ( • )⊗R R

′ : C(R) −→ C(R′) has the *restriction property.

14.1.9 Graded Flat Base Change Theorem. Assume that a is graded, andlet R′ =

⊕g∈GR

′g be a second commutative Noetherian G-graded ring.

Let f : R −→ R′ be a ring homomorphism which is homogeneous (see14.1.3) and flat.

(i) Both the negative (strongly) connected sequences of covariant functors

(Hia( • )⊗R R

′)i∈N0 and (HiaR′(( • )⊗R R

′))i∈N0

from C(R) to C(R′) have the *restriction property.(ii) The isomorphism of connected sequences

(ρi)i∈N0 : (Hia( • )⊗R R

′)i∈N0

∼=−→ (HiaR′(( • )⊗R R

′))i∈N0

of 4.3.2 has the *restriction property, so that, for each i ∈ N0 and eachgraded R-module M , there is a homogeneous R′-isomorphism

ρiM : Hia(M)⊗R R

′ ∼=−→ HiaR′(M ⊗R R

′).

Proof. (i) Since f is homogeneous, the functor ( • ) ⊗R R′ : C(R) −→C(R′) has the *restriction property: see 14.1.8. Since the extension aR′ of ato R′ under f is a graded ideal, it follows from 13.4.3 and 13.4.4 that theconnected sequence (Hi

aR′)i∈N0 (from C(R′) to C(R′)) has the *restrictionproperty. Hence (Hi

aR′(( • )⊗RR′))i∈N0 has the *restriction property. It is just

as easy to see that (Hia( • )⊗R R

′)i∈N0 has the *restriction property.(ii) Note that, by 4.3.1, the natural equivalence ρ0 has the *restriction prop-

erty.For each i ∈ N0 and each graded R-module M , we can use the grading

on HiaR′(M ⊗R R′) of (i) to define a grading on Hi

a(M) ⊗R R′ in such a


way that the isomorphism ρiM : Hia(M) ⊗R R′

∼=−→ HiaR′(M ⊗R R′) is

homogeneous: we recover the grading of (i) on the Γa(M)⊗RR′. With respect

to these gradings, (Hia( • )⊗RR

′)i∈N0 also has the *restriction property. SinceHi

a(I)⊗RR′ = 0 for all i ∈ N whenever I is a *injective gradedR-module (by

13.2.6), it follows from 13.3.15 that these gradings coincide with the naturalones used in part (i). All the remaining claims follow from this.

Next, we explore the behaviour of graded local cohomology with respect toa shift functor.

14.1.10 Remarks. Let g0 ∈ G and j ∈ N0. Let rg0 ∈ Rg0 be a homogeneouselement of degree g0. Let L,M be graded R-modules.

(i) The graded R-modules

*HomR(L,M(g0)) and (*HomR(L,M))(g0)

are equal. Since *ExtjR( • , N) is the j-th right derived functor in *C(R)of *HomR(•, N), we see that the gradedR-modules *ExtjR(L,M(g0))

and (*ExtjR(L,M))(g0) are again equal (and not just isomorphic in*C(R)). Hence the graded R-modules

lim−→n∈N

ExtjR(R/an,M(g0)) and

(lim−→n∈N

ExtjR(R/an,M)

)(g0)

are equal.(ii) If we forget the gradings onM andM(g0), we obtain the same ungraded

R-module M . By 13.4.6(ii), the grading on Hja(M) can be defined from

the grading of 13.3.13(ii) on lim−→n∈N

ExtjR(R/an,M) simply by requiring

that the isomorphism φja M be homogeneous. Similarly, we can obtainthe grading on Hj

a(M(g0)) by using φja M to ‘lift across’ the grading of13.3.13(ii) on lim

−→n∈N

ExtjR(R/an,M(g0)). It therefore follows from part

(i) that the graded modules Hja(M(g0)) and (Hj

a(M))(g0) are equal.(iii) The fact that RjDa(M(g0)) and (RjDa(M))(g0) are the same graded

R-module can be deduced in a similar way from 13.5.7(ii).

14.1.11 �Exercise. Obtain a ‘graded’ version of Proposition 8.1.2.In detail, assume that a is graded; let b be a homogeneous element of R. Let

f :M −→ N be a homogeneous homomorphism of graded R-modules.(i) Show that there is a long exact sequence of graded R-modules and


homogeneous R-homomorphisms

0 H0a+Rb(M) H0

a(M) H0a(Mb)

H1a+Rb(M) H1

a(M) H1a(Mb)

· · · · · ·

Hia+Rb(M) Hi

a(M) Hia(Mb)

Hi+1a+Rb(M) · · ·

� � �

� � �

�

� � �

� �


Hia+Rb(M) Hi

a(M) Hia(Mb) Hi+1

a+Rb(M)� � �

Hia+Rb(N) Hi

a(N) Hia(Nb) Hi+1

a+Rb(N)� � �

Hia+Rb(f) Hi

a(f) Hia(fb) Hi+1

a+Rb(f)

� � � �

commutes for all i ∈ N0.(ii) Let i ∈ N0. Show that there is a commutative diagram

0 H1Rb(H

ia(M)) Hi+1

a+Rb(M) ΓRb(Hi+1a (M)) 0� � � �

0 H1Rb(H

ia(N)) Hi+1

a+Rb(N) ΓRb(Hi+1a (N)) 0� � � �

Hi+1a+Rb(f)H1

Rb(Hia(f)) ΓRb(H

i+1a (f))

� � �

(in the category *C(R)) with exact rows. The top row is referred to as thecomparison exact sequence for M .

14.1.12 Lemma. Assume that G = Zn and R =⊕

g∈N0n Rg is positively

graded, and that the ideal a is generated by homogeneous elements of degree0. Let a0 = a ∩ R0. Let h ∈ G. We denote by ( • )h : *C(R) −→ C(R0) thefunctor which assigns, to each graded R-module, and to each homogeneoushomomorphism of graded R-modules, the h-th component.

There is an isomorphism (Hia( • )h)i∈N0

∼=−→ (Hia0(( • )h))i∈N0 of negative

strongly connected sequences of covariant functors from *C(R) to C(R0).

Proof. Observe that Γa( • )h and Γa0(( • )h) are the same functor. Let I =⊕g∈Zn Ig be a *injective graded R-module. Impose the trivial Zn-grading


on the commutative Noetherian ring R0. Since a = a0R, it follows from theGraded Independence Theorem 14.1.7 and Exercise 3.4.5 that, for all i ∈ N,

0 = Hia(I) = Hi

a0R(I)∼= Hi

a0(I� R0) = Hi

a0

(⊕g∈GIg)∼=⊕

g∈GHia0(Ig).

Hence Ih is Γa0-acyclic. The result is now an easy consequence of 13.3.5.

14.1.13 Example. Assume that that G = Zn and R =⊕

g∈N0n Rg is posi-

tively graded. Let h ∈ Zn and let L be an R0-module. We can define a gradedR-module hL such that, for all g ∈ G,

(hL)g =

{L if g = h,

0 if g = h.

(These conditions necessitate that rgm = 0 for all m ∈ hL and all rg ∈ Rg

whenever g ∈ G \ {0}.)Assume that the ideal a is graded, and let a0 = a∩R0. Let i ∈ N0. We show

how to use Lemma 14.1.12 to calculate Hia(

hL). Let R+ =⊕

g∈N0n\{0}Rg;

observe thatR+ annihilates hL and that a+R+ = a0R+R+. The argument ofExample 4.2.2 can be modified to our ‘Zn-graded’ situation to produce homo-geneous R-isomorphisms Hi

a(hL) ∼= Hi

a+R+(hL) ∼= Hi

a0R(hL). We can now

use 14.1.12 to deduce that, for g ∈ Zn,

Hia(

hL)g ∼= Hia0R(

hL)g ∼= Hia0((hL)g) =

{Hi

a0(L) if g = h,

0 if g = h.

Hence there is a homogeneous R-isomorphism hHia0(L) ∼= Hi

a(hL).

We end this section with some results that concern the special case of Z-graded rings, that is, the special case where G = Z. In this case, much basictheory has been developed by Bruns and Herzog in [7, §1.5 and §3.6]. We nowprepare for a Z-graded analogue of (a special case of) the local Lichtenbaum–Hartshorne Vanishing Theorem 8.2.1. Before we come to the theorem itself,however, we provide two preparatory exercises.

14.1.14 �Exercise. Assume that G = Z and that (R,m) is *local; assumefurther that m is actually a maximal ideal of R. Let M be a graded R-module.Use [7, 1.5.6 and 1.5.8] to show that dimM = dimRm

Mm.

14.1.15 �Exercise. Assume that G = Z, that R =⊕

j∈N0Rj is positively

graded, and that the subring R0 is a local ring having maximal ideal m0.

(i) Show that R is *local with unique *maximal ideal

m := m0 ⊕R1 ⊕R2 ⊕ · · · ⊕Rn ⊕ · · · .


(ii) Show that, for all j, i ∈ N, the j-th component of mj+i is contained inmi

0Rj .(iii) Assume that R = R0. By [7, 1.5.4], there exist non-zero homogeneous

elements y1, . . . , yt of R of positive degrees such that

R = R0[y1, . . . , yt].

Let d := max{deg yi : 1 ≤ i ≤ t}. Show that R(i−1)d+j ⊆ mi for alli, j ∈ N.

(iv) Show that the multiplication in R induces a natural ring structure on theAbelian group

∏j∈N0

Rj . (The following hint can ease the checking ofthe ring axioms. For a := (a0, a1, . . . , aj , . . .) ∈

∏j∈N0

Rj and h ∈ N0,define

a≤h := (a0, a1, . . . , ah, 0, 0, . . .);

show that (a≤hb≤h)≤h = (ab)≤h for a, b ∈∏

j∈N0Rj .)

(v) Assume in addition that R0 is complete. Prove that the inclusion mapR −→

∏j∈N0

Rj provides the m-adic completion of R.

14.1.16 Graded Lichtenbaum–Hartshorne Vanishing Theorem. Assumethat G = Z, and that R =

⊕j∈N0

Rj is positively graded and an integral do-main; assume also that the subringR0 is a complete local ring having maximalideal m0.

Assume that the ideal a is graded and proper, and that dimR/a > 0. Setd := dimR. Then Hd

a (R) = 0.

Proof. Let m := m0 ⊕ R1 ⊕ R2 ⊕ · · · ⊕ Rj ⊕ · · · . By 14.1.15, our gradedring R is *local with unique *maximal ideal m. Let Rm denote the completionof the local ring Rm. In view of the natural ring isomorphisms

R/mj ∼=−→ Rm/(mRm)j for j ∈ N,

Rm is isomorphic to the m-adic completion of R, which is an integral domainby 14.1.15.

Note that dimRm = d, by 14.1.14, since m is actually a maximal ideal ofR. Since a is graded, all its minimal primes are graded, and so are containedin m. Since dimR/a > 0, it therefore follows that m is not a minimal primeof a. Hence dim

(Rm/aRm

)= dim (Rm/aRm) > 0. We can therefore use

the local Lichtenbaum–Hartshorne Vanishing Theorem 8.2.1 to deduce thatHd

aRm(Rm) = 0. Thus (Hd

a (R))m = 0, by 4.3.3. Therefore Hda (R) = 0, by

14.1.2(ii).

14.2 *Indecomposable *injective modules 295

Two other results to which we would like to add ‘graded frills’ are Faltings’Annihilator Theorem 9.5.1 and Grothendieck’s Finiteness Theorem 9.5.2. Weshall have to defer graded versions of these results until the end of §14.3, bywhich point we shall have presented more results about the behaviour of G-graded modules.

14.2 *Indecomposable *injective modules

One of our major aims in this chapter is to provide versions of local dualitywhich apply in ‘graded’ situations, including versions which involve ‘gradedcanonical modules’. To prepare for this, we shall develop the ‘graded’ ana-logue of the decomposition theory for injective modules over a commutativeNoetherian ring.

Recall that we are assuming that R is G-graded throughout this chapter. In13.2.4, we showed that each graded R-module M has a *injective envelope,and between any two *injective envelopes of M there is a homogeneous iso-morphism which restricts to the identity map on M ; we agreed to denote by*E(M) or *ER(M) one choice of *injective envelope of M ; and we provedthat *E(M), with its grading forgotten, is an essential extension of M .

14.2.1 �Exercise. Consider a non-empty family (Mλ)λ∈Λ of graded R-mod-ules.

(i) Show that⊕

λ∈ΛMλ is *injective if and only if Mλ is *injective for allλ ∈ Λ. (Here is a hint: you might find the ‘graded Baer criterion’ 13.2.7useful.)

(ii) Show that the obvious map⊕

λ∈ΛMλ −→⊕

λ∈Λ *E(Mλ) providesthe *injective envelope of

⊕λ∈ΛMλ.

14.2.2 �Exercise. Let I be a graded submodule of the graded R-module M ,and suppose that I is *injective. Show that I is a direct summand of M withgraded complement.

14.2.3 Definition. A graded R-module is said to be *indecomposable pre-cisely when it is non-zero and cannot be written as the direct sum of two propergraded submodules.

14.2.4 Proposition. Let p ∈ *Spec(R).

(i) The *injective graded R-module *E(R/p)(g) is *indecomposable foreach g ∈ G.


(ii) A non-zero *injective gradedR-module I has a *indecomposable *injec-tive graded submodule which must, by 14.2.2, be a direct summand (withgraded complement). In fact, for each q ∈ Ass I , there exists g0 ∈ G anda homogeneous element m of I of degree g0 for which (0 :R m) = q,and then I has a graded submodule that is homogeneously isomorphicto *E(R/q)(−g0).

(iii) Each *indecomposable *injective graded R-module is isomorphic (inthe category *C(R)) to *E(R/q)(−g0) for some q ∈ *Spec(R) andg0 ∈ G.

(iv) Let r be a homogeneous element of degree g inR\p. Then multiplicationby r provides a homogeneous automorphism of degree g of *E(R/p).Also, each element of *E(R/p) is annihilated by some power of p.

(v) Let q ∈ * Spec(R). If *E(R/p) ∼= *E(R/q)(−g) (in *C(R)) for someg ∈ G, then p = q.

Proof. (i) Let m be a homogeneous generator of the graded submodule R/pof *E(R/p). Suppose that L, N are non-zero graded submodules of *E(R/p)

such that *E(R/p) = L⊕N . Then L∩Rm = 0 and N ∩Rm = 0; thus thereexist homogeneous elements a, b ∈ R such that 0 = am ∈ L and 0 = bm ∈N . Since (0 :R m) = p, prime, we must have a, b ∈ R \ p, so that ab ∈ p.Therefore 0 = abm ∈ L ∩N , and this is a contradiction. Therefore *E(R/p)

is *indecomposable. It follows easily that *E(R/p)(g) is *indecomposable foreach g ∈ G.

(ii) Since I = 0, it must have an associated prime, q say, which must begraded, by 13.1.6(ii); moreover, q = (0 :R m) for some homogeneous elementm of I . Let degm = g0. Thus there is a homogeneous isomorphism ϕ :

(R/q)(−g0)∼=−→ Rm such that ϕ(1 + q) = m.

Since I is *injective, we can extend ϕ to a homogeneous homomorphismψ : *E((R/q)(−g0)) = *E(R/q)(−g0) −→ I; as

Kerψ ∩ (R/q)(g0) = 0,

it follows that ψ is monomorphic; therefore J := Imψ is a *injective gradedsubmodule of I and there is a homogeneous isomorphism

J ∼= *E(R/q)(−g0).

It follows from (i) that J is *indecomposable.Since J is *injective, there is a homogeneousR-homomorphism ξ : I −→ J

that extends the identity map IdJ on J ; then Ker ξ is a graded submodule of Iand J⊕Ker ξ = I . Thus J is a direct summand of I with graded complement.


(iii) Apply (ii) to a *indecomposable *injective graded R-module I , and thedesired conclusion is immediate.

(iv) Multiplication by r provides a homogeneous R-homomorphism μr :

*E(R/p) −→ *E(R/p)(g); since Kerμr∩(R/p) = 0, the map μr is monom-orphic. Therefore Imμr is a *injective graded submodule of the *indecompos-able *injective graded R-module *E(R/p)(g), and it follows from 14.2.2 thatμr must be surjective.

For the second claim, it is enough for us to show that an arbitrary non-zerohomogeneous element m ∈ *E(R/p) is annihilated by some power of p. Letq ∈ AssRm, and recall from 13.1.6(ii) that q is graded. Since Rm∩ (R/p) =0, we see that q ∈ Ass(R/p) = {p}, so that q = p. Therefore (0 :R Rm) isp-primary.

(v) Let h(R) denote the set of non-zero homogeneous elements of R, andlet g′ ∈ G. By (iv), the set of homogeneous elements r of R for which multi-plication by r provides an automorphism (of some degree) of *E(R/p)(−g′)is precisely h(R) \ p. The desired conclusion is now immediate.

The next lemma can be proved by making straightforward modifications tothe proof of the corresponding ‘ungraded’ result in 10.1.12.

14.2.5 Lemma. Let S be a multiplicatively closed subset of homogeneouselements ofR, and letM be a (G-)graded S−1R-module. ThenM is *injectiveover R if and only if it is *injective over S−1R.

14.2.6 Lemma. Let S be a multiplicatively closed subset of homogeneouselements of R, and let p ∈ Spec(R) be such that p ∩ S = ∅. By 14.2.4(iv),the *indecomposable *injective graded R-module *ER(R/p) has a naturalstructure as a G-graded S−1R-module.

In the category *C(S−1R), we have

*ER(R/p) ∼= *ES−1R(S−1R/S−1p).

Furthermore, *ES−1R(S−1R/S−1p), when considered as a G-graded R-

module by means of the natural homomorphism R −→ S−1R, is homoge-neously isomorphic to *ER(R/p).

Proof. By Lemma 14.2.5, the graded S−1R-module *ER(R/p) is *injectiveover S−1R. Since a graded S−1R-submodule of *ER(R/p) is automatically agraded R-submodule, it is immediate from 14.2.4(i) that *ER(R/p) is *indec-omposable as S−1R-module.

There is a homogeneous generatorm of degree 0 of the gradedR-submoduleR/p of *ER(R/p). When the latter module is considered as a graded S−1R-module, (0 :S−1R m) = S−1p, and m still has degree 0. It therefore follows


from 14.2.4(ii) that there is a homogeneous S−1R-isomorphism *ER(R/p) ∼=*ES−1R(S

−1R/S−1p).The final claim is now immediate.

14.2.7 �Exercise. Suppose that our G-graded ring R is *simple, that is, Rhas exactly two graded ideals, namely 0 andR. Use the ‘graded Baer criterion’13.2.7 to show that every graded R-module is *injective.

14.2.8 �Exercise. Assume that a is graded, and that p ∈ *Spec(R) is suchthat a ⊆ p. Show that the graded R/a-module (0 :*ER(R/p) a) is homoge-neously isomorphic to *ER/a((R/a)/(p/a)).

14.2.9 �Exercise. Assume that (R,m) is *local. Show that there are homo-geneous isomorphisms

*HomR(R/m, *E(R/m)) ∼= (0 :*E(R/m) m) ∼= *ER/m(R/m) = R/m.

14.2.10 �Exercise. Show that each *injective graded R-module I is a directsum of *indecomposable *injective graded submodules. (Here is a hint: adaptthe argument in the proof of 10.1.8 to our G-graded situation, that is, applyZorn’s Lemma to the set of all sets of *indecomposable *injective graded sub-modules of I whose sum is direct.)

14.2.11 Definition and �Exercise. Let M be a graded R-module. A minimal*injective resolution of M is a *injective resolution

I• : 0 −→ I0d0

−→ I1 −→ · · · −→ Iidi

−→ Ii+1 −→ · · ·

of M (in the category *C(R)) such that Ii is a *essential extension of Ker di

for every i ∈ N0.Show that M has a minimal *injective resolution, and that the i-th term in

such a resolution is uniquely determined, up to isomorphism in *C(R), by M .We denote this i-th term by *Ei(M), or by *Ei

R(M).

With the notation of 14.2.11, and for i ∈ N0, it follows from 14.2.4(iii) and14.2.10 that there is a family (pα)α∈Λ of graded prime ideals ofR and a family(gα)α∈Λ of elements of G for which there is a homogeneous isomorphism

*Ei(M)∼=−→⊕

α∈Λ*E(R/pα)(−gα).

Of course, we would like, as in the analogous ungraded situation (see 11.1.4)to be able to show that, for p ∈ * Spec(R), the cardinality of the set

{α ∈ Λ : pα = p}

depends only on *Ei(M) and p (and therefore only on i, M and p) and not on


the particular decomposition of *Ei(M) (as a direct sum of *indecomposable*injective submodules) chosen. This is indeed the case, and, interestingly, thecardinality in question turns out to be equal to the ordinary Bass numberμi(p,M) of 11.1.4. We are, in the spirit of this book, going to establish this;however, instead of trying to imitate the standard ‘ungraded’ argument, weshall employ Theorem 13.2.4(iii).

14.2.12 Proposition. Let M be a graded R-module. By 14.2.10 and 14.2.4,there is a family (pα)α∈Λ of graded prime ideals of R and a family (gα)α∈Λ

of elements of G for which there is a homogeneous isomorphism *E(M)∼=−→⊕

α∈Λ *E(R/pα)(−gα).Let p ∈ *Spec(R). Then the cardinality of the set {α ∈ Λ : pα = p} is

equal to the ordinary Bass number μ0(p,M) (see 11.1.4), and so depends onlyon M and p and not on the particular decomposition of *E(M) (as a directsum of *indecomposable *injective submodules) chosen.

Proof. If (Nα)α∈Λ is a family of R-modules, then, by 14.2.1(ii) (applied inthe case where R is considered to be trivially G-graded, that is, R0 = R andRg = 0 for all g ∈ G \ {0}),

E(⊕

α∈ΛNα

) ∼=⊕α∈ΛE(Nα).

Now consider the given G-grading on R. For a graded R-module U andg ∈ G, it follows from 13.2.4(iii) that E(*E(U)(g)) ∼= E(U). Hence, on useof this and the preceding paragraph, we see that

E(M) ∼= E(*E(M)) ∼= E(⊕

α∈Λ*E(R/pα)(gα))

∼=⊕

α∈ΛE(*E(R/pα)(gα)) ∼=⊕

α∈ΛE(R/pα),

from which the claim is clear.

We want to extend the result of 14.2.12 to the ‘higher’ terms in minimal*injective resolutions. The following exercise concerns an important point inthe theory of Bass numbers for ungraded situations.

14.2.13 �Exercise. Let R′ be a commutative Noetherian ring, let M be anR′-module, and let

0 −→ E0(M)d0

−→ E1(M) −→ · · · −→ Ei(M)di

−→ Ei+1(M) −→ · · ·

be the minimal injective resolution of M , so that there is an augmentation R′-homomorphism α :M → E0(M) such that the sequence

0 −→Mα−→ E0(M) −→ · · · −→ Ei(M)

di

−→ Ei+1(M) −→ · · ·


is exact.Deduce from 11.1.7 that, for each p ∈ Spec(R′), the induced homomor-

phism HomR′p(R′p/pR

′p, αp) is an isomorphism.

It will also be convenient if we record now some further consequences ofLemma 13.2.4(iii).

14.2.14 Remarks. Let M be a graded R-module. Let

0 −→ *E0(M)e0−→ *E1(M) −→ · · · −→ *Ei(M)

ei−→ *Ei+1(M) −→ · · ·

be the minimal *injective resolution of M , with associated (necessarily ho-mogeneous) augmentation homomorphism β : M −→ *E0(M). Also, letα :M −→ E0(M) provide the injective envelope of M .

(i) Since E0(M) is injective, there is a commutative diagram

Mβ

*E0(M)�

φ0

αM E0(M)�

�

of R-modules and R-homomorphisms. Now *E0(M) is an essential ex-tension of Imβ, by 13.2.4(iii), and so, since Kerφ0 ∩ Imβ = 0, itfollows that φ0 is actually a monomorphism.

(ii) Let p ∈ Spec(R) and set k(p) := Rp/pRp. Now

HomRp(k(p), φ0p) ◦HomRp

(k(p), βp) = HomRp(k(p), αp),

which is an isomorphism by 14.2.13; however, the left-exactness of the‘Hom’ functor ensures that HomRp

(k(p), φ0p) and HomRp(k(p), βp) are

both monomorphisms, and so it follows that they are both isomorphisms.

Our need for a proof of the following lemma is explained by our ‘first vari-able only’ approach to the functors *ExtiR( • , N) (i ∈ N0): see 13.1.8.

14.2.15 Lemma. Let L,M be graded R-modules with M finitely generated.Let

I• : 0 −→ I0f0

−→ I1 −→ · · · −→ Iifi

−→ Ii+1 −→ · · ·

be a *injective resolution of L (in the category *C(R)). Then, for each i ∈ N0,there is a homogeneous isomorphism

*ExtiR(M,L) ∼= Hi(*HomR(M, I•)).


Proof. For each i ∈ N0, set Ki := Ker f i. Note that there is a homogeneousisomorphismK0 ∼= L. Suppose that j ∈ N with j > 1. For all i = 1, . . . , j−1,there is an exact sequence

0 −→ Ki−1 −→ Ii−1 −→ Ki −→ 0

of gradedR-modules and homogeneous homomorphisms. SinceM is a finitelygenerated gradedR-module, it follows from (13.2.5 and) 13.3.11 that there arehomogeneous isomorphisms

*ExtjR(M,L) ∼= *ExtjR(M,K0) ∼= *Extj−1R (M,K1) ∼= · · ·

∼= *Ext2R(M,Kj−2) ∼= *Ext1R(M,Kj−1).

This means that it is now enough for us to prove the claim in the statement ofthe lemma in the special cases in which i = 0 and i = 1. However, the claimsin these two cases follow easily from the exact sequences

0 −→ *HomR(M,L) −→ *HomR(M, I0) −→ *HomR(M,K1)

−→ *Ext1R(M,L) −→ 0

and 0 −→ *HomR(M,K1) −→ *HomR(M, I1) −→ *HomR(M,K2) ofgraded R-modules and homogeneous homomorphisms which can also be ob-tained from 13.2.5 and 13.3.11.

14.2.16 Theorem. LetM be a gradedR-module. Let j ∈ N0, and denote thej-th term in the minimal *injective resolution of M by *Ej(M), as in 14.2.11.By 14.2.10 and 14.2.4(iii), there is a family (pα)α∈Λ of graded prime ideals ofR and a family (gα)α∈Λ of elements of G for which there is a homogeneousisomorphism

*Ej(M)∼=−→⊕α∈Λ

*E(R/pα)(−gα).

Let p ∈ *Spec(R). Then the cardinality of the set {α ∈ Λ : pα = p} isequal to the ordinary Bass number μj(p,M).

Proof. The claim was proved in 14.2.12 in the special case in which j = 0.We consider next the case in which j = 1.

Use the notation introduced in 14.2.14, and setKi = Ker ei for each i ∈ N0.Also, let

0 −→ E0(M)d0

−→ E1(M) −→ · · · −→ Ei(M)di

−→ Ei+1(M) −→ · · ·

be the (ordinary, ungraded) minimal injective resolution ofM (with associated


augmentation homomorphism α : M −→ E0(M)). For each i ∈ N0, setCi := Ker di. Consider the commutative diagram

0 M *E0(M) K1 0� � � �

φ0 φ0

0 M E0(M) C1 0� � � ��

with exact rows, in which φ0 is the homomorphism induced by φ0. Localizethis diagram at p, and then apply the functor HomRp

(k(p), • ), noting thatExt1R(R/p, *E

0(M)) = 0 by 13.2.5, and that HomRp(k(p), φ0p) is an isomor-

phism by 14.2.14(ii). The Five Lemma shows that

HomRp(k(p), φ0p) : HomRp

(k(p), (K1)p)∼=−→ HomRp

(k(p), (C1)p)

is an isomorphism. Now *E1(M) = *E(K1), and so, by 14.2.12, the cardinal-ity of the set {α ∈ Λ : pα = p} is equal to μ0(p,K1); we have just shown thatthis is μ0(p, C1), which is equal to μ1(p,M) by the theory of Bass numbers inthe ungraded case.

Now suppose that j > 1. By 14.2.15, there is a homogeneous isomorphism*Ext1R(R/p,K

j−1) ∼= *ExtjR(R/p,M), so that μ1(p,Kj−1) = μj(p,M).However,

0 −→ Kj−1 ⊆−→ *Ej−1(M)dj−1

−→ *Ej(M)

is the start of the minimal *injective resolution ofKj−1, and so it follows fromwhat we have already proved in the case in which j = 1 that the cardinality ofthe set {α ∈ Λ : pα = p} is equal to μ1(p,Kj−1), which we now know to beequal to μj(p,M).

14.3 A graded version of the Annihilator Theorem

The purpose of this section is to present graded versions of Faltings’ Annihi-lator Theorem 9.5.1 and Grothendieck’s Finiteness Theorem 9.5.2. To preparefor this, we are going to present some results due to S. Goto and K.-i. Watanabe[22]. Recall that we are assuming that R is G-graded throughout this chapter.The key points are that, ifM is a non-zero finitely generated gradedR-module,then, for each p ∈ SuppM , we have p∗ ∈ SuppM and

dimMp = dimMp∗ + ht p/p∗ and depthMp = depthMp∗ + ht p/p∗.

14.3 A graded version of the Annihilator Theorem 303

We begin with a description of the structure of *simple G-graded commuta-tive Noetherian rings.

14.3.1 Example. Let K be a field and let G′ be a subgroup of G. Then thegroup ring K[G′] is a G-graded commutative ring with

K[G′]g =

{Kg if g ∈ G′,0 if g ∈ G′.

Let e1, . . . , eh be a free base for G′ as Abelian group, and, for each i =

1, . . . , h, use Ti to denote the element ei = 1ei in K[G′]. Then T1, . . . , Thare algebraically independent over K, and

K[G′] ∼= K[T1, . . . , Th, T−11 , . . . , T−1

h ]

under an isomorphism which maps ei to Ti (for i ∈ {1, . . . , h}). ThusK[G′] isa (Noetherian) regular unique factorization domain. Note also that every non-zero homogeneous element of K[G′] is a unit, and so K[G′] is a *simple G-graded commutative Noetherian ring. Our first aim in this section is to establishthe converse statement.

14.3.2 Theorem. (See S. Goto and K.-i. Watanabe [22, Theorem 1.1.4].)(Recall that R is G-graded.) The following statements are equivalent.

(i) There is a fieldK and a subgroupG′ ofG such thatR is homogeneouslyisomorphic to K[G′], where the latter is G-graded in the manner de-scribed in 14.3.1.

(ii) The G-graded ring R is *simple.(iii) Every graded R-module is *injective and *free and R is non-trivial.

Proof. (i)⇒ (ii) This was proved in 14.3.1.(ii)⇒ (iii) Assume thatR is *simple. This means that the only graded ideals

of R are 0 and R itself. Let M be a graded R-module. It is immediate from the‘graded Baer criterion’ 13.2.7 that M is *injective. Therefore, by 14.2.10 and14.2.4(iii), there is a family (pα)α∈Λ of graded prime ideals of R and a family(gα)α∈Λ of elements of G for which there is a homogeneous isomorphismM

∼=−→⊕

α∈Λ *E(R/pα)(−gα). However, the only proper graded ideal of Ris 0, and so, since every graded R-module is *injective, we have *E(R/pα) =

R for all α ∈ Λ. Therefore M is *free.(iii) ⇒ (i) Assume that every graded R-module is *injective and *free. Let

a be a proper graded ideal of R. By assumption, the graded R-module R/a isfree, and so has zero annihilator. Therefore a = 0. Thus R is *simple.


Let g ∈ G. Each non-zero homogeneous element ug ∈ Rg must be in-vertible in R, with u−1

g ∈ R−g . In particular, each non-zero element of R0 isinvertible in R0, so that R0 is a field, K say, and, for each g′ ∈ G for whichRg′ = 0, the R0-module Rg′ , that is, the K-vector space Rg′ , is generated byany one of its non-zero elements, for if u, v ∈ Rg′\{0}, then vu−1 ∈ R0 = K,so that v ∈ Ku. Also in particular, the product of two non-zero homogeneouselements of R is again non-zero. Let G′ := {g ∈ G : Rg = 0}, a subgroupof G. Let e1, . . . , eh be a free base for G′ as Abelian group, and, for eachi = 1, . . . , h, let Ti be a non-zero element of Rei .

Consider K[G′] as G-graded in the manner described in 14.3.1. There isa homogeneous ring homomorphism φ : K[G′] −→ R which acts as theidentity on K = R0 and maps ei to Ti (for each i = 1, . . . , h). The choice ofG′ ensures that φ is surjective, and since Kerφ is a proper graded ideal of the*simple G-graded ring K[G′], we must have that φ is an isomorphism.

14.3.3 Remark. It follows from 14.3.1 and 14.3.2 that, if R is *simple, thenR is a regular unique factorization domain with dimR ≤ rankG.

14.3.4 Corollary. Each *maximal ideal of R is *prime.

Proof. Let m be a *maximal ideal of R. Then the G-graded ring R/m hasexactly two graded ideals, and so is *simple. Therefore R/m is an integraldomain, by 14.3.3.

14.3.5 Proposition. Assume that the G-graded ring R is an integral domain,and let p ∈ Spec(R). Then ht p = ht p* + ht p/p*.

Proof. Recall that p* ∈ Spec(R). It is clear that ht p ≥ ht p* + ht p/p*; weshall prove the opposite inequality by induction on h := ht p*. When h = 0,we have p* = 0 (because R is an integral domain), so that ht p = ht p/p* andthe desired inequality is clear.

So suppose, inductively, that h > 0 and that the result has been proved insituations where ht p* < h. Denote ht p by n. Let 0 = r be a homogeneouselement of R contained in p*. Then dimRp/rRp = n − 1 because R is anintegral domain, and so there exists a chain of prime ideals p0 ⊂ p1 ⊂ · · · ⊂pn−1 = p of R with r ∈ p0. Now p0, being a minimal prime ideal of thegraded ideal Rr, must be graded, and so p0 ⊆ p*. In fact, (p/p0)* = p*/p0.Note that ht p/p0 = n− 1 and ht(p/p0)* ≤ h− 1.

Therefore, if we apply the inductive hypothesis to the prime ideal p/p0 inthe G-graded integral domain R/p0, we see that

ht p/p0 = ht(p/p0)* + ht(p/p0)/(p/p0)* = ht(p/p0)* + ht p/p*,


so that n − 1 ≤ h − 1 + ht p/p*. Therefore ht p ≤ ht p* + ht p/p*, and theinductive step is complete.

The next theorem, again due to Goto and Watanabe, is a strengthening of14.3.5.

14.3.6 Theorem. (See S. Goto and K.-i. Watanabe [22, Proposition 1.2.2].)(The reader is reminded that R is G-graded.) Let p ∈ SuppM where M is agraded R-module. Recall from 13.1.6(i) that p* ∈ SuppM . We have

dimMp = dimMp∗ + ht p/p*.

Proof. For q ∈ SuppM , we shall denote dimMq by htM q. It is clear thathtM p ≥ htM p*+ht p/p*; we shall prove the opposite inequality by inductionon h := htM p*.

First suppose that h = 0. After homogeneous localization at p, we can, anddo, assume that (R, p*) is *local. Since h = 0, it follows that p* is the one andonly graded prime ideal in SuppM ; therefore AssM = {p*}. Hence eachq ∈ SuppM such that q ⊆ p must have q* = p*, and it is clear from this thathtM p = ht p/p*.

Now suppose, inductively, that h > 0 and the desired result has been provedfor smaller values of h. We can again assume that (R, p*) is *local. DenotehtM p by n and ht p/p* by d. Since n ≥ h+ d, we have n > d.

There exists a chain of prime ideals p0 ⊂ p1 ⊂ · · · ⊂ pn = p in SuppM .Set q := pn−d; then htM q = n− d and ht p/q = d. If q ⊆ p*, then n− d =

htM q ≤ htM p* = h and n ≤ d+ h, as required. So we suppose that q ⊆ p*,so that q is not graded because (R, p*) is *local. We do have q* ⊆ p* andq* ∈ SuppM . Let t := ht q/q* and u := ht p*/q*. Note that t > 0 because qis not graded.

By 14.3.5 applied to the G-graded integral domain R/q*, we have

ht p/q* = ht(p/q*)* + ht(p/q*)/(p/q*)* = ht p*/q* + ht p/p* = u+ d.

But ht p/q* ≥ ht p/q+ht q/q* = d+t. Therefore u ≥ t > 0. This means thatq* is strictly contained in p*, so that htM q* < h. The inductive hypothesistherefore yields that n− d = htM q ≤ htM q* + ht q/q* = htM q* + t. NowhtM q* + u = htM q* + ht p*/q* ≤ htM p* = h. Therefore

n− d ≤ htM q* + t ≤ h− u+ t,

so that

n ≤ d+ h+ (t− u) ≤ d+ h

since u ≥ t. This completes the inductive step.


In addition to Theorem 14.3.6, we would like to have available the compan-ion result, also due to Goto and Watanabe, that, if M is a finitely generatedgraded R-module and p ∈ SuppM , then depthMp = depthMp∗ + ht p/p*.We shall obtain this as a corollary of the following result about Bass numbers.

14.3.7 Theorem. (See S. Goto and K.-i. Watanabe [22, Theorem 1.2.3].)(Recall that R is G-graded.) Let M be a graded R-module. Let p ∈ Spec(R)

and let d := ht p/p*. Then

μi(p,M) =

{0 if 0 ≤ i < d,

μi−d(p*,M) if i ≥ d.

Proof. By homogeneous localization at p, and use of the ‘invariance’ of Bassnumbers under fraction formation (see [50, p.150]), we reduce to the casewhere (R, p*) is *local. Then R/p* is a *simple G-graded ring, and so, by14.3.2, every graded R/p*-module is *free. Also, d is the dimension of thelocal ring Rp/p*Rp, and this is regular, by 14.3.3. Let u1, . . . , ud ∈ p be suchthat their natural images in Rp/p*Rp, which we denote by α1, . . . , αd respec-tively, generate the maximal ideal of that regular local ring. Thus

pRp = p*Rp + (α1, . . . , αd)Rp.

For each j = 0, . . . , d, set Qj := p*Rp + (α1, . . . , αj)Rp, and note thatthis is a prime ideal of Rp. Therefore, if j < d there is an exact sequence of(ungraded) Rp-modules and Rp-homomorphisms

0 � � � �Rp/Qj Rp/Qj Rp/Qj+1

αj+1

0,

and this induces a long exact sequence

0 HomRp(Rp/Qj+1,Mp) HomRp

(Rp/Qj ,Mp)� �

HomRp(Rp/Qj ,Mp) Ext1Rp

(Rp/Qj+1,Mp)� �

· · ·�

ExtiRp(Rp/Qj+1,Mp) ExtiRp

(Rp/Qj ,Mp)� �

ExtiRp(Rp/Qj ,Mp) Exti+1

Rp(Rp/Qj+1,Mp)� �

· · · .�

αj+1

αj+1

Note that Q0 = p*Rp. Let i ∈ N0. Now ExtiR(R/p*,M) is the R-moduleunderlying the graded R-module *ExtiR(R/p*,M). The latter has a naturalstructure as graded R/p*-module, and we observed above that every graded


R/p*-module is *free. Consequently, ExtiR(R/p*,M) is a freeR/p*-module;therefore α1 is a non-zerodivisor on ExtiRp

(Rp/Q0,Mp) and, in view of theabove long exact sequence (in the case where j = 0), we have

ExtiRp(Rp/Q1,Mp) ∼= Exti−1

Rp(Rp/Q0,Mp)/α1 Ext

i−1Rp

(Rp/Q0,Mp)

(as Rp-modules) for all i ∈ Z. (When i is negative, the statement is obviouslytrue.) Our immediate aim is to show that

ExtiRp(Rp/Qj ,Mp)

∼= Exti−jRp

(Rp/Q0,Mp)/(α1, . . . , αj) Exti−jRp

(Rp/Q0,Mp)

for all i ∈ Z for all j = 2, . . . , d.So suppose, inductively, that j ∈ {1, . . . , d− 1} and we have shown that

ExtiRp(Rp/Qj ,Mp)

∼= Exti−jRp

(Rp/Q0,Mp)/(α1, . . . , αj) Exti−jRp

(Rp/Q0,Mp)

(as Rp-modules) for all i ∈ Z. Since ExtkRp(Rp/Q0,Mp) is a free Rp/Q0-

module for all k ∈ Z, it follows that ExtiRp(Rp/Qj ,Mp) is a free Rp/Qj-

module for all i ∈ Z. For each i ∈ Z it therefore follows that αj+1 is a non-zerodivisor on Exti−1

Rp(Rp/Qj ,Mp), so that, in view of the above long exact

sequence,

ExtiRp(Rp/Qj+1,Mp) ∼= Exti−1

Rp(Rp/Qj ,Mp)/αj+1 Ext

i−1Rp

(Rp/Qj ,Mp)

(as Rp-modules). We can now use the inductive hypothesis to deduce that

ExtiRp(Rp/Qj+1,Mp)

∼= Exti−j−1Rp

(Rp/Q0,Mp)/(α1, . . . , αj+1) Exti−j−1Rp

(Rp/Q0,Mp)

(as Rp-modules) for all i ∈ Z. This completes the inductive step.Since Qd = pRp, it therefore follows, by induction, that

ExtiRp(Rp/pRp,Mp)

∼= Exti−dRp

(Rp/Q0,Mp)/(α1, . . . , αd) Exti−dRp

(Rp/Q0,Mp)

= Exti−dRp

(Rp/p*Rp,Mp)/(α1, . . . , αd) Exti−dRp

(Rp/p*Rp,Mp)

(as Rp-modules) for all i ∈ Z.For each prime ideal q of R, denote the field Rq/qRq by k(q). Let i ∈ Z.

Recall from [50, Theorem 18.7] that

μi(q,M) = dimk(q) ExtiRq

(k(q),Mq) = dimk(q)

(ExtiR(R/q,M)

)q.

(We interpret μj(q,M) for negative j as zero.) Now Exti−dR (R/p*,M) is


a free R/p*-module, and it follows from that that Exti−dRp

(Rp/p*Rp,Mp)

is a free Rp/p*Rp-module of the same rank. At this point, recall also thatpRp = p*Rp + (α1, . . . , αd)Rp. It therefore follows from the above inductiveargument that

μi−d(p*,M) = dimk(p∗)(Exti−d

R (R/p*,M))p∗

= rankR/p∗ Exti−dR (R/p*,M)

= rankRp/p∗RpExti−d

Rp(Rp/p*Rp,Mp)

= rankk(p) ExtiRp

(Rp/pRp,Mp)

= dimk(p) ExtiRp

(k(p),Mp) = μi(p,M).

In particular, μi(p,M) = 0 for all i < d.

When discussing depths of localizations of a finitely generated R-module,we shall follow the notation and conventions of 9.2.1.

14.3.8 Corollary. Let M be a finitely generated graded R-module. Let p ∈Spec(R). Then

depthMp = depthMp∗ + ht p/p*.

Proof. Recall from from [50, Theorems 16.7 and 18.7] that, for q ∈ Spec(R),depthMq is the least integer i such that μi(q,M) = 0 (if any such integersexist, and∞ otherwise). The claim is therefore immediate from 14.3.7.

14.3.9 �Exercise. (Recall that R is G-graded.)

(i) Show that the following statements are equivalent:

(a) R is Cohen–Macaulay;(b) R(p) is Cohen–Macaulay for all p ∈ *Spec(R);(c) Rp is Cohen–Macaulay for all p ∈ *Spec(R).

(ii) Show that the following statements are equivalent:

(a) R is Gorenstein;(b) R(p) is Gorenstein for all p ∈ *Spec(R);(c) Rp is Gorenstein for all p ∈ *Spec(R).

We are now in a position to present G-graded versions of Faltings’ Anni-hilator Theorem 9.5.1 and Grothendieck’s Finiteness Theorem 9.5.2. Loosely,these graded versions say that, in the G-graded case, both theorems can be re-formulated so that only graded prime ideals need be considered. As the GradedFiniteness Theorem has important geometric significance in connection withthe cohomology of projective schemes, we provide a proof.

14.4 Graded local duality 309

14.3.10 Graded Annihilator and Finiteness Theorems. (Recall that R isG-graded.) Assume that R is a homomorphic image of a regular (commutativeNoetherian) ring. Assume that the ideal a is graded; let b be a second gradedideal of R. Let M be a finitely generated graded R-module. Then

fba (M) = inf {adja depthMp : p ∈ *Spec(R) \Var(b)}= inf {depthMp + ht(a+ p)/p : p ∈ *Spec(R) \Var(b)} .

In particular,

fa(M) = inf {depthMp + ht(a+ p)/p : p ∈ *Spec(R) \Var(a)} .

Proof. By Faltings’ Annihilator Theorem 9.5.1,

fba (M) = inf {depthMp + ht(a+ p)/p : p ∈ Spec(R) \Var(b)} .

It is therefore sufficient for us to show that, for each non-graded prime idealp ∈ Spec(R) \ Var(b) for which depthMp is finite, the graded ideal p* issuch that p* ∈ *Spec(R) \Var(b) and

depthMp∗ + ht(a+ p*)/p* ≤ depthMp + ht(a+ p)/p.

This we do.Of course p* is a graded prime ideal ofR (see 13.1.3(vi)), and it is clear that

p* ⊇ b. Let d := ht p/p*; by 14.3.8, depthMp = depthMp∗ + d.Let t := ht(a+p)/p and let q be a minimal prime of a+p such that ht q/p =

ht(a+ p)/p. Since R is a homomorphic image of a regular ring, it is catenary,and therefore ht q/p* = ht q/p+ht p/p* = t+ d. Since a+ p* ⊆ a+ p ⊆ q,it follows that ht(a+ p*)/p* ≤ ht q/p* = t+ d. Therefore

depthMp∗ + ht(a+ p*)/p* ≤ depthMp − d+ t+ d

= depthMp + ht(a+ p)/p,

and this completes the proof.

14.4 Graded local duality

The purpose of this section is to develop a G-graded analogue of the LocalDuality Theorem 11.2.6. This will concern the situation where (R,m) is a*local G-graded ring that can be expressed as a homomorphic image of aGorenstein *local G-graded (commutative Noetherian) ring by means of a ho-mogeneous homomorphism. Many of the G-graded rings that occur in appli-cations are finitely generated algebras over fields, and so we are only imposinga mild restriction.


Recall that we are assuming that R is G-graded throughout this chapter.

14.4.1 Graded Local Duality Theorem. Assume that (R,m) is *local withhtm = n. Assume also that there is a Gorenstein *local G-graded com-mutative Noetherian ring (R′,m′) and a surjective homogeneous ring ho-momorphism f : R′ −→ R. Let htm′ = n′. Let *D denote the functor*HomR( • , *E(R/m)) from *C(R) to itself.

Let M be a graded R-module, let N ′ be a graded R′-module, and let j ∈N0. Now M can be regarded as a graded R′-module by means of f ; then,*ExtjR′(M,N ′) has a natural structure as graded R-module.

There exists g ∈ G such that Hn′m′(R′) ∼= *ER′(R′/m′)(−g) in *C(R′). For

any such g, there is a homomorphism

Ψ := (ψi)i∈N0 :(Hi

m

)i∈N0

−→(

*D(*Extn′−i

R′ ( • , R′(g))))i∈N0

of negative connected sequences of covariant functors from *C(R) to *C(R)which is such that ψi

M is a (necessarily homogeneous) isomorphism for alli ∈ N0 whenever M is a finitely generated graded R-module.

Proof. We shall first deal with the special case where R = R′ (and f isthe identity map), so that (R,m) itself is a Gorenstein *local G-graded ring.Since dimRm = n , it follows from Grothendieck’s Vanishing Theorem 6.1.2that (Hj

m(N))m ∼= HjmRm

(Nm) = 0 for all j > n and all R-modules N .Therefore, by 14.1.2(ii), we have Hj

m(M) = 0 for all j > n and all gradedR-modules M . Consequently, Hn

m is a right exact functor from *C(R) to it-self, and

(Hn−i

m

)i∈N0

is a positive strongly connected sequence of covariantfunctors from *C(R) to *C(R).

Let g ∈ G. Note that the long complex resulting from application of thepositive connected sequence

(*D(*ExtiR( • , R(g)))

)i∈N0

to a short exact se-quence of graded R-modules and homogeneous homomorphisms

0 −→ L −→M −→ N −→ 0

is exact. Thus(*D(*ExtiR( • , R(g)))

)i∈N0

is a positive strongly connectedsequence of covariant functors from *C(R) to itself.

Our next aim is to find a natural equivalence between the functors Hnm and

*D(*HomR( • , R(g))) (from *C(R) to itself) for a suitable g ∈ G. Since Ris Gorenstein, it follows from 14.2.16 that there is a family (gp)p∈∗Spec(R) ofelements of G such that there is a homogeneous isomorphism

*EiR(R)

∼=⊕

p∈∗Spec(R)ht p=i

*ER(R/p)(−gp) for each i ∈ N0.

14.4 Graded local duality 311

We can calculate the graded module Hnm(R) by applying Γm to the minimal

*injective resolution of R and then taking cohomology; in view of 14.2.4(iv),the result is that Hn

m(R)∼= *ER(R/m)(−gm) in *C(R).

We can now modify the ideas of 11.2.4 to establish the existence of a naturaltransformation

φ : Hnm −→ *HomR(*HomR( • , R), Hn

m(R))

of functors from *C(R) to itself, for which, for each graded R-module M , wehave (φM (y))(f) = Hn

m(f)(y) for all y ∈ Hnm(M) and f ∈ *HomR(M,R).

One easily checks that φR is an isomorphism.Choose g ∈ G such that Hn

m(R)∼= *ER(R/m)(−g) in *C(R). (For exam-

ple, we could take g := gm.) We obtain from φ a natural transformation

ψ0 : Hnm −→ *HomR(*HomR( • , R), *ER(R/m)(−g))= *HomR(*HomR( • , R(g)), *ER(R/m))

of functors from *C(R) to itself, for which ψ0R is an isomorphism. Note that,since R is Gorenstein, (Hn−i

m (R))m ∼= Hn−imRm

(Rm) = 0 for all i > 0, so thatHn−i

m (R) = 0 for all i > 0 by 14.1.2(ii). Also *ExtiR′(P,R′(g)) = 0 forall i > 0 for all *projective graded R-modules P . It follows from the gradedversion of the analogue of 1.3.4 for positive connected sequences that ψ0 canbe incorporated into a (uniquely determined) homomorphism


m

)i∈N0

−→(*D(*ExtiR( • , R(g)))

)i∈N0

of positive connected sequences of covariant functors from *C(R) to *C(R).Furthermore, it is easy to prove by induction that, for each i ∈ N, the homo-morphism φiM is an isomorphism whenever M is a finitely generated gradedR-module: use the fact that such an M can be included in an exact sequence0 −→ K −→ F −→ M −→ 0 in *C(R) in which F is a finitely generated*free R-module. Note that we can interpret

(φn−i)i∈N0 :(Hi

m

)i∈N0

−→(*D(*Extn−i

R ( • , R(g))))i∈N0

as a homomorphism of negative connected sequences.We have thus established the claims of the theorem in the case where R =

R′. We now deal with the general case, where there is a Gorenstein *local G-graded commutative Noetherian ring (R′,m′) and a surjective homogeneousring homomorphism f : R′ −→ R. Let *D′ := *HomR′( • , *ER′(R′/m′)).Our work so far in this proof establishes the existence of a g ∈ G such thatHn′

m′(R′) ∼= *ER′(R′/m′)(−g) in *C(R′) and, for such a g, the existence of a


homomorphism

Ψ := (ψi)i∈N0 :(Hi

m′)i∈N0

−→(

*D′(*Extn′−i

R′ ( • , R′(g))))i∈N0

of negative connected sequences of covariant functors from *C(R′) to *C(R′)such that φiM is a (necessarily homogeneous) isomorphism whenever M is afinitely generated graded R′-module. To complete the proof, we use the argu-ment in the proof of the Local Duality Theorem 11.2.6, modified for the G-graded context. Important ingredients include the Graded Independence The-orem 14.1.7, and the fact that the graded submodule (0 :*ER′ (R′/m′) Ker f)

of *ER′(R′/m′) is, when viewed as a graded R-module, homogeneously iso-morphic to *ER(R/m).

Note. It should be noted that the element g in the statement of the GradedLocal Duality Theorem 14.4.1 need not be uniquely determined. The interestedreader is referred to [6, Lemma 1.5].

In the situation of 14.4.1, the functor *D = *HomR( • , *E(R/m)) wouldseem to be an obvious graded analogue of the functor of 10.2.1 used to con-struct Matlis duals over a local ring. However, there is another approach to *Dwhich is particularly useful when the 0-th component R0 of the *local ring Ris a field, and this approach is the subject of the next exercise. The exercise isbased on [7, Proposition 3.6.16], but our approach is a little different from thatof Bruns and Herzog; also, we are working in the G-graded context, whereasBruns’ and Herzog’s treatment is for Z-graded rings.

14.4.2 �Exercise: Graded Matlis Duality. Suppose (R,m) is *local. Notethat R0 is local with maximal ideal m0 := m ∩R0; let E0 := ER0(R0/m0).

(i) Let M =⊕

g∈GMg be a graded R-module. Now R0 can be consid-ered as a G-graded ring with trivial grading, and any R0-module canbe considered as a graded R0-module concentrated in degree 0. Also,the grading on our graded R-module M provides a structure as gradedR0-module on M .

Show that M∨ := *HomR0(M,E0) has a natural structure as agraded R-module with g-th component (M∨)g = HomR0(M−g, E0)

for all g ∈ G. (Remember that *HomR0(M,E0) is a submodule ofHomR0(M,E0), and the latter has a natural structure as an R-module.)

Deduce that ( • )∨ is an exact, additive functor from *C(R) to itself.(ii) Show that the functors ( • )∨ and *HomR( • , R∨), that is, the functors

*HomR0( • , E0) and *HomR( • , *HomR0(R,E0)), from *C(R) toitself, are naturally equivalent. (Here is a hint: if we forget the gradings,

14.5 *Canonical modules 313

there is a natural equivalence

κ : HomR0( • , E0) −→ HomR( • ,HomR0(R,E0));

for a graded R-module M , consider an appropriate restriction of κM .)(iii) Let φ0 : R −→ E0 be the R0-homomorphism whose restriction to R0

is the composition of the natural epimorphism R0 → R0/m0 and theinclusion R0/m0 → E0 = ER0(R0/m0), and whose restriction to Rg

for all g ∈ G \ {0} is zero. Show that R∨ is a *essential extension of itsgraded submodule Rφ0.

(iv) Show that R∨ ∼= *ER(R/m) (in the category *C(R)), and deduce thatthe functors ( • )∨ and *D := *HomR( • , *E(R/m)) (from *C(R) toitself) are naturally equivalent.

(v) Show that, if the local ringR0 is complete, then, wheneverM is a finitelygenerated graded R-module, there is a homogeneous R-isomorphismM ∼= (M∨)∨ =:M∨∨.

Thus, in the situation of Exercise 14.4.2, it is reasonable for us to regardthe functor ( • )∨ = *HomR0( • , ER0(R0/m0)) as the ‘graded Matlis dualityfunctor’. In the particular case in whichR0 is a fieldK (and this is the situationin many practical applications of graded ring theory),ER0(R0/m0) = K, and,for a graded R-module M =

⊕g∈GMg , the grading of the ‘graded Matlis

dual’ is given by the attractively simple formula

M∨ = *HomK(M,K) =⊕g∈G

HomK(M−g,K).

14.5 *Canonical modules

Recall that we are assuming that R is G-graded throughout this chapter. Also,recall from 12.1.2 that a canonical module for a local ring (R′,m′) is a finitelygenerated R′-module K such that HomR′(K,ER′(R′/m′)) ∼= HdimR′

m′ (R′).The obvious graded analogue is given in the following definition.

14.5.1 Definition. Suppose that (R,m) is *local; let htm = n. A *canonicalmodule for R is a finitely generated graded R-module C for which there is ahomogeneous isomorphism

*HomR(C, *ER(R/m)) ∼= Hnm(R).

14.5.2 Example. Assume that (R,m) is *local with htm = n, and that thereis a Gorenstein *local G-graded commutative Noetherian ring (R′,m′) withhtm′ = n′ and a surjective homogeneous ring homomorphism f : R′ −→ R.


It follows from the Graded Local Duality Theorem 14.4.1 that, for someg ∈ G, there is a homogeneous isomorphism

*Hom(*Extn′−n

R′ (R,R′(g)), *ER(R/m)) ∼= Hnm(R).

Therefore *Extn′−n

R′ (R,R′(g)) is a *canonical module for R.

14.5.3 Proposition. Assume that (R,m) is *local with a *canonical moduleC. Then Cm is a canonical module for Rm.

Proof. By 13.2.4(iii), the R-module *ER(R/m) (with its grading forgot-ten) is an essential extension of R/m, and so there is a monomorphism α :

*ER(R/m) −→ ER(R/m). Denote *ER(R/m) by *E and ER(R/m) by E.Let K := Cokerα, and let k(m) denote the residue field of the local ring Rm.When the grading on *E is forgotten, we have

Ext1Rm(k(m), (*E)m) ∼= (Ext1R(R/m, *E))m = 0,

by 13.2.5. The exact sequence 0 −→ *E α−→ E −→ K −→ 0 thereforeinduces an exact sequence

0 −→ HomRm(k(m), (*E)m) −→ HomRm

(k(m), Em)

−→ HomRm(k(m),Km) −→ 0.

Since HomRm(k(m), Em) is a 1-dimensional vector space over k(m) and

HomRm(k(m), (*E)m) = 0,

we must have HomRm(k(m),Km) = 0; therefore Km = 0, since Km is an

Artinian Rm-module.Therefore αm is an isomorphism.Since C is a *canonical module forR, there is a homogeneousR-isomorph-

ism *HomR(C, *ER(R/m)) ∼= Hnm(R), where n = htm. Now forget the

gradings, localize at m, and use the isomorphism αm: there result isomor-phisms of Rm-modules

HomRm(Cm, ERm

(Rm/mRm)) ∼= HomRm(Cm, (ER(R/m))m)

∼= HomRm(Cm, (*ER(R/m))m)

∼= (HomR(C, *ER(R/m)))m∼= (Hn

m(R))m∼= Hn

mRm(Rm).

Therefore Cm is a canonical module for Rm.

Limitations on space mean that we are not able, in this book, to develop thetheory of *canonical modules in generality similar to that of Chapter 12. As


a fairly short treatment is possible in the Cohen–Macaulay case, we contentourselves with that.

14.5.4 Theorem. Assume that (R,m) is *local and Cohen–Macaulay andthat C is a finitely generated graded R-module such that μi(m, C) = δi,htmfor all i ∈ N0. Then Cp is a canonical module for Rp for all (graded orungraded) p ∈ Spec(R); that is, C is a canonical module for R in the sense of12.1.28. Consequently, for all p ∈ Spec(R), we have μi(p, C) = δi,ht p for alli ∈ N0.

Proof. It follows from 12.1.27 that Cm is a canonical module for Rm. Letp ∈ Spec(R), so that p* ∈ *Spec(R) and p* ⊆ m. It also follows from12.1.27 that Cp∗ is a canonical module for Rp∗ , so that μi(p*, C) = δi,ht p∗

for all i ∈ N0. It now follows from the Goto–Watanabe Theorems 14.3.6 and14.3.7 that μi(p, C) = δi,ht p for all i ∈ N0. Therefore Cp is a canonicalmodule for Rp, by 12.1.27 again.

14.5.5 Corollary. Assume that (R,m) is *local and Cohen–Macaulay andthat C is a *canonical module for R. Then C is a canonical module for R inthe sense of 12.1.28.

Proof. It follows from 14.5.3 that Cm is a canonical module for Rm. There-fore, by 12.1.26, we have μi(m, C) = μi(mRm, Cm) = δi,htm for all i ∈ N0.Now 14.5.4 shows that C is a canonical module for R.

14.5.6 Lemma. Assume that R is Cohen–Macaulay and has a canonicalmodule C. We saw in 12.1.30 that the trivial extension R′ = R ∝ C of R by Cis a Gorenstein ring, so that R is a homomorphic image of a Gorenstein ring.

If C = ⊕g∈GCg is graded, then R′ is G-graded in such a way that thecanonical homomorphism φ : R′ −→ R is homogeneous. If, in addition, R is*local, then so too is R′.

Proof. The decomposition R′ = ⊕g∈G(Rg ⊕ Cg) provides a grading on R′,and φ is homogeneous with respect to this grading and the original grading onR. Furthermore, if m is the unique *maximal graded ideal of R, then a routinecheck shows that m⊕ C is the unique *maximal graded ideal of R′.

The following exercise will be very helpful in our development of the theoryof *canonical modules.

14.5.7 �Exercise. Let α : M −→ N be a homogeneous homomorphism ofgraded R-modules.

(i) Show that, if αp :Mp −→ Np is an isomorphism for all p ∈ *Spec(R),then α is an isomorphism.


(ii) Show that, if (R,m) is *local and αm :Mm −→ Nm is an isomorphism,then α is an isomorphism.

When (R,m) is *local and Cohen–Macaulay, a *canonical module C for Ris a canonical module for R (by 14.5.5), and so satisfies μi(p, C) = δi,ht p forall i ∈ N0 and p ∈ Spec(R). We are now going to study a general finitely gen-erated graded R-module with the property that, when its grading is forgotten,it is a canonical module for R.

14.5.8 Lemma. Assume (R,m) is *local and Cohen–Macaulay; set n :=

htm. Suppose that C is a finitely generated graded R-module which is acanonical module forR in the sense of 12.1.28, so that, for all p ∈ *Spec(R),we have μi(p, C) = δi,ht p.

(i) There is a family (gp)p∈∗Spec(R) of elements of G for which there existhomogeneous isomorphisms

*Ei(C) ∼=⊕

p∈∗Spec(R)ht p= i

*E(R/p)(−gp) for all i ∈ N0.

(ii) For any family (gp)p∈∗Spec(R) of elements of G as in part (i), there arehomogeneous isomorphisms

(a) *ExtiR(R/m, C) ∼= 0 for i = n,(b) *ExtnR(R/m, C) ∼= (R/m)(−gm), and(c) Hn

m(C)∼= *E(R/m)(−gm).

Proof. (i) This is immediate from Theorem 14.2.16.(ii) Let E• denote the minimal *injective resolution of C.For each p ∈ * Spec(R) \ {m}, there exists a homogeneous element rp ∈

m \ p, and so it follows from 14.2.4(iv) that *HomR(R/m, *E(R/p)) = 0

and Γm(*E(R/p)) = 0. Hence the complex *HomR(R/m, E•) has all terms

other than its n-th equal to 0, while its n-th term is isomorphic (in *C(R))to *HomR(R/m, *E(R/m)(−gm)). Parts (a) and (b) therefore follow from14.2.15 and 14.2.9, while part (c) is a consequence of the fact (see 13.4.3) thatwe can calculate the graded R-module Hn

m(C) by application of the functorΓm to E•.

Recall that, in 12.1.6, we proved that any two canonical modules for a localring are isomorphic. We are now going to address the analogous issue in theCohen–Macaulay G-graded *local case.

14.5.9 Theorem. Suppose (R,m) is Cohen–Macaulay and *local,


and let C,C ′ be graded R-modules that are canonical modules for R in thesense of 12.1.28. Let n := htm. Then

(i) the map β : R −→ *HomR(C,C) defined by β(r) = r IdC for allr ∈ R is a homogeneous isomorphism; and

(ii) there exist g ∈ G and a homogeneous isomorphism φ : C∼=−→ C ′(g).

Proof. (i) For each p ∈ Spec(R), the Rp-module Cp is a canonical modulefor Rp. We can now deduce from 12.2.6 that βp is an isomorphism for allp ∈ Spec(R). Hence β is an isomorphism, and it is clearly homogeneous.

(ii) LetH denote the gradedR-module *HomR(C,C′). It follows from part

(i) above and 12.1.6 that Hp is a free Rp-module of rank 1, for all (graded orungraded) p ∈ Spec(R).

Let {φ1, . . . , φt} be a generating set, consisting of t homogeneous elements,for H , that is minimal in the sense that no proper subset of it also generates H .Let deg φi = gi for i = 1, . . . , t.

There is a graded *free R-module F , of the form⊕t

i=1R(−gi), and a ho-mogeneous R-epimorphism ψ : F −→ H which maps the generator, ei say,of degree gi in R(−gi) to φi.

Let K := Kerψ, a graded submodule of F . Our immediate aim is toshow that K ⊆ mF . Suppose that this is not the case; then there exists ahomogeneous element k ∈ K \ mF . Since k is homogeneous, we can writek =∑t

i=1 riei, where, for i ∈ {1, . . . , t}, the element ri ∈ R is homogeneousof degree deg k − gi. Then there exists j ∈ {1, . . . , t} such that rj ∈ m, sothat, since rj is homogeneous, it must be a unit of R. Application of ψ there-fore yields that 0 = ψ(k) =

∑ti=1 riφi, so that φj is an R-linear combination

of the other φi and we have a contradiction to the minimality.Therefore K ⊆ mF , and it follows from this that the maps F/mF −→

H/mH and Fm/mRmFm −→ Hm/mRmHm induced by ψ are both isomor-phisms. Therefore Fm/mRmFm and Hm/mRmHm have equal dimensions asvector spaces over Rm/mRm.

Now Ext1R(H,K)p ∼= Ext1Rp(Hp,Kp) = 0 for all p ∈ Spec(R), and so

*Ext1R(H,K) = 0. Therefore the exact sequence

0 −→ K⊆−→ F

ψ−→ H −→ 0

splits in *C(R). It follows that there is an Rm-isomorphism

Fm/mRmFm∼= Km/mRmKm ⊕Hm/mRmHm,

and therefore our calculations above with vector space dimensions show that


Km/mRmKm = 0. ThereforeKm = 0 by Nakayama’s Lemma, so thatK = 0

by 14.1.2(ii).Hence H is *free. Its rank must be 1, so that there exists a homogeneous

element φ ∈ H , of degree g say, which forms a base for H . Thus φ : C −→C ′(g) is a homogeneous R-homomorphism. For each p ∈ Spec(R), the Rp-homomorphism φp : Cp −→ C ′p generates HomRp

(Cp, C′p). By 12.1.6, there

is an Rp-isomorphism λ : Cp −→ C ′p in HomRp(Cp, C

′p); therefore, there

exist r ∈ R and s ∈ R \ p such that λ = (r/s)φp. It follows that φp issurjective, so that λ−1 ◦ φp : Cp −→ Cp is a surjective endomorphism, andtherefore an isomorphism. Consequently, φp is an isomorphism. As this is truefor each p ∈ Spec(R), it follows that φ : C −→ C ′(g) is a homogeneousisomorphism.

14.5.10 Theorem. Assume that (R,m) is Cohen–Macaulay and *local withhtm = n, and admits a *canonical module C. Set *E := *E(R/m), and let*D := *HomR( • , *E).

There is a natural transformation

φ0 : Hnm −→ *D(*HomR( • , C))

of functors from *C(R) to itself which is such that φ0M is an isomorphismwhenever M is a finitely generated graded R-module.

There is a unique extension of φ0 to a homomorphism


m

)i∈N0

−→(*D(*ExtiR( • , C))

)i∈N0

of (positive strongly) connected sequences of covariant functors from *C(R) to*C(R). Furthermore, φiM is an isomorphism for all i ∈ N0 whenever M is afinitely generated graded R-module.

In particular, for each finitely generated graded R-module M , there arehomogeneous isomorphisms

Hn−im (M) ∼= *D(*ExtiR(M,C)) for all i ∈ Z.

Proof. As in the proof of the Graded Local Duality Theorem 14.4.1, we canshow that Hj

m(M) = 0 for all j > n and all graded R-modules M , that Hnm

is a right exact functor from *C(R) to itself, and that(Hn−i

m

)i∈N0

is a positivestrongly connected sequence of covariant functors from *C(R) to *C(R).

Next, the ideas of 6.1.9 and 6.1.10 can be modified to show that the func-tors Hn

m and ( • ) ⊗R Hnm(R), from *C(R) to itself, are naturally equivalent.

Since C is a *canonical module for R, there is a homogeneous isomorphism*HomR(C, *E) ∼= Hn

m(R); therefore, the functors

Hnm and ( • )⊗R *HomR(C, *E),


from *C(R) to itself, are naturally equivalent.Next, recall the natural transformation of functors

ξ • , • , • : ( • )⊗R HomR( • , • ) −→ HomR(HomR( • , • ), • )

(from C(R)×C(R)×C(R) to C(R)) of 10.2.16: it is such that, for R-modulesM , I and J , we have (ξM,I,J (m⊗ f)) (g) = f(g(m)) for m ∈ M , f ∈HomR(I, J) and g ∈ HomR(M, I). Take C for I and *E for J ; then, whenM is graded, ξM,C,*E maps M ⊗R *HomR(C, *E) into

*HomR(*HomR(M,C), *E);

one can easily check that degrees are preserved. Set ψ0M := ξM,C,*E for eachgraded R-module M . Then

ψ0 : ( • )⊗R *HomR(C, *E) −→ *HomR(*HomR( • , C), *E)

is a natural transformation of functors from *C(R) to itself. Moreover, one canmodify the argument in the proof of 10.2.16 to show that ψ0M is an isomor-phism whenever M is a finitely generated graded R-module: use of 13.2.5(i)enables one to see that, whenever A −→ B −→ L is an exact sequence in thecategory *C(R), then the induced sequence

*HomR(L, *E) −→ *HomR(B, *E) −→ *HomR(A, *E)

is again exact.We can compose ψ0 with a natural equivalence from the second paragraph

of this proof to obtain a natural transformation

φ0 : Hnm −→ *D(*HomR( • , C))

of functors (from *C(R) to itself) with the property that φ0M is an isomor-phism whenever M is a finitely generated graded R-module.

We now reason as in the proof of 14.4.1. Since R is Cohen–Macaulay,(Hn−i

m (R))m ∼= Hn−imRm

(Rm) = 0 for all i > 0, so that Hn−im (R) = 0

for all i > 0 by 14.1.2(ii). Also *ExtiR(P,C) = 0 for all i > 0 for all*projective graded R-modules P . It follows from the graded version of theanalogue of 1.3.4 for positive connected sequences that φ0 can be incorporatedinto a (uniquely determined) homomorphism


m

)i∈N0

−→(*D(*ExtiR( • , C))

)i∈N0

of positive connected sequences of covariant functors from *C(R) to *C(R).Furthermore, it is easy to prove by induction that, for each i ∈ N, the homo-morphism φiM is an isomorphism whenever M is a finitely generated gradedR-module: use the fact that such an M can be included in an exact sequence


0 −→ K −→ F −→ M −→ 0 in *C(R) in which F is a finitely generated*free R-module.

In order to complete the proof of the final claim, one should note that, inview of 14.1.2(ii), the graded module *ExtjR(M,C) = 0 for all j > n foreach finitely generated graded R-module M , because (*ExtjR(M,C))m = 0

since inj dimRmCm = n by 12.1.21.

14.5.11 Corollary. Assume that (R,m) is Cohen–Macaulay and *local withhtm = n, and that it admits a *canonical module C. Then there is a homoge-neous isomorphism Hn

m(C)∼=−→ *E(R/m).

Proof. By 14.5.10, there is a homogeneous isomorphism

Hnm(C)

∼= *HomR(*HomR(C,C), *E(R/m)).

Since C is a canonical module for R (by 14.5.5), it follows from 14.5.9(i) thatthere is a homogeneous isomorphism R ∼= *HomR(C,C), and the desiredresult follows.

In the case where (R,m) is a Cohen–Macaulay *local Z-graded ring, Brunsand Herzog in [7, 3.6.8] gave a definition of *canonical module for R differentfrom ours. We are now in a position to reconcile these two approaches.

14.5.12 Corollary. Assume the G-graded ring (R,m) is *local and Cohen–Macaulay; set htm = n. Let C be a finitely generated graded R-module.Then C is a *canonical module for R if and only if there are homogeneousisomorphisms

*ExtiR(R/m, C) ∼={0 for i = n,

R/m for i = n.

Proof. (⇒) When C is a *canonical module for R, it follows from 14.5.10that there are homogeneous isomorphisms

Hn−im (R/m) ∼= *HomR(*ExtiR(R/m, C), *E(R/m)) for all i ∈ Z.

Since, for i = n, we have (Hn−im (R/m))m = 0, it follows from 14.1.2(ii) that

Hn−im (R/m) = 0, so that *ExtiR(R/m, C) = 0. Also, there is a homogeneous

isomorphism

*HomR(*ExtnR(R/m, C), *E(R/m)) ∼= R/m.

Since the G-graded ring R/m is *simple, every graded R/m-module is *freeand *injective, by 14.3.2. Now (0 :*E(R/m) m) = R/m by 14.2.9, and so thegraded R/m-module *ExtnR(R/m, C) must be *free of rank 1. Thus

*ExtnR(R/m, C) ∼= (R/m)(g)


in *C(R) for some g ∈ G, so that

R/m ∼= *HomR(*ExtnR(R/m, C), *E(R/m))

∼= *HomR((R/m)(g), *E(R/m)) ∼= (R/m)(−g)

in *C(R). Thus there is a homogeneous isomorphism (R/m)(−g) ∼= R/m;application of the shift functor ( • )(g) then shows that R/m ∼= (R/m)(g), andthis part of the proof is complete.

(⇐) Suppose that there are homogeneous isomorphisms


R/m for i = n.

It follows from this that μi(m, C) = δi,htm for all i ∈ N0, so that C is acanonical module for R by 14.5.4. Therefore, by 14.5.6, there is a Gorenstein*local G-graded ring R′ and a homogeneous surjective ring homomorphismR′ −→ R. It now follows from 14.5.2 that there is a *canonical module C ′ forR. Since a *canonical module forR is automatically a canonical module forR(by 14.5.5), we can now use 14.5.9(ii) to see that there exists g ∈ G for whichthere is a homogeneous isomorphism C ′ ∼= C(g).

It follows from the ‘(⇒)’ part of this proof that there is a homogeneousisomorphism *ExtnR(R/m, C

′) ∼= R/m. We can now make use of 13.3.11 tosee that there are homogeneous isomorphisms

R/m ∼= *ExtnR(R/m, C′) ∼= *ExtnR(R/m, C(g))

= *ExtnR(R/m, C)(g) ∼= (R/m)(g).

The homogeneous isomorphism R/m ∼= R/m(g) leads to a homogeneousisomorphism *E(R/m) ∼= *E(R/m)(g), and application of the shift functor( • )(−g) yields a homogeneous isomorphism *E(R/m)(−g) ∼= *E(R/m).By 14.5.10, there are homogeneous isomorphisms

Hnm(R)

∼= *HomR(*HomR(R,C′), *E(R/m)) ∼= *HomR(C

′, *E(R/m)),

and use of our homogeneous isomorphisms obtained above then yield furtherhomogeneous isomorphisms

Hnm(R)

∼= *HomR(C(g), *E(R/m)) = *HomR(C, *E(R/m)(−g))∼= *HomR(C, *E(R/m)).

(We have used 13.3.11 again.) Therefore C is a *canonical module for R.

14.5.13 �Exercise. Assume that (R,m) is *local.


(i) Show that GR := {g ∈ G : R/m ∼= (R/m)(g) in *C(R)} is a subgroupof G.

(ii) Suppose that R is Cohen–Macaulay and that C is a *canonical modulefor R. Show that

GR = {g ∈ G : C(g) is a *canonical module for R}.

(iii) In the case where G = Zn for some n ∈ N and R is positively graded,show that GR = 0.

(iv) LetH be an arbitrary subgroup of Z. Give an example of a (commutativeNoetherian) Z-graded *local ring S such that GS = H .

14.5.14 Theorem. Suppose that G = Zn, and that (R,m) is *local, Cohen–Macaulay, and positively (Zn-)graded, and has a *canonical module. Then anytwo *canonical modules for R are isomorphic in the graded category *C(R).

Proof. Let C,C ′ be *canonical modules for R. It follows from 14.5.5 thatC and C ′ are canonical modules for R in the sense of 12.1.28. Therefore, by14.5.9(ii), there exist g ∈ G and a homogeneous isomorphism φ : C

∼=−→C ′(g). Hence C ′(g) is a *canonical module forR, so that, sinceR is positively(Zn-)graded, we must have g = 0 by 14.5.13.

14.5.15 Lemma. Assume that (R,m) is *local and Cohen–Macaulay withhtm = n, and let C be a finitely generated graded R-module such thatμi(m, C) = δi,n for all i ∈ N0, that is (by 12.1.25), such that Cm is canonicalfor the Cohen–Macaulay local ring Rm.

Then there exists g ∈ G such that *En(C) ∼= *E(R/m)(−g) (in *C(R));furthermore, for any such g, the shifted graded module C(g) is a *canonicalmodule for R.

Proof. It follows from 14.5.4 that C is a canonical module for R in the senseof 12.1.28. It follows from Lemma 14.5.8(i) that there exists g ∈ G suchthat *En(C) ∼= *E(R/m)(−g) (in *C(R)); furthermore, for any such g, parts(ii)(a),(b) of the same lemma show that there are homogeneous isomorphisms


(R/m)(−g) for i = n.

The result therefore follows from 14.5.12 and 14.1.10(i).

14.5.16 Corollary. Suppose (R,m) is *local and Gorenstein. Then there ex-ists g ∈ G such that R(g) is a *canonical module for R.

Proof. This is immediate from 14.5.15 becauseRm is a canonical module forRm.


In the case where G = Zn and (R,m) is *local, Gorenstein and posi-tively (Zn-)graded, it follows from 14.5.16 and 14.5.13 that there is a uniqueg ∈ Zn such that R(g) is a *canonical module for R. Note that, since m =

(m ∩ R0)⊕(⊕

g∈N0n\{0}Rg

), the *simple R-module R/m is concentrated

in degree 0. By 14.5.11, there is a homogeneous isomorphism Hnm(R(g))

∼=*E(R/m), so that Hn

m(R)∼= *E(R/m)(−g) in *C(R). This means that g

can be identified as the degree in which the *simple submodule of Hnm(R) is

concentrated. This g is an important invariant that we shall discuss further (inspecial cases) below; for the present we content ourselves with a calculation ofthe invariant in the case of a polynomial ring over a field.

14.5.17 Example. Let K be a field and let R := K[X1, . . . , Xn], the ringof polynomials over K in n indeterminates (where n ∈ N). Consider R to beZn-graded as in 13.1.4(ii) (so we are taking G to be Zn here).

Note that R is Gorenstein and *local, and is positively graded by Zn, withunique *maximal ideal m := (X1, . . . , Xn). As was explained just above,there is a unique g ∈ Zn such that R(g) is a *canonical module for R. We nowcalculate g.

We use 14.5.11, 14.1.10(ii) and 13.5.3 to see that there are homogeneousisomorphisms

*E(R/m) ∼= Hnm(R(g)) = (Hn

m(R))(g)∼= K[X−1 , . . . , X

−n ](g).

Now the graded submoduleR/m of *E(R/m) is generated by a homogeneouselement of degree 0 which has annihilator m; furthermore, the only homoge-neous elements ofK[X−1 , . . . , X

−n ] which have annihilator m are the elements

αX−11 . . . X−1

n where α ∈ K \ {0}. It follows that g = (−1, . . . ,−1).As in 13.5.3, we can also regardK[X1, . . . , Xn] as Z-graded, where degXi

= 1 for all i = 1, . . . , n; then our polynomial ring is just the result Rφ ofapplying the φ-coarsening functor of 13.5.1 to R, where φ : Zn −→ Z is theAbelian group homomorphism for which

φ((i1, . . . , in)) = i1 + · · ·+ in for all (i1, . . . , in) ∈ Zn.

Since Hnmφ(R

φ) ∼= Hnm(R)

φ in *CZ(R) by 13.5.2, it follows that, when weregard R as Z-graded in this way, the unique integer a for which R(a) is a*canonical module for R is the degree in which the Z-graded *simple sub-module of Hn

m(R)φ is concentrated, namely −n.

14.5.18 Example. Here we review graded local duality for a ring of polyno-mials over a field K in n indeterminates X1, . . . , Xn. Let R =

⊕g∈Zn Rg =

K[X1, . . . , Xn], considered to be Zn-graded as in 13.1.4, and let m denote


the unique *maximal ideal (X1, . . . , Xn) of R. We apply 14.4.1 to R: wecan take R′ = R and f : R′ −→ R to be the identity ring homomorphism;also, R0 = K and m0 := m ∩ R0 = 0, so that E0 := ER0(R0/m0) is justK. We saw in 14.5.17 that −1 = (−1, . . . ,−1) is the unique g ∈ Zn suchthat R(g) is a *canonical module for R, that is, the unique g ∈ Zn such thatHn

m(R)∼= *ER(R/m)(−g) in *C(R).

It therefore follows from 14.4.1 and 14.4.2 that graded local duality for thisR takes the following form: there is a homomorphism

Ψ := (ψi)i∈N0 :(Hi

m

)i∈N0

−→(*HomK(*Extn−i

R ( • , R(−1)),K))i∈N0

of (negative strongly) connected sequences of covariant functors from *C(R)to itself which is such that ψi

M is a (homogeneous) isomorphism for all i ∈ N0

whenever M is a finitely generated graded R-module. In particular, for such iand M , there are homogeneous isomorphisms

Him(M) ∼= *HomK(*Extn−i

R (M,R(−1)),K)

and (since R0 = K is a complete local ring)

*HomK(Him(M),K) ∼= *Extn−i

R (M,R(−1)).

Similarly, when we regard R = K[X1, . . . , Xn] as Z-graded, where degXi

= 1 for all i = 1, . . . , n (so that m = (X1, . . . , Xn) is again the unique*maximal ideal of R), graded local duality yields a homomorphism

Φ := (φi)i∈N0 :(Hi

m

)i∈N0

−→(*HomK(*Extn−i

R ( • , R(−n)),K))i∈N0

of (negative strongly) connected sequences of covariant functors from *CZ(R)to itself which is such that φiM is a (homogeneous) isomorphism for all i ∈ N0

whenever M is a finitely generated (Z-)graded R-module.

14.5.19 Example. Let K and R = K[X1, . . . , Xn], considered to be Zn-graded, be as in 14.5.17. We now describe the structure of the *indecomposable*injective R-modules. Recall that each of these is homogeneously isomorphicto a shift of *ER(R/p) for some graded prime ideal p of R, and that we calcu-lated *ER(R/m) (where m = (X1, . . . , Xn)) in 14.5.17.

It follows immediately from 14.2.6 and 14.2.7 that there are homogeneousR-isomorphisms

*ER(R/0) ∼= *ER((0))(R((0))/0) ∼= K[X1, . . . , Xn, X

−11 , . . . , X−1

n ].

Now let t ∈ {1, . . . , n − 1} and p := (Xt+1, . . . , Xn). We now describe*ER(R/p), and note that similar calculations (and shifts) will then provide acomplete description of all *indecomposable *injective R-modules.


Note that ht p = n − t. Since R is Gorenstein, μi(qR(p), R(p)) = δi,ht q(Kronecker delta) for all i ∈ N0 and q ∈ * Spec(R) with q ⊆ p. Thus, if we usethe minimal *injective resolution to calculate Hn−t

pR(p)(R(p)), the properties de-

scribed in 14.2.4(iv) yield a homogeneous R(p)-isomorphism Hn−tpR(p)

(R(p)) ∼=*ER(p)

(R(p)/pR(p))(−g) for some g ∈ Zn. By 14.1.9, there is a homoge-neous R(p)-isomorphism Hn−t

pR(p)(R(p)) ∼= (Hn−t

p (R))(p). Also, when viewedas an R-module, *ER(p)

(R(p)/pR(p))(−g) is homogeneously isomorphic to*ER(R/p)(−g). There is therefore a homogeneous R-isomorphism

(Hn−tp (R))(p) ∼= *ER(R/p)(−g),

and so our strategy is to calculate the Zn-gradedR-moduleHn−tp (R) and then

homogenously localize it at p.We consider R as K[X1, . . . , Xt][Xt+1, . . . , Xn] and use the calculations

in 13.5.3, with S taken to be the Zt-graded ring K[X1, . . . , Xt]. There resultsa Zn-homogeneous R-isomorphism

Hn−tp (K[X1, . . . , Xn]) ∼= K[X1, . . . , Xt][X

−t+1, . . . , X

−n ],

where deg(Xi11 . . . Xit

t Xjt+1

t+1 . . . Xjnn ) = (i, j) for all i = (i1, . . . , it) ∈ N0

t

and j = (jt+1, . . . , jn) ∈ (−N)n−t. Thus, after homogenous localization at p,we see that a K-basis for *ER(R/p)(−g) is(

Xi11 . . . Xit

t Xjt+1

t+1 . . . Xjnn

)(i1,...,it)∈Zt,(jt+1,...,jn)∈(−N)n−t

,

where deg(Xi11 . . . Xit

t Xjt+1

t+1 . . . Xjnn ) = (i, j) for all i = (i1, . . . , it) ∈ Zt

and j = (jt+1, . . . , jn) ∈ (−N)n−t, and that the Zn-graded R-module struc-ture is such that

Xk(Xi11 . . . Xit

t Xjt+1

t+1 . . . Xjnn )

=

⎧⎪⎪⎨⎪⎪⎩Xi1

1 . . . Xik+1k . . . Xit

t Xjt+1

t+1 . . . Xjnn if 1 ≤ k ≤ t,

Xi11 . . . Xit

t Xjt+1

t+1 . . . Xjk+1k . . . Xjn

n if t+ 1 ≤ k ≤ n, jk < −1,0 if t+ 1 ≤ k ≤ n, jk = −1.

Furthermore, such an Xi11 . . . Xit

t Xjt+1

t+1 . . . Xjnn is annihilated by p if and only

if j = (−1, . . . ,−1) ∈ Zn−t, that is, if and only if it has the form

Xi11 . . . Xit

t X−1t+1 . . . X

−1n .

Note that multiplication by X1 provides a homogeneous automorphism of de-gree (1, 0, . . . , 0) of *ER(R/p); similar comments apply to X2, . . . , Xt. Itfollows that there is a Zn-homogeneous isomorphism

*ER(R/p) ∼= K[X1, . . . , Xt][X−t+1, . . . , X

−n ]((k, (−1, . . . ,−1)))


for any k ∈ Zt. A more combinatorial approach to this *indecomposable*injective module is provided by Miller and Sturmfels in [53, Chapter 11].

We can use 14.2.8 and 14.5.19 to find the *indecomposable *injective mod-ules over a homomorphic image of the polynomial ring of 14.5.19. In partic-ular, this applies to the Stanley–Reisner rings (with respect to the field K) ofsimplicial complexes on {1, . . . , n} that were introduced in 13.1.5.

14.5.20 Exercise. Let Δ be the simplicial complex on {1, 2, 3, 4} consistingof all the subsets of {1, 2, 3} and all the subsets of {1, 4}. Let K be a field andwork in the polynomial ring K[X1, X2, X3, X4].

(i) Show that aΔ = (X2X4, X3X4).(ii) Identify *EK[Δ] (K[Δ]/ ((X2, X3, X4)/aΔ)) with a shift of a submod-

ule of K[X1][X−2 , X

−3 , X

−4 ] via 14.2.8 and 14.5.19. Show that

B := {Xi11 X

i22 X

i33 X

i44 : i1 ∈ Z, i2, i3, i4 ∈ −N

and i2 = i3 = −1 or i4 = −1}

is aK-basis for *EK[Δ] (K[Δ]/ ((X2, X3, X4)/aΔ)). What can you sayabout the degrees of the elements of B?

14.5.21 Definition and Remarks. Suppose G = Z and (R,m) is Cohen–Macaulay and *local, and is a positively (Z-)graded ring; assume also that Rhas a *canonical module. By 14.5.14, this is uniquely determined up to homo-geneous isomorphism: we denote by ωR one choice of *canonical module forR. The a-invariant of R is defined to be

a(R) := − beg(ωR) = −min{n ∈ Z : (ωR)n = 0}.

(See 14.1.1 for the definition of the beginning of a Z-graded R-module.)

(i) Note that, when R is Gorenstein, a(R) is the unique integer a for whichR(a) is a *canonical module for R. Thus the a-invariant of a polyno-mial ring over a field, Z-graded so that each variable has degree 1, wascalculated in 14.5.17.

(ii) In the general case, we can use graded local duality, as described in14.5.10, to see that (with the notation *D of that result) there are ho-mogeneous isomorphisms

Hnm(R)

∼= *D(*HomR(R,ωR)) ∼= *D(ωR),

and so it follows from Graded Matlis Duality 14.4.2 that

a(R) = − beg(ωR) = end(*D(ωR)) = end(Hnm(R)).


The following proposition gives some hints about ways in which gradedlocal duality and graded Matlis duality can be used in tandem.

14.5.22 Proposition. Suppose that (R,m) is Cohen–Macaulay and *local,and has a *canonical module. Set n := htm.

Then R is Gorenstein if and only if there is a homogeneous isomorphism(0 :Hn

m(R) m) ∼= (R/m)(g) for some g ∈ G.

Proof. (⇒) There exists g ∈ G such that R(g) is a *canonical module forR, by 14.5.16. By 14.5.5 and 14.5.8, there exist h ∈ G and a homogeneousisomorphism Hn

m(R(g))∼= *E(R/m)(−h). It follows that there are homoge-

neous isomorphisms Hnm(R)

∼= *E(R/m)(−h− g) and

(0 :Hnm(R) m) ∼= (0 :*E(R/m)(−h−g) m) = (R/m)(−h− g).

(We have used 14.2.9 here.)(⇐) Let C be a *canonical module for R, and let *D denote the functor

*HomR( • , *E(R/m)) from *C(R) to itself. By definition, there is a homo-geneous isomorphism *D(C) ∼= Hn

m(R). There are therefore homogeneousisomorphisms

(0 :Hnm(R) m) ∼= *HomR(R/m, H

nm(R))

∼= *HomR(R/m, *HomR(C, *E(R/m)))

∼= *HomR(R/m⊗R C, *E(R/m))

∼= *HomR(C/mC, *E(R/m)).

Application of *D and use of 14.2.9 therefore yield further homogeneous iso-morphisms

*D(*D(C/mC)) ∼= *D((0 :Hnm(R) m)) ∼= *D((R/m)(g)) ∼= (R/m)(−g).

However, the canonical R-homomorphism C/mC −→ *D(*D(C/mC)) ishomogeneous and monomorphic. Since Cm/mRmCm = 0 (because Cm iscanonical for Rm by 14.5.3), it follows that Cm is a cyclic Rm-module, so thatRm is Gorenstein. Therefore Rp is Gorenstein for all p ∈ *Spec(R), so thatR is Gorenstein by 14.3.9(ii).

In §12.3, we discussed the concept of S2-ification, and we showed that anS2-ification exists in a local ring R that has a faithful canonical module. Weare now going to guide the reader to some results about S2-ifications in gradedsituations.

14.5.23 Notation. Suppose that (R,m) is G-graded and *local, and that C


is a *canonical module for R. Recall that there is a minimal primary decom-position 0 =

⋂ni=1 qi for the zero ideal in which each term is graded. Sup-

pose that the qi are indexed so that htm/qi = htm for all i = 1, . . . , t

and htm/qi < htm for all i = t + 1, . . . , n. (Of course, t could be n.) SetuR(0) =

⋂ti=1 qi. Note that uR(0) is graded, so that uR(0) = 0 if and only if

uR(0)Rm = 0, and that uR(0)Rm = uRm(0) in the notation of 12.1.12.

Note also that, by 14.5.3, the localization Cm is a canonical module forRm. We have (0 :Rm

Cm) = uRm(0), by 12.1.15. Therefore the graded ideal

(0 :R C) is equal to uR(0).As in 12.3.8, we shall denote by S the system of all ideals s of R such that

Var(s) is contained in the non-S2 locus of R. We shall guide the reader, inExercise 14.5.24 below, to the result that, when the *canonical R-module C isfaithful, there is an S2-ification ηSR : R −→ DS(R) in the sense of 12.3.9, andthis R-algebra is G-graded with homogeneous structural homomorphism.

14.5.24 �Exercise. Let the situation and notation be as in 14.5.23, and denoteby *H the system of all graded ideals of R of height at least 2. Assume thatuR(0) = 0, so that uRm

(0) = 0. Let hR : R −→ *HomR(C,C) =: H denotethe natural homogeneous R-homomorphism for which hR(r) = r IdC for allr ∈ R.

(i) Show that there is an isomorphism of Rm-algebras

Hm

∼=−→ HomRm(Cm, Cm),

and recall that HomRm(Cm, Cm) is the endomorphism ring of the canon-

ical Rm-module Cm.(ii) Show that the natural R-homomorphism H −→ Hm is injective. (Here

is a hint: note that an associated prime of the kernel of the specifiedhomomorphism would have to be an associated prime of H .)

(iii) Show that KerhR = 0; use 12.2.7 to show that CokerhR is *H-torsion.(iv) Use the Goto–Watanabe results 14.3.6 and 14.3.8, part (i) and 12.1.18(i)

to show that the R-module H is S2 (see 12.1.16).(v) Show that Γ*H(H) = 0 and use part (ii) together with 12.3.10(ii) to

show that H is a commutative G-graded Noetherian ring.(vi) Use 13.5.4(ii), 2.2.15, 2.2.17 and 12.3.2 to show that there is a unique

homogeneous R-algebra isomorphism

ψ′ : H = *HomR(C,C)∼=−→ D*H(R).

(vii) Use part (vi) to show that H1*H(R) is a finitely generated R-module; let

a be its annihilator, and note that a is graded. Show that


(a) ht a ≥ 2;(b) there is a uniquely determined homogeneous R-algebra isomor-

phism Da(R)∼=−→ D*H(R); and

(c) Var(a) is equal to the non-S2 locus of R, so that every S-torsionR-module is a-torsion.

(viii) Deduce that R has an S2-ification ηSR : R −→ DS(R), that DS(R) isa G-graded commutative Noetherian ring, that ηSR is homogeneous, andthat there are unique homogeneous R-algebra isomorphisms

*HomR(C,C)∼=−→ Da(R)

∼=−→ D*H(R)∼=−→ DS(R).

We shall consider some examples of S2-ifications in graded situations in§15.2.

The result established in the following exercise is due to J. Herzog and E.Kunz [34].

14.5.25 Exercise. Let a1, . . . , ah ∈ N \ {1} satisfy GCD(a1, . . . , ah) = 1.

Let S := a1N0 + · · · + ahN0, the additive subsemigroup of N0 generatedby a1, . . . , ah. Let K be a field and let R be the subring of the polynomialring K[X] given by R := K[Xa1 , . . . , Xah ]. Of course, R is a 1-dimensionalCohen–Macaulay ring; furthermore, it is positively Z-graded, by virtue of thegrading inherited from the usual Z-grading onK[X] in which degX = 1, and,with this grading, R is *local. We denote by R+ the unique graded maximalideal of R.

(i) Show that there exists c ∈ N such that n ∈ S for all n ∈ N withn ≥ c. Thus L := N0 \ S is a non-empty finite set, which we refer to asthe set of non-members of S, and so has a greatest member, e say. Thesemigroup S is said to be symmetric precisely when, for all integers nwith 0 ≤ n ≤ e, we have n ∈ S if and only if e− n ∈ S.

(ii) Show that R has a *canonical module.(iii) Show that K[X]X = K[X,X−1] can be naturally identified with the

ideal transformDR+(R) in the sense that there is a unique homogeneousR-isomorphism φ′ : K[X,X−1] −→ DR+

(R) such that the diagram

Rφ

K[X,X−1]�

ηRφ′∼=

DR+(R) ,

��

��


in which φ is the inclusion map, commutes.(iv) Use 13.5.4(i) to show that, for n ∈ Z,

dimK(H1R+

(R)n) =

{1 if n < 0 or n ∈ L,0 if n ∈ S,

and deduce that end(H1R+

(R)) = e.(v) Show that, if R is Gorenstein, then S is symmetric.

(vi) Now suppose that S is symmetric. Show that, if n ∈ L ∪ {−i : i ∈ N},then s := e−n ∈ S andXsH1

R+(R)n = 0. Deduce thatR is Gorenstein.

14.5.26 �Exercise. Let K be a field and let R := K[X1, . . . , Xn], the ring ofpolynomials over K in n indeterminates (where n ∈ N), graded by Z so thatR0 = K and degXi = 1 for all i = 1, . . . , n. Let r ∈ N, and consider the r-thVeronesean subringR(r) ofR, as in 13.5.9. In Exercise 13.5.13, the reader wasasked to show that R(r) is Cohen–Macaulay. Prove that R(r) is Gorenstein ifand only if n ≡ 0 (mod r).

14.5.27 Exercise. Assume that G = Z and that (R,m) is Cohen–Macaulayand *local, and positively (Z-)graded; assume also that R has a *canonicalmodule. Let b ∈ R be a homogeneous element of positive degree d which isa non-zerodivisor on R. Show that R/bR has a *canonical module and thata(R/bR) = a(R) + d.

15

Links with projective varieties

One of the reasons for the interest in graded local cohomology is provided bythe numerous applications to projective algebraic geometry. This short chapteris intended to provide a little geometric insight, with the aim of motivating thework on Castelnuovo regularity in Chapter 16.

In 2.3.2, we saw that the ideal transform has a geometric meaning in certaincases: if V is an affine variety over K, an algebraically closed field, b is anon-zero ideal of O(V ), and U denotes the open subset of V determined byb, then the ideal transform Db(O(V )) is isomorphic, as an O(V )-algebra, tothe ring of regular functions on U . One of our first aims for this chapter is theestablishment of a graded analogue of this result. This graded analogue appliesto irreducible affine algebraic cones.

Throughout this chapter, all graded rings and modules are to be understoodto be Z-graded.

15.1 Affine algebraic cones

15.1.1 Notation and Terminology. We shall employ the notation and termi-nology concerning graded rings and modules described in 13.1.1, 13.1.3 and14.1.1, but, in accordance with our convention for the whole of this chapter,restricted to the special case in which G = Z. In addition, when R is gradedand the ideal a is graded, and M is a graded R-module, so that (see 13.4.3and 13.4.4) the Hi

a(M) (i ∈ N0) are all graded R-modules, we use Hia(M)n

to denote the n-th component of the graded module Hia(M) (for i ∈ N0 and

n ∈ Z); also Da(M) is graded (by 13.3.14), and we use Da(M)n to denote itsn-th component.

332 Links with projective varieties

Now assume that R =⊕

n∈N0Rn is positively graded. We set

R+ :=⊕n∈N

Rn =⊕n>0

Rn = 0⊕R1 ⊕R2 ⊕ . . .⊕Rn ⊕ . . . ,

the irrelevant ideal of R. Of course, R+ is graded, and so, for a graded R-module M , we can define the R0-modules Hi

R+(M)n (i ∈ N0, n ∈ Z).

15.1.2 Geometric Notation and Reminders. Let K be an algebraically clo-sed field, and let r ∈ N. We shall use the notation of 2.3.1, and we shall denotethe origin (0, . . . , 0) of Ar(K) simply by 0. For a subset C of a quasi-affinevarietyW overK, we shall extend the notation of 6.4.1 to denote the vanishingideal

{f ∈ O(W ) : f(q) = 0 for all q ∈ C}

of C by IW (C).

(i) By a cone (with vertex 0) in Ar(K) we mean a set C ⊆ Ar(K) such that0 ∈ C and, whenever q ∈ C, then λq ∈ C for all λ ∈ K. Such a coneC is called an affine algebraic cone in Ar(K) if and only if it is also anaffine algebraic set; also C is said to be non-degenerate precisely whenC = {0}.

Since a graded ideal can be generated by homogeneous elements, it isclear that, if b is a proper graded ideal ofK[X1, . . . , Xr], then VAr(K)(b)

is an affine algebraic cone in Ar(K). It is easy to see that, conversely,if C is an affine algebraic cone in Ar(K), then IAr(K)(C) is a propergraded ideal. Thus the affine algebraic cones in Ar(K) are precisely thealgebraic sets in Ar(K) which have proper graded vanishing ideals.

(ii) LetC be an irreducible affine algebraic cone in Ar(K). Since IAr(K)(C)

is a graded prime ideal ofK[X1, . . . , Xr], the ringO(C) of regular func-tions onC inherits a grading fromK[X1, . . . , Xr], in such a way that therestriction homomorphism K[X1, . . . , Xr] −→ O(C) is homogeneous(see 14.1.3). Of course, for each n ∈ N0, we denote the n-th componentofO(C) byO(C)n. Note thatO(C)0 can be identified withK. It is alsoworth noting that O(C) is *local with unique *maximal graded idealO(C)+; furthermore, O(C)+ is actually a maximal ideal in this case.

(iii) With the notation of part (ii), let b be a non-zero graded ideal of O(C),let VC(b) denote the closed subset of C determined by b, and let Ube the open subset C \ VC(b) of C. In fact, if c denotes the inverseimage of b in K[X1, . . . , Xr] under the restriction homomorphism, thenVC(b) = VAr(K)(c), and U is obtained from C by removal of anotheraffine algebraic cone (or the empty set).

15.1 Affine algebraic cones 333

Let n ∈ Z. A regular function f ∈ O(U) is said to be homogeneousof degree n precisely when, for each p ∈ U , there exists an open setW ⊆ U with p ∈W , an integer d ∈ N0, and g ∈ O(C)d, h ∈ O(C)n+d

such that, for each q ∈ W , we have g(q) = 0 and f(q) = h(q)/g(q).The set of all regular functions on U which are homogeneous of degreen is denoted by O(U)n. The fact that this definition is not ambiguous inthe case when U = C is one consequence of the next proposition.

15.1.3 Proposition. Let the situation be as in 15.1.2(iii). Thus K is an al-gebraically closed field, r ∈ N, C is an irreducible affine algebraic cone inAr(K), b is a non-zero graded ideal of O(C), and U = C \ VC(b).

The subsets O(U)n (n ∈ Z) defined in 15.1.2(iii) provide O(U) with astructure as a graded ring with respect to which the homomorphisms � U andνC,b in the commutative diagram

O(C) U O(U)�

ηO(C)

νC,b∼=

Db(O(C))

��

of 2.3.2 are homogeneous. (It should be noted that, since b is graded, it fol-lows from 13.3.14 that Db(O(C)) is a graded O(C)-module and that ηO(C)

is homogeneous.)

Proof. Let us abbreviate νC,b by ν and ηO(C) by η. Since ν is a ring isomor-phism (by 2.3.2), it is enough for us to show thatO(U)n = ν−1(Db(O(C))n)

for each n ∈ Z, and this is what we shall do.Let y ∈ Db(O(C))n. Since Coker η, being isomorphic to H1

b(O(C)), isb-torsion, there exists t ∈ N such that bty ∈ Im η. Let p ∈ U . Since U =

C \ VC(b) and b is graded, there exists a homogeneous element g ∈ bt, ofdegree d ∈ N0 say, such that g(p) = 0. Then W := U \ VC(gO(C)) is anopen subset of U which contains p.

As gy ∈ Im η, there is an element h ∈ O(C) with gy = η(h). As gy ∈Db(O(C))n+d and as η is homogeneous and injective, we must have h ∈O(C)n+d. Now

g� Uν−1(y) = ν−1(η(g))ν−1(y) = ν−1(η(g)y) = ν−1(gy)

= ν−1(η(h)) = h� U .

Therefore, for each q ∈ W , we have g(q) = 0 and ν−1(y)(q) = h(q)/g(q).Hence ν−1(y) ∈ O(U)n. We have proved that O(U)n ⊇ ν−1(Db(O(C))n).


Now let f ∈ O(U)n. Choose p ∈ U ; then there exists an open set W ⊆ U

with p ∈ W , an integer d ∈ N0, and g ∈ O(C)d, h ∈ O(C)n+d such that,for each q ∈ W , we have g(q) = 0 and f(q) = h(q)/g(q). This shows, inparticular, that g� W .f� W = h� W . As W is a non-empty open subset of theirreducible topological space U , it follows that g� U .f = h� U . Now apply ν:we obtain

η(g)ν(f) = ν(g� U )ν(f) = ν(g� U .f) = ν(h� U ) = η(h).

As η is homogeneous, η(g) ∈ Db(O(C))d and η(h) ∈ Db(O(C))n+d. As ηis injective and g = 0, we have η(g) = 0; also, as Db(O(C)) is a domain, itfollows that ν(f) must be homogeneous of degree n. We have therefore shownthat O(U)n ⊆ ν−1(Db(O(C))n), and so the proof is complete.

Theorem 2.3.2, and its graded refinement 15.1.3, provide a link between lo-cal cohomology and algebraic varieties. Towards the end of the book, we shallencounter more general versions of these two results, because they are relatedto the Deligne Isomorphism Theorem 20.1.14 (see 20.1.17) and its graded ver-sion 20.2.7.

We explore now the special case of Proposition 15.1.3 in which dimC > 0

and the graded ideal b is the irrelevant idealO(C)+ (see 15.1.1) ofO(C): thenthe open set U = C \ {0} is just the punctured cone C

◦of the given irreducible

affine algebraic cone C.

15.1.4 Corollary. Consider the special case of 15.1.3 where the irreducibleaffine algebraic cone C has dimC > 1 and b = O(C)+. Then

U = C \ VC(O(C)+) = C \ {0} = C◦,

the punctured cone of C; also

(i) the restriction ring homomorphism �C◦ : O(C) −→ O(C

◦) makesO(C

◦)

into a finitely generated graded O(C)-module;(ii) end(H1

O(C)+(O(C))) <∞ (see 14.1.1 for the definition of the end of a

graded module);(iii) H1

O(C)+(O(C))n is a finite-dimensional vector space over K, for all

n ∈ Z; and(iv) H1

O(C)+(O(C))n = 0 for all n ≤ 0.

Proof. Set R := O(C). Now (R,R+) is *local, by 15.1.2(ii), and a domain.Hence, by 14.1.14, we have htR+ = dimR = dimC > 1. Therefore, for allp ∈ *Spec(R) \Var(R+), we have p ⊂ R+ and

depthRp + htR+/p ≥ 2 > 1.

15.1 Affine algebraic cones 335

It thus follows from the Graded Finiteness Theorem 14.3.10 that H1R+

(R) isfinitely generated. By 2.2.6(i)(c), 13.5.4 and 15.1.3, there is an exact sequence

R

C◦

−→ O(C◦) −→ H1

R+(R) −→ 0 of graded R-modules and homogeneous

R-homomorphisms. It is now immediate that �C◦ makes O(C

◦) into a finitely

generated graded R-module.Note also that, since H1

R+(R) is finitely generated and R+-torsion, we must

have H1R+

(R)n = 0 for all n � 0 (that is, for all n greater than some fixedinteger n0), and so end(H1

R+(R)) < ∞. We have therefore completed the

proofs of parts (i) and (ii), while part (iii) is now immediate from the facts thatR0 = K and, for each n ∈ Z, the component H1

R+(R)n is a finitely generated

R0-module.It remains for us to prove part (iv). Use �

C◦ to identify R as a subring of

O(C◦). It is enough for us to show that K = R0 = O(C)0 = O(C

◦)0 and that

O(C◦)n = 0 for all n < 0, and this we do.

First, by part (i), the integral domain O(C◦)0 is an integral extension of the

algebraically closed field K = R0, and so R0 = O(C◦)0. Second, since O(C

◦)

is a finitely generated graded R-module and R is positively graded, O(C◦)n =

0 for all n � 0. Hence, for each n < 0 and each γ ∈ O(C◦)n, there exists

t ∈ N such that γt = 0, so that γ = 0 sinceO(C◦) is a domain. This completes

the proof.

15.1.5 Exercise. Calculate O(C), O(C◦) and H1

O(C)+(O(C)) for an irre-

ducible affine algebraic cone C of dimension 1 in Ar(K), where K is an al-gebraically closed field and r ∈ N. (Here is a hint. Let c := (c1, . . . , cr) ∈C \ {0}. Without loss of generality, one can assume that c1 = 0: with thisassumption, show that

IAr(K)(C) = (c2X1 − c1X2, c3X1 − c1X3, . . . , crX1 − c1Xr).)

15.1.6 Exercise. Assume that R =⊕

n∈N0Rn is positively graded and an

integral domain, and that the ideal a is graded and non-zero.

(i) Show that, if a ∩R0 = 0, then Da(R)n = H1a(R)n = 0 for all n < 0.

(ii) Assume that the subring R0 is a homomorphic image of a regular ring,and that ht a > 1. Show that ηa makes Da(R) into a finitely generatedR-module, and deduce that Da(R)n and H1

a(R)n are finitely generatedR0-modules for all n ∈ Z. Show further that Da(R)n = H1

a(R)n = 0

for all n < 0.(iii) Assume that R0 is an algebraically closed field K, and that ht a > 1.


Using ηa to identify R as a subring of Da(R), show that Da(R)0 = K

and H1a(R)0 = 0.

15.1.7 Exercise. Let the situation and notation be as in 15.1.3, and assumethat ht b > 1. Use 15.1.6(ii),(iii) to show that O(U) is a finitely generatedO(C)-module for which O(U)0 = K.

15.2 Projective varieties

In the situation of 15.1.3, we have seen (see 2.3.3) that we can regard non-zeroelements of the local cohomology module H1

b(O(C)) as obstructions to theextension of regular functions on U to regular functions on C; we have alsoobtained, in 15.1.4, in the case when U is the punctured cone C

◦, information

about some of the components H1O(C)+

(O(C))n = H1b(O(C))n. In the light

of the connections between irreducible affine algebraic cones and projectivevarieties (reviewed in 15.2.1 below), it would be reasonable for one to suspectthat there are links between graded local cohomology and projective varieties.Such suspicions would be well founded, and we plan to expose some of thelinks in this and subsequent chapters.

15.2.1 Reminders and Notation. Here we specify the notation and termi-nology that we shall use for discussion of projective varieties. Let K be analgebraically closed field, and let r ∈ N. We shall find it convenient to varyslightly the notation of 2.3.1 and regard the polynomial ringK[X0, . . . , Xr] inr + 1 indeterminates X0, X1, . . . , Xr as the coordinate ring O(Ar+1(K)) ofaffine (r+1)-space Ar+1(K) over K. As in 15.1.2, we shall denote the origin(0, . . . , 0) of Ar+1(K) simply by 0.

(i) For c := (c0, . . . , cr) ∈ Ar+1(K) \ {0}, we use (c0 : · · · : cr) todenote the line {λc : λ ∈ K} in Ar+1(K) through c and 0. We shall usePr(K) to denote projective r-space over K, that is, the set{

(c0 : · · · : cr) : (c0, . . . , cr) ∈ Ar+1(K) \ {0}}

of all lines through the origin in Ar+1(K), endowed with the Zariski topology.We remind the reader that the closed sets in this topology are precisely theprojective algebraic sets, that is, the sets of the form

VPr(K)(a) := {(c0 : · · · : cr) ∈ Pr(K) : f(c0, . . . , cr) = 0 for all f ∈ a} ,

where a is a proper graded ideal ofK[X0, . . . , Xr]. (When a = (X0, . . . , Xr),the corresponding projective algebraic set is the empty one.) If f1, . . . , ft are

15.2 Projective varieties 337

non-constant homogeneous polynomials in K[X0, . . . , Xr], then the projec-tive algebraic set VPr(K)((f1, . . . , ft)) is denoted by VPr(K)(f1, . . . , ft). Ofcourse, every projective algebraic set in Pr(K) can be represented in this form.

We shall use Pr to denote complex projective r-space Pr(C). All unex-plained mentions of topological notions, including ‘open’ and ‘closed’ subsets,in connection with projective spaces will refer to the Zariski topology.

(ii) By the statement ‘V ⊆ Pr(K) is a projective variety’ we shall meanthat V is an irreducible closed subset of Pr(K) (with the induced topology),and by the statement ‘W ⊆ Pr(K) is a quasi-projective variety’ we shall meanthat W is a non-empty open subset of a projective variety V ⊆ Pr(K) (againwith the induced topology). (It is wise for us to include the ‘⊆ Pr(K)’ in thenotation because (and some of our examples below will remind the reader ofthis) isomorphic projective varieties, embedded in projective spaces in differ-ent ways, can have non-isomorphic homogeneous coordinate rings! (See parts(vi) and (viii) below.) This problem does not occur with affine varieties.)

By a variety we shall mean an affine, quasi-affine, projective, or quasi-projective variety.

(iii) Let V be the closed subset of Pr(K) given by V = VPr(K)(a), wherea is a proper graded ideal of K[X0, . . . , Xr]. The affine cone Cone(V ) ⊆Ar+1(K) over V in Ar+1(K) is defined by

Cone(V ) ={(c0, . . . , cr) ∈ Ar+1(K) \ {0} : (c0 : · · · : cr) ∈ V

}∪ {0}

= VAr+1(K)(a),

an affine algebraic cone in Ar+1(K).On the other hand, for the affine algebraic cone C in Ar+1(K) defined by

C = VAr+1(K)(a′), where a′ is a proper graded ideal of K[X0, . . . , Xr], we

define the projectivization C+ of C to be the closed subset

C+ = {(c0 : · · · : cr) ∈ Pr(K) : (c0, . . . , cr) ∈ C \ {0}} = VPr(K)(a′)

of Pr(K). Thus V �→ Cone(V ) provides a bijective map from the set T :=

{V ⊆ Pr(K) : V is closed} of all closed subsets of Pr(K) to the set{C ⊆ Ar+1(K) : C is an affine algebraic cone

}of all affine algebraic cones in Ar+1(K); the inverse map is given byC �→ C+.Note that these two maps both preserve inclusion relations.

(iv) It follows from part (iii) and standard facts from affine algebraic geom-etry that there is a bijective correspondence between the set T of all closed sub-sets of Pr(K) and the set of all radical proper graded ideals of K[X0, . . . , Xr]


under which a closed subset V ∈ T corresponds to

IAr+1(K)(Cone(V )) = {f ∈ K[X0, . . . , Xr] : f(c0, . . . , cr) = 0 for all

(c0, . . . , cr) ∈ Cone(V )}= {f ∈ K[X0, . . . , Xr]+ : f(c0, . . . , cr) = 0 for all

(c0, . . . , cr) ∈ Ar+1(K) \ {0}with (c0 : · · · : cr) ∈ V } .

This is also denoted by IPr(K)(V ), and is called the vanishing ideal of V .(v) Note that, if V1, . . . , Vt are closed subsets in Pr(K), then

Cone

(t⋃

i=1

Vi

)=

t⋃i=1

Cone(Vi) and Cone

(t⋂

i=1

Vi

)=

t⋂i=1

Cone(Vi).

Also, the minimal prime ideals of a proper graded ideal of K[X0, . . . , Xr] areagain graded (by 13.1.6(ii)), and so it follows that a closed subset V of Pr(K)

is irreducible if and only if Cone(V ) is irreducible and non-degenerate, andthat this is the case if and only if IAr+1(K)(Cone(V )) is a graded prime idealof K[X0, . . . , Xr] properly contained in (X0, . . . , Xr).

Thus, in the bijective correspondence of part (iv), the projective varietiesin Pr(K) correspond to the graded prime ideals of K[X0, . . . , Xr] properlycontained in (X0, . . . , Xr).

(vi) Let V ⊆ Pr(K) be a projective variety. By parts (iv) and (v), theideal p := IAr+1(K)(Cone(V )) is a graded prime ideal ofK[X0, . . . , Xr] withp ⊂ (X0, . . . , Xr). We refer to the positively graded *local ring

O(Cone(V )) = K[X0, . . . , Xr]/p = K[X0, . . . , Xr]/IAr+1(K)(Cone(V ))

as the homogeneous coordinate ring of V ⊆ Pr(K).(vii) Remember that the dimension dimV of a variety V is defined as the

maximum length l of a strictly descending chain C0 ⊃ C1 ⊃ · · · ⊃ Cl ofclosed irreducible subsets of V , and that such a V is called a curve if and only ifdimV = 1 and a surface if and only if dimV = 2. Using the correspondencedescribed in part (iii) and the observation made in part (v), we obtain, for aprojective variety V ′ ⊆ Pr(K), that

dimV ′ = dim(Cone(V ′))− 1 = dim(O(Cone(V ′)))− 1.

(viii) We mentioned in (ii) above that isomorphic projective varieties, em-bedded in projective spaces in different ways, can have non-isomorphic ho-mogeneous coordinate rings. Properties of the homogeneous coordinate ringO(Cone(V )) of a projective variety V ⊆ Pr(K) are referred to as arith-metic properties of V . For example, we say that V ⊆ Pr(K) is arithmetically


Cohen–Macaulay (respectively arithmetically Gorenstein, . . .) if and only ifO(Cone(V )) is a Cohen–Macaulay (respectively Gorenstein, . . .) ring. Wealso define the arithmetic depth arithdepthV of V to be the grade of theunique *maximal graded ideal of the homogeneous coordinate ring of V . ThusarithdepthV = gradeO(Cone(V ))+.

15.2.2 Reminder and �Exercise: Veronesean tranformations. Let K bean algebraically closed field, and let r, d ∈ N. Let V ⊆ Pr(K) be a projectivevariety, and let p denote (the graded prime) kernel IPr(K)(V ) of the restrictionhomomorphism K[X0, . . . , Xr] −→ O(Cone(V )). Set

T = T (d)r ={(ν0, . . . , νr) ∈ N0

r+1 : ν0 + · · ·+ νr = d},

a set with cardinality(r+dr

). We shall use a family of (algebraically indepen-

dent) indeterminates (Yν)ν∈T indexed by T . Let

φ : K[Yν : ν ∈ T ] −→ K[X0, . . . , Xr]

be the K-algebra homomorphism for which φ(Y(ν0,...,νr)) = Xν00 . . . Xνr

r forall (ν0, . . . , νr) ∈ T . Let θ be the composition

K[Yν : ν ∈ T ] K[X0, . . . , Xr] O(Cone(V )),� �φ Cone(V )

and let q := Ker θ, a graded prime ideal of K[Yν : ν ∈ T ].

(i) Show that, with the notation of 13.5.9, Imφ = K[X0, . . . , Xr](d), the

d-th Veronesean subring of K[X0, . . . , Xr], that φ(q) = p(d), and thatIm θ = O(Cone(V ))(d).

Note that, when Veronesean subrings are given the grading described at thebeginning of 13.5.9, both φ : K[Yν : ν ∈ T ] −→ K[X0, . . . , Xr]

(d) and

θ : K[Yν : ν ∈ T ] −→ O(Cone(V ))(d)

are homogeneous ring homomorphisms.If we consider K[Yν : ν ∈ T ] as O

(A(

r+dr )(K)

), then the graded prime

ideal q of this polynomial ring defines a projective variety

V (d) := VP(r+d

r )−1(K)

(q) ⊆ P(r+dr )−1(K),

called the d-th Veronesean of V . Note that there is a homogeneous isomor-phism of graded K-algebras O(Cone(V (d)))

∼=−→ O(Cone(V ))(d).The main aim of this exercise is to show that the varieties V and V (d)

are isomorphic. Some additional notation will be helpful. Given an elementβ = (βν)ν∈T ∈ A(

r+dr )(K) \ {0}, we use β = (βν)ν∈T to denote the line


{λβ : λ ∈ K}, considered as an element of P(r+dr )−1(K). Also, if y =

(y0, . . . , yr) is an (r + 1)-tuple of elements of some commutative K-algebraand ν = (ν0, . . . , νr) ∈ N0

r+1, then we shall use yν to denote yν00 . . . yνr

r .For each i = 0, . . . , r, let ei = (0, . . . , 0, 1, 0, . . . , 0) be the element of N0

r+1

whose only non-zero component is a ‘1’ in the i-th position. In order to sim-plify notation, we shall also employ the Z-module structure on Zr+1. For ex-ample, with this notation, we can write that dei ∈ T for all i = 0, . . . , r.

(ii) Let s ∈ N and let ω(1), . . . , ω(s), μ(1), . . . , μ(s) ∈ T be such that∑si=1 ω

(i) =∑s

i=1 μ(i).

Show that Yω(1) . . . Yω(s) − Yμ(1) . . . Yμ(s) ∈ q.

(iii) Let ν = (ν0, . . . , νr) ∈ T . Deduce from part (ii) that, for all i, j, k ∈{0, . . . , r}, we have

(a) Yek+(d−1)eiYei+(d−1)ej − YdeiYek+(d−1)ej ∈ q;(b)∏r

α=0(Yeα+(d−1)ei)να − Yν(Ydei)d−1 ∈ q; and

(c) Y dν −∏r

α=0(Ydeα)να ∈ q.

(iv) Show that there is a morphism of varieties ϑ(d) : V −→ V (d) which issuch that

ϑ(d)((α0 : · · · : αr)) = (αν)ν∈T for all α = (α0 : · · · : αr) ∈ V.

(v) For i = 0, . . . , r, let Ui := V (d) \ VP(r+d

r )−1(K)

(Ydei). Use part (iii)(c)

to show that U0, . . . , Ur form an open covering of V (d).(vi) Show that there is a morphism of varieties ω : V (d) −→ Pr(K) which

is such that, for each i = 0, . . . , r and for all (βν)ν∈T ∈ Ui,

ω((βν)ν∈T ) = (βe0+(d−1)ei : · · · : βej+(d−1)ei : · · · : βer+(d−1)ei).

(vii) Show that Imω ⊆ V . (Here are some hints. Suppose i ∈ {0, . . . , r} and(βν)ν∈T ∈ Ui. Let α := (βe0+(d−1)ei , . . . , βer+(d−1)ei). It is enoughto show that, for each homogeneous g ∈ p, we have gd(α) = 0 (as thiswould imply that g(α) = 0). Now gd ∈ p(d); by part (i), we have p(d) =φ(q), and so there exists a homogeneous f ∈ q such that φ(f) = gd.Now use part (iii)(b).)

(viii) Show that ω : V (d) −→ V and ϑ(d) : V −→ V (d) are inverse isomor-phisms of varieties. The isomorphism ϑ(d) is called the d-th Veroneseantransformation of V or the d-th Veronesean map on V .

15.2.3 Example: rational normal curves. Consider the particular case ofExercise 15.2.2 in which r = 1, V = P1(K), but d is still arbitrary. The d-th


Veronesean P1(K)(d) of P1(K) has the property that there is a homogeneousisomorphism

O(P1(K)(d)) ∼= O(P1(K))(d) = K[X0, X1](d)

= K[Xd0 , X

d−10 X1, . . . , X0X

d−11 , Xd

1 ] ⊆ K[X0, X1].

The Veronesean transformation ϑ(d) : P1(K) −→ P1(K)(d) is an isomor-phism of varieties, given by ϑ(d)((σ : τ)) = (σd : σd−1τ : · · · : στd−1 : τd)

for all (σ : τ) ∈ P1(K). Thus P1(K)(d) ⊆ Pd(K) is a curve which is iso-morphic to the projective line P1(K); it is called the rational normal curvein projective d-space, and we shall denote it by N(d) ⊆ Pd(K). The curveN(3) ⊆ P3(K) is also called the twisted cubic.

The phenomenon mentioned in 15.2.1(ii),(viii) is illustrated by these ratio-nal normal curves: it follows from 14.5.26 that, provided d ≥ 3, the rationalnormal curveN(d) ⊆ Pd(K) is not arithmetically Gorenstein, whereas P1(K),to which it is isomorphic, is.

15.2.4 Proposition. Let K be an algebraically closed field, and let r, d ∈ N.Let V ⊆ Pr(K) be a projective variety of positive dimension, and consider thed-th Veronesean V (d) ⊆ P(

r+dr )−1(K) of V . Set

e := end(H1O(Cone(V ))+

(O(Cone(V )))).

Then

arithdepthV (d)

{> 1 if d > e,

= 1 if d = e.

In particular, if the projective variety V ⊆ Pr(K) is a curve, then V (d) isarithmetically Cohen–Macaulay if d > e, and is not arithmetically Cohen–Macaulay if d = e.

Proof. Let R := O(Cone(V )). By 15.1.4(ii),(iv), we have e < ∞ andH1

R+(R)n = 0 for all n ≤ 0. Hence the (d, 0)-th Veronesean submodule

H1R+

(R)(d,0) of H1R+

(R)� R(d) (see 13.5.9) is zero if d > e and is non-zero ifd = e. By 13.5.9(v), there is a homogeneous R(d)-isomorphism

H1R+

(R)(d,0)∼=−→ H1

(R+)(d)(R(d));

furthermore, (R+)(d) = (R(d))+. Note also that H0

(R(d))+(R(d)) = 0 since

R(d) is a domain and R(d)+ = 0. Since there is a homogeneous ring isomor-

phism O(Cone(V (d)))∼=−→ R(d) by 15.2.2, it follows that arithdepthV (d) >

1 if d > e and arithdepthV (d) = 1 if d = e.


If V is a curve, then dimR = 2, so that, since R is an integral extension do-main of R(d), it follows that the *local graded ring O(Cone(V (d))) is Cohen–Macaulay if and only if arithdepthV (d) > 1.

15.2.5 �Exercise. Let K be an algebraically closed field, let d ∈ N withd ≥ 3, and let A(d) be the subring of the ring K[X,Y ] of polynomials overK in two indeterminates X and Y described in 13.5.12 and given by A(d) :=

K[Xd, Xd−1Y,XY d−1, Y d]. Recall from 13.5.12 that A(d) inherits a gradingfrom K[X,Y ](d), the d-th Veronesean subring of K[X,Y ].

Let Y0, Y1, Y2, Y3 be independent indeterminates over K and consider thepolynomial ring K[Y0, Y1, Y2, Y3] as the coordinate ring O(A4(K)). Let

ψ : K[Y0, Y1, Y2, Y3] −→ A(d)

be the (surjective, homogeneous) K-algebra homomorphism for which

ψ(Y0) = Xd, ψ(Y1) = Xd−1Y, ψ(Y2) = XY d−1 and ψ(Y3) = Y d.

Let p := Kerψ; this is a graded prime ideal ofK[Y0, Y1, Y2, Y3] and so definesa projective variety Σ(d) := VP3(K)(p) ⊆ P3(K).

(i) Show that there is a morphism ρ : P1(K) −→ Σ(d) of varieties forwhich ρ((σ : τ)) = (σd : σd−1τ : στd−1 : τd) for all (σ : τ) ∈ P1(K).

(ii) Show that Y d2 − Y d−1

3 Y0 ∈ p.(iii) Set U := Σ(d) \VP3(K)((Y0)) and U ′ := Σ(d) \VP3(K)((Y3)). Show that

U and U ′ form an open covering of Σ(d).(iv) Show that there is a morphism μ : Σ(d) −→ P1(K) of varieties for

which

μ((α : β : γ : δ)) =

{(α : β) if (α : β : γ : δ) ∈ U,(γ : δ) if (α : β : γ : δ) ∈ U ′.

(v) Show that ρ and μ are inverse isomorphisms of varieties, so that Σ(d) isagain a curve isomorphic to P1(K).

(vi) Let t ∈ N. Use 13.5.12 to show that, when d > 3, the curve Σ(d) ⊆P3(K) is not arithmetically Cohen–Macaulay, but that the t-th Verone-sean (Σ(d))

(t) ⊆ P(t+33 )−1(K) is arithmetically Cohen–Macaulay pro-

vided that t ≥ d−2. Is (Σ(d))(d−3) ⊆ P(

d3)−1(K) arithmetically Cohen–

Macaulay?

15.2.6 Remarks. Let the situation and notation be as in Exercise 15.2.5,where the curve Σ(d) ⊆ P3(K) was constructed, for each integer d ≥ 3. Asa variety, Σ(d) is isomorphic to the projective line P1(K). Note that Σ(3) ⊆


P3(K) is just the twisted cubic N(3) ⊆ P3(K) of 15.2.3. The curve Σ(4) ⊆P3(K) is called the twisted quartic or Macaulay’s curve.

Exercise 15.2.5(vi) shows that the arithmetic structure of Σ(d) ⊆ P3(K),which reflects the way in which the curve is embedded in P3(K), is morecomplicated for d > 3 then it is when d = 3. This is another illustration of thephenomenon mentioned in 15.2.1(ii),(viii).

15.2.7 Exercise. Let K be a field, and let U,W, S, T be indeterminates overK. In the 5-th Veronesean subring K[U,W, S, T ](5) of the polynomial ringK[U,W,S, T ], set f1 := U4T, f2 :=WS4 and f3 := U4S −WT 4. Considerthe four K-subalgebras A, B, C and R of K[U,W,S, T ](5) given by

A := K[U4S,U4T,WS4,WS3T,WS2T 2,WST 3,WT 4],

B := K[U4S,U4T,WS4,WS3T,WST 3,WT 4] ⊆ A,

C := K[U4S,U4T,WS4,WS2T 2,WST 3,WT 4] ⊆ A,

R := K[f1, f2, f3] = K[U4T,WS4, U4S −WT 4] ⊆ B ∩ C ⊆ A.

Observe that each of these subrings inherits a grading from K[U,W, S, T ](5)

that turns it into a homogeneous positively graded *local integral domain. Itfollows from 14.5.2 and 14.5.24 that each of them has a *canonical moduleand an S2-ification.

(i) Show (by induction on n) that, for all n ∈ N0, the set of monomials

{U4iWn−iSkT 4n−3i−k : 0 ≤ i ≤ n, 0 ≤ k ≤ 4n− 3i}

is a K-basis of An and conclude that dimK An = (n+ 1)(5n+ 2)/2.

(ii) Show that

A2 = (f1K + f2K + f3K)A1

and deduce that

An = (f1K + f2K + f3K)An−1 for all n ≥ 2.

Deduce that the R-module A is generated by the five homogeneous ele-ments

g1 := 1, g2 := U4S, g3 :=WS3T, g4 :=WS2T 2, g5 :=WST 3.

(iii) Show that dimA = 3 = dimR, that f1, f2, f3 are algebraically inde-pendent over K, that f1, f2, f3 is an R-sequence, and that dimK Rn =

(n+ 1)(n+ 2)/2 for all n ∈ N0.


(iv) For each n ∈ N0 let

Mn := {fν11 fν2

2 fν33 : ν1, ν2, ν3 ∈ N0, ν1 + ν2 + ν3 = n},

the set of all monomials of degree n in f1, f2, f3, and let Mngi, forn ∈ N0 and i ∈ {1, . . . , 5}, denote {mngi : mn ∈ Mn}. Use part (ii)to show that, for each n ∈ N, the members of the set

Mng1 ∪Mn−1g2 ∪Mn−1g3 ∪Mn−1g4 ∪Mn−1g5

span the K-space An, and use the formula in part (i) for dimK An todeduce that the above-displayed set is actually aK-basis forAn. Deducethat the five homogeneous elements g1, g2, g3, g4, g5 form a base for theR-module A, so that A is a *free graded R-module of rank 5.

(v) Deduce that f1, f2, f3 is an A-sequence and conclude that the ring A isCohen–Macaulay.

(vi) Show that A = B +WS2T 2B and B+A ⊆ B. Deduce that there is ahomogeneous B-isomorphism A/B ∼= 1K, where 1K is as defined in14.1.13.

(vii) Show that that there is a homogeneousB-isomorphismH1B+

(B) ∼= 1K;deduce that gradeB+ = 1 and that the non-S2 locus of B is {B+} =

Var(B+).(viii) Use 2.2.15(iii), 2.2.17 and 13.5.4(ii) to show that there is a homoge-

neous isomorphism of B-algebras A ∼= DB+(B), and that both theseB-algebras are isomorphic to the S2-ification of B.

Let p be the ideal (U4S,U4T,WS2T 2,WST 3,WT 4) of

C = K[U4S,U4T,WS4,WS2T 2,WST 3,WT 4].

(ix) Show that A = C +WS3TC, and that pA ⊆ C.(x) Show that, for all n ∈ N, we have (WS4)nWS3T /∈ C.

(xi) Show that C/p = K[WS4 + p], and that WS4 + p is transcendentaloverK (in the usual sense that the only polynomial f ∈ K[X] for whichf(WS4 + p) = 0 is the zero polynomial). Deduce that p ∈ *Spec(C),that dimC/p = 1 and that ht p = 2.

(xii) Show that the C-module A/C is generated by WS3T + C, and that theannihilator of this element is p. Deduce that there is a homogeneous C-isomorphism H1

p(C)∼= (C/p)(−1). Deduce that grade p = 1 and that

the non-S2 locus of C is Var(p).(xiii) Show that there is a homogeneous isomorphism of C-algebras A ∼=

Dp(C), and that both theseC-algebras are isomorphic to the S2-ificationof C.


15.2.8 Exercise. Let K be an algebraically closed field, and consider thehomogeneous, N0-gradedK-algebrasA,B and C defined in 15.2.7. Let VA ⊂P6(K) be the irreducible projective variety whose homogeneous coordinatering satisfies O(Cone(VA)) = A; also let VB , VC ⊂ P5(K) be the irreducibleprojective varieties such that O(Cone(VB)) = B and O(Cone(VC)) = C

respectively.

(i) Show that

VA = {(αs : αt : βs4 : βs3t : βs2t2 : βst3 : βt4) ∈ P6(K) :

(α, β), (s, t) ∈ K2 \ {(0, 0)}},

so that VA is the disjoint union of the lines in P6(K) joining the points(s : t : 0 : 0 : 0 : 0 : 0) and (0 : 0 : s4 : s3t : s2t2 : st3 : t4),where (s : t) runs through the projective line P1(K). Readers with somebackground in classical algebraic geometry might recognize VA as thestandard rational normal surface scroll S(1, 4) ⊂ P6(K) (see [27, p.94]).

(ii) Find descriptions for VB and VC similar to that for VA in part (a).(iii) Let e4 = (0 : 0 : 0 : 0 : 1 : 0 : 0) ∈ P6(K). Consider the projection

map (see [30, p. 22]) π4 : P6(K) \ {e4} −→ P5(K) for which

π4((x0 : x1 : x2 : x3 : x4 : x5 : x6)) = (x0 : x1 : x2 : x3 : x5 : x6)

for all (x0 : x1 : x2 : x3 : x4 : x5 : x6) ∈ P6(K) \ {e4}. Show thatπ4(VA) = VB .

(iv) Let e3 = (0 : 0 : 0 : 1 : 0 : 0 : 0) ∈ P6(K). Consider the projectionmap π3 : P6(K) \ {e3} −→ P5(K) for which

π3((x0 : x1 : x2 : x3 : x4 : x5 : x6)) = (x0 : x1 : x2 : x4 : x5 : x6)

for all (x0 : x1 : x2 : x3 : x4 : x5 : x6) ∈ P6(K) \ {e3}. Show thatπ3(VA) = VC .

Although the two varieties VB and VC are both obtained by projecting VAfrom a point, the two projections turn out to be quite different, as can be seenfrom the properties of their homogeneous coordinate rings developed in 15.2.7.

In the next Chapter 16, we shall discuss the situation whereR =⊕

n∈N0Rn

is positively graded; we shall study in considerable depth the componentsHi

R+(M)n (n ∈ Z) of the i-th (i ∈ N0) local cohomology module, with re-

spect to the irrelevant ideal R+ of R, of a finitely generated graded R-moduleM . There is very strong motivation from projective algebraic geometry forsuch study, and we hope that our discussions of projective varieties in thischapter have given the reader some hints of this motivation.

16

Castelnuovo regularity

In Chapter 15, we have seen that, when (K is an algebraically closed field,r ∈ N and) R is the homogeneous coordinate ring of a projective varietyV ⊆ Pr(K) of positive dimension, the end of the (necessarily graded) firstlocal cohomology module H1

R+(R) is of interest: see 15.2.4. This is one moti-

vation for our work in this chapter, where we shall study, in the case whenR =⊕n∈N0

Rn is positively graded, the ends of the local cohomology modulesHi

R+(M) for a finitely generated graded R-module M . Perhaps the most im-

portant invariant related to these ends is the so-called (Castelnuovo–Mumford)regularity of M . This invariant is of great significance in algebraic geometry,and, as we shall see in 16.3.7 and 16.3.8, it provides links between local co-homology theory and the syzygies of finitely generated graded modules over apolynomial ring over a field.

Throughout this chapter, all graded rings and modules are to be understoodto be Z-graded.

16.1 Finitely generated components

Our first goal in this chapter is to establish the basic facts that, in the notation ofthe above introduction, for each i ∈ N0, the R0-module Hi

R+(M)n is finitely

generated for all n ∈ Z, and is zero for all sufficiently large n. These facts,which generalize 15.1.4(iii),(ii), are the basis for much of the work in this andthe next chapter.

16.1.1 Notation and Terminology. We shall employ the notation and ter-minology concerning graded rings and modules described in 13.1.1, 13.1.3,14.1.1 and 15.1.1, but, in accordance with our convention for the whole of thischapter, restricted to the special case in whichG = Z. In particular, all polyno-

16.1 Finitely generated components 347

mial rings R0[X1, . . . , Xn] (in n indeterminates X1, . . . , Xn over a commuta-tive Noetherian ring R0) considered in this chapter will be positively Z-gradedso that R0 is the component of degree 0 and degXi = 1 for all i = 1, . . . , n.

Recall [7, p. 29] that, when R =⊕

n∈N0Rn is positively graded, we say

that R is homogeneous if and only if R is generated as an R0-algebra by itsforms of degree 1, that is, if and only if R = R0[R1].

Also, when R is merely positively graded, L =⊕

n∈Z Ln is a graded R-module and t ∈ Z is fixed, we define

L≥t :=⊕n∈Zn≥t

Ln,

a graded submodule of L. For example, in 16.1.6 below, we shall be concernedwith the graded R-modules DR+(M)≥t (for a graded R-module M ).

We shall need to use the following graded version of the Prime AvoidanceTheorem; for a proof, we refer the reader to [7].

16.1.2 Homogeneous Prime Avoidance Lemma. (See [7, 1.5.10].) Let Rbe graded and assume the ideal a is graded and generated by elements ofpositive degree. Let p1, . . . , pn ∈ Spec(R) be such that, for all i = 1, . . . , n,we have a ⊆ pi. Then there exists a homogeneous element in a\(p1∪· · ·∪pn).

�

We can improve on the result of 16.1.2 whenR0 is local with infinite residuefield and a is generated by elements of degree 1.

16.1.3 Lemma. (See [7, 1.5.12].) Assume that R =⊕

n∈ZRn is graded,that R0 is local with infinite residue field, and that the ideal a is graded andgenerated by elements of degree 1. Let b1, . . . , bn be ideals of R such that, forall i = 1, . . . , n, we have a ⊆ bi. Then there exists a homogeneous element ofdegree 1 in a \ (b1 ∪ · · · ∪ bn).

Proof. Set a1 := a ∩ R1, and let m0 denote the maximal ideal of R0. Thehypotheses ensure that, for all i = 1, . . . , n,

(bi ∩ a1 +m0a1)/m0a1 = a1/m0a1.

Since R0/m0 is infinite, there exists r1 ∈ a1 \⋃n

i=1(bi ∩ a1 +m0a1).

16.1.4 Lemma. Assume that R =⊕

n∈ZRn is graded and that the ideal ais graded and generated by elements of positive degree. Let M be a finitelygenerated R-module such that aM =M and gradeM a > 0.

(i) There is a homogeneous element in a which is a non-zerodivisor on M .

348 Castelnuovo regularity

(ii) If, in addition,R0 is local with infinite residue field and a is generated byelements of degree 1, then there exists a homogeneous element of degree1 in a which is a non-zerodivisor on M .

Proof. Since a ⊆⋃

p∈AssM p, part (i) is immediate from 16.1.2, while part(ii) follows from 16.1.3.

We are now ready to prove the basic finiteness and vanishing theorem whichwas mentioned in the first paragraph of this section.

16.1.5 Theorem. Assume that R =⊕

n∈N0Rn is positively graded; let M

be a finitely generated graded R-module.

(i) For all i ∈ N0 and all n ∈ Z, the R0-module HiR+

(M)n is finitelygenerated.

(ii) There exists r ∈ Z such that HiR+

(M)n = 0 for all i ∈ N0 and alln ≥ r.

Proof. Let i ∈ N0. We prove, by induction on i, that HiR+

(M)n is a finitelygenerated R0-module for all n ∈ Z, and is zero for all sufficiently large valuesof n. This will prove not only part (i) but also part (ii), because, in view of3.3.3, there can only be finitely many integers i for whichHi

R+(M) = 0, since

HiR+

(M) = 0 for all i > ara(R+).We consider first the case where i = 0. Since H0

R+(M) is a submodule

of M , it is finitely generated as an R-module, and so there exists u ∈ Nsuch that (R+)

uH0R+

(M) = 0. Now (R+)iH0

R+(M)/(R+)

i+1H0R+

(M) isa Noetherian R/R+-module, and so is a Noetherian R0-module (for eachi = 0, . . . , u − 1). Therefore H0

R+(M) is a Noetherian R0-module. Hence

H0R+

(M)n is a finitely generated R0-module for all n ∈ Z, and only finitelymany of the H0

R+(M)n can be non-zero.

Now suppose that i > 0 and our desired result has been proved for smallervalues of i (and for all choices of the finitely generated graded R-module M ).Since ΓR+(M) is a graded submodule of M , it follows from 2.1.7(iii) that, forall i ∈ N, there is a (homogeneous) isomorphism

HiR+

(M) ∼= HiR+

(M/ΓR+(M)).

Thus, for the purpose of this inductive step, we can replaceM byM/ΓR+(M)

and so (in view of 2.1.2 and 2.1.1) assume that R+ contains a non-zerodivisoron M . This we do.

We can assume that M = 0; we therefore assume that M = R+M . Then,by Lemma 16.1.4, there exists a homogeneous element r ∈ R+ which is anon-zerodivisor on M . Let t denote the degree of r.

16.1 Finitely generated components 349

The exact sequence 0 −→Mr−→M(t) −→ (M/rM)(t) −→ 0 of graded

R-modules and homogeneous homomorphisms induces a long exact sequence(in *C(R)) of local cohomology modules, from which we deduce (with the aidof 14.1.10(ii)) an exact sequence

Hi−1R+

(M/rM)n+t −→ HiR+

(M)nr−→ Hi

R+(M)n+t

of R0-modules for all n ∈ Z.By the inductive hypothesis, there exists s ∈ Z such that

Hi−1R+

(M/rM)j = 0 for all j ≥ s.

Hence, for all n ≥ s− t, the sequence 0 −→ HiR+

(M)nr−→ Hi

R+(M)n+t is

exact; therefore, since HiR+

(M) is R+-torsion and r ∈ R+, we have

HiR+

(M)n = 0 for all n ≥ s− t.

The inductive hypothesis also yields that Hi−1R+

(M/rM)j is a finitely gen-erated R0-module for all j ∈ Z. Fix n ∈ Z and let k ∈ N0 be such thatn+kt ≥ s− t, so thatHi

R+(M)n+kt = 0 by the last paragraph. Now, for each

j = 0, . . . , k − 1, there is an exact sequence

Hi−1R+

(M/rM)n+(j+1)t −→ HiR+

(M)n+jtr−→ Hi

R+(M)n+(j+1)t

of R0-modules, and Hi−1R+

(M/rM)n+(j+1)t is finitely generated over R0. Wecan therefore deduce, successively, that Hi

R+(M)n+jt is finitely generated

over R0 for j = k − 1, k − 2, . . . , 1, 0; we thus conclude that HiR+

(M)nis finitely generated over R0, and the inductive step is complete.

16.1.6 Corollary. Assume that R =⊕

n∈N0Rn is positively graded; let

M =⊕

n∈ZMn be a finitely generated graded R-module. Then (with thenotation of 16.1.1)

(i) DR+(M)≥t is a finitely generated R-module for all t ∈ Z;(ii) DR+(M)n is a finitely generated R0-module for all n ∈ Z; and

(iii) for all sufficiently large n, the restriction to Mn of the map ηM :M −→DR+(M) of 2.2.6(i)(c) provides an R0-isomorphism

(ηM )n :Mn −→ DR+(M)n.

Proof. These claims are all immediate consequences of Theorem 16.1.5 andthe exact sequence

0 −→ ΓR+(M)ξM−→M

ηM−→ DR+(M)ζM−→ H1

R+(M) −→ 0

of graded R-modules and homogeneous homomorphisms: see 13.5.4(i).


Theorem 16.1.5 shows that, when R is positively graded, the theory of localcohomology with respect to the irrelevant ideal ofR is particularly satisfactory.It is natural to ask whether local cohomology with respect to other homoge-neous ideals exhibits similar properties. The next three exercises concern ex-amples where the answer is clearly negative: the conclusions of these exercisesshould be compared with those of Theorem 16.1.5.

16.1.7 Exercise. Let (R0, πR0) be a discrete valuation ring, and letR denotethe polynomial ring R0[X]. Let m := πR +XR, the unique maximal gradedideal of R. Let M be a non-zero, torsion-free, finitely generated, graded R-module.

(i) Use the Graded Finiteness Theorem 14.3.10 to show that H1m(M) is

finitely generated.(ii) Show that H2

m(M/XM) = 0.(iii) Use 13.5.5 to see that there exists n0 ∈ Z for which H1

m(M/XM)n0 isnot finitely generated as R0-module.

(iv) Use the exact sequence 0 → MX−→ M(1) → (M/XM)(1) → 0 to

show that there exists n1 ∈ Z such that H2m(M)n is not finitely gener-

ated for all n < n1 and there exists n2 ∈ Z such that H2m(M)n = 0 for

all n > n2.

16.1.8 Exercise. Let R0 be a commutative Noetherian ring, and let R denotethe polynomial ringR0[X,Y ]. Use Exercise 13.5.5 to show that, for all n ∈ Z,the R0-module H1

XR(R)n is free but not finitely generated.Compare this with the conclusions of Theorem 16.1.5.

16.1.9 Exercise. Let (R0, πR0) be a discrete valuation ring as in Exercise16.1.7, and let R denote the graded ring R0[X,Y ], as in Exercise 16.1.8. Leta := πR + XR. Use 16.1.8 and 14.1.11 to show that, for all n ∈ Z, theR0-module H2

a(R)n is not finitely generated.

16.1.10 �Exercise. Assume that R =⊕

n∈N0Rn is positively graded and

such that (R0,m0) is local. Set m = m0R + R+, the unique graded maximalideal of R. Let M be a finitely generated graded R-module such that d :=

dim(M/ΓR+(M)) > 0.

(i) Show that there is a homogeneous element a of positive degree t whichis a non-zerodivisor on M :=M/ΓR+

(M).(ii) Use Theorem 6.1.4 to show that Hd

m(M) = 0, and part (i) to deduce thatthere are infinitely many negative n ∈ Z such that Hd

m(M)n = 0.(iii) Use 14.1.7 and 14.1.12 to show that, for each i ∈ N0, there are only

finitely many integers n ∈ Z for which Him(ΓR+(M))n = 0.

16.2 The basics of Castelnuovo regularity 351

(iv) Prove that Hdm(M)n = 0 for infinitely many negative integers n.

16.2 The basics of Castelnuovo regularity

Assume that R is positively graded and homogeneous, and let M be a finitelygenerated graded R-module. It is clear from 3.3.3 and 16.1.5 that there is aninteger r such that, for all s ∈ Z with s > r, we have Hi

R+(M)s−i = 0 for all

i ∈ Z (or, equivalently, for all i ∈ N0). The Castelnuovo regularity of M is theinfimum of the set of integers r with this property (interpreted as −∞ if thisinfimum does not exist). This is a very important invariant of M , and a topicwhich has featured in much recent research. In this section, we develop thebasic theory of this and some related, slightly more complicated, invariants.

16.2.1 Definitions. Assume that R is positively graded and homogeneous(see 16.1.1). Let M be a finitely generated graded R-module, and let r ∈ Zand l ∈ N0.

(i) We follow A. Ooishi in [65, Definition 1] and say that M is r-regularin the sense of Castelnuovo–Mumford if and only if Hi

R+(M)s−i = 0

for all i, s ∈ Z with s > r. In practice, the phrase ‘in the sense ofCastelnuovo–Mumford’ is usually omitted.

As pointed out just before this definition, there does exist an r ∈ Z such thatM is r-regular. Note that M is r-regular if and only if Hi

R+(M)s−i = 0 for

all i, s ∈ Z with s > r and i ≥ 0. This observation leads to the following moregeneral definition.

(ii) We say that M is r-regular at and above level l if and only if

HiR+

(M)s−i = 0 for all i, s ∈ Z with s > r and i ≥ l.

Thus M is r-regular if and only if it is r-regular at and above level 0.

Motivation for Definition 16.2.1 comes from Theorem 16.2.5 below, whichis a version of a proposition of D. Mumford [54, p. 99], adapted to the con-text of local cohomology. (See also Ooishi [65, Theorem 2].) However, beforepresenting Theorem 16.2.5, we provide some technical comments which willhelp us to make appropriate reductions in the proof of the theorem and otherresults.

16.2.2 Remarks. Assume that R =⊕

n∈ZRn is graded, and that the ideal ais graded. Let R′0 be a commutative Noetherian ring, and let f0 : R0 −→ R′0


be a flat ring homomorphism. Set R′ := R ⊗R0R′0, and let f : R −→ R′ be

the natural ring homomorphism.

(i) Since R is a finitely generated R0-algebra by [7, Theorem 1.5.5], itfollows that R′ is Noetherian. Furthermore, R′ has the structure of agraded ring given by R′ =

⊕n∈ZR

′n, where R′n is the natural image of

Rn ⊗R0 R′0 in R′ for each n ∈ N. Thus the ring homomorphism f is

homogeneous (in the sense of 14.1.3), and flat.(ii) It therefore follows from the Graded Flat Base Change Theorem 14.1.9

that the sequences (Hia( • ) ⊗R R′)i∈N0 and (Hi

aR′(( • ) ⊗R R′))i∈N0

(restricted to the ‘graded’ category *C(R)) are isomorphic negativeconnected sequences of covariant functors from *C(R) to *C(R′).

(iii) Let L =⊕

n∈Z Ln be a graded R-module. Then L ⊗R R′ is a gradedR′-module: if we use the natural isomorphisms

L⊗R R′ = L⊗R (R⊗R0 R

′0)

∼=−→ (L⊗R R)⊗R0 R′0

∼=−→ L⊗R0 R′0

to identify L ⊗R R′ with L ⊗R0 R′0 =: L′, then the grading on L′ is

given by L′ =⊕

n∈Z L′n, where L′n is the natural image of Ln ⊗R0 R

′0

in L′.(iv) It follows from part (ii), and part (iii) applied to the graded R-module

Hia(M) (where M is a graded R-module), that, for all i ∈ N0 and all

n ∈ Z, there is a natural equivalence of functors (from *C(R) to C(R′0))

Hia( • )n ⊗R0 R

′0 −→ Hi

aR′(( • )⊗R R′)n.

(v) We can deduce from 4.3.5, 13.3.14 and the formula in 2.2.15(ii) thatthere is a natural equivalence of functors

ε : Da( • )⊗R R′ −→ DaR′(( • )⊗R R

′)

from the ‘graded’ category *C(R) to *C(R′).(vi) If we now use part (v), and part (iii) applied to the graded R-module

Da(M) (where M is a graded R-module), we obtain, for each n ∈ Z, anatural equivalence of functors (from *C(R) to C(R′0))

Da( • )n ⊗R0 R′0 −→ DaR′(( • )⊗R R

′)n.

We give two examples where the ideas of 16.2.2 can be used.

16.2.3 Example. Assume that R =⊕

n∈N0Rn is positively graded and ho-

mogeneous (see 16.1.1), let M be a graded R-module, and let p0 ∈ Spec(R0).We can apply the techniques (and notation) of 16.2.2 with the choices a =

R+ and R′0 = (R0)p0 , a flat R0-algebra. The ring R′ = R ⊗R0 (R0)p0 isagain positively graded and homogeneous, and M ′ = M ⊗R0 (R0)p0 is a


graded R′-module. Also, R+R′ = R′+, the irrelevant ideal of R′. It follows

from 16.2.2(iv) that, for each i ∈ N0 and each n ∈ Z, there is an isomorphismof (R0)p0 -modules

(Hi

R+(M)n)p0

∼= HiR′

+(M ′)n. Finally, note that the 0-th

component of R′ is a local ring isomorphic to (R0)p0 .

16.2.4 Example. Assume that R =⊕


mogeneous, and such that (R0,m0) is local; let M be a graded R-module.We can apply the techniques (and notation) of 16.2.2 with the choices a =

R+ and R′0 = R0[X]m0R0[X], the localization of the polynomial ring R0[X]

at the prime ideal m0R0[X]. This R′0 is a flat R0-algebra, and

R′ = R⊗R0 R0[X]m0R0[X]

is positively graded and homogeneous, M ′ = M ⊗R0 R0[X]m0R0[X] is agraded R′-module, and R+R

′ = R′+, the irrelevant ideal of R′.It follows from 16.2.2(iv) that, for each i ∈ N0 and each n ∈ Z, there is an

isomorphism of R0[X]m0R0[X]-modules

HiR+

(M)n ⊗R0 R0[X]m0R0[X]∼= Hi

R′+(M ′)n;

thus, sinceR0[X]m0R0[X] is a faithfully flatR0-algebra,HiR+

(M)n = 0 if andonly if Hi

R′+(M ′)n = 0.

Finally, note that the 0-th component of R′ is a local ring isomorphic toR0[X]m0R0[X], having infinite residue field.



mogeneous; let M be a finitely generated graded R-module. Let r ∈ Z andl ∈ N: the reader should note that we are assuming that l is positive.

Assume that HiR+

(M)r+1−i = 0 for all i ≥ l. Then M is r-regular at andabove level l (see 16.2.1(ii)).

Proof. It suffices to show that, for each p0 ∈ Spec(R0), the (R0)p0 -module(Hi

R+(M)s−i)p0 vanishes for all i ≥ l and all s > r.

We can apply Examples 16.2.3 and 16.2.4 in turn to see that it is enough forus to establish the claim in the statement of the theorem under the additionalassumption that (R0,m0) is a local ring with infinite residue field, and we shallmake this assumption in what follows.

It follows from [7, 1.5.4] that dimM is finite, and we are going to argue byinduction on dimM . When dimM = −1 there is nothing to prove, and whendimM = 0 the result is an easy consequence of Grothendieck’s VanishingTheorem 6.1.2.

Now suppose that dimM > 0 and our desired result has been proved forall finitely generated graded R-modules of smaller dimension. Since ΓR+(M)


is a graded submodule of M , it follows from 2.1.7(iii) that, for each i ∈ N,there is a homogeneous isomorphism Hi

R+(M) ∼= Hi

R+(M/ΓR+(M)). In

particular, since l is positive, HiR+

(M/ΓR+(M))r+1−i = 0 for all i ≥ l. Nowwe must have dim(M/ΓR+(M)) ≤ dimM : if this inequality is strict, we canuse the inductive assumption to achieve our aim; otherwise, for the purposeof this inductive step, we can replace M by M/ΓR+(M) and so (in view of2.1.2 and 2.1.1) assume that M is R+-torsion-free and that R+ contains anon-zerodivisor on M . This we do.

Now M = R+M (since otherwise M = 0), and so, by Lemma 16.1.4(ii),there exists a homogeneous element a ∈ R1 (and so of degree 1) which is anon-zerodivisor on M .

The exact sequence 0 −→Ma−→M(1) −→ (M/aM)(1) −→ 0 of graded

R-modules and homogeneous homomorphisms induces a long exact sequence(in *C(R)) of local cohomology modules, from which we deduce (with the aidof 14.1.10(ii)) an exact sequence

HiR+

(M)n −→ HiR+

(M/aM)n −→ Hi+1R+

(M)n−1a−→ Hi+1

R+(M)n

ofR0-modules, for all i ∈ N and all n ∈ Z. Use of this with n = r+1−i showsthat Hi

R+(M/aM)r+1−i = 0 for all i ≥ l. Now dim(M/aM) < dimM

(since a lies outside every minimal prime ideal of SuppM ), and so it followsfrom the inductive hypothesis that M/aM is r-regular at and above level l.

Fix an integer i ≥ l. For each n ∈ Z, there is an exact sequence

HiR+

(M)n−1−ia−→ Hi

R+(M)n−i −→ Hi

R+(M/aM)n−i

of R0-modules. We know that HiR+

(M)r+1−i = HiR+

(M/aM)r+2−i =

0, and so it follows from the above exact sequence (with n = r + 2) thatHi

R+(M)r+2−i = 0. We can now repeat this argument, using induction, to

deduce that HiR+

(M)r+j−i = 0 for all j ∈ N. It follows that M is r-regularat and above level l.

In the statement (and proof) of Theorem 16.2.5, we stressed that the integerl is assumed to be positive. In fact, if l in that theorem is replaced by 0, then theresulting statement is no longer always true: this is illustrated by the followingexercise.

16.2.6 Exercise. Let the situation and notation be as in Theorem 16.2.5. Finda counterexample to the statement ‘if Hi

R+(M)r+1−i = 0 for all i ≥ 0, then

M is r-regular at and above level 0 (that is, M is r-regular)’ by consideringR := R0[X1, . . . , Xn], the ring of polynomials in n (≥ 1) indeterminates overa commutative Noetherian ring R0, and M := tR0 ⊕ R for an appropriatevalue of the integer t, where tR0 is as defined in 14.1.13.


The result of Theorem 16.2.5 can be rephrased thus: for a fixed positiveinteger l, if Hi

R+(M)r+1−i = 0 for all i ≥ l, then Hi

R+(M)s−i = 0 for all

i ≥ l and s > r. Some readers might find the interest in ‘reverse diagonalvanishing’ surprising, and they might find the following diagram helpful. Thediagram concerns the special case of Theorem 16.2.5 in which R0 is Artinian,so that, for all i, n ∈ Z, the R0-module Hi

R+(M)n has finite length hiM (n).

In the diagram, hiM (n) is plotted at the position (n, i) in the Oni co-ordinateplane. Theorem 16.2.5 shows that, if in the diagram

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0

0

....0

0

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0

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0

0

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0

0

0

0

....0

0

0

....

.

.

.

.

..

.

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.

..

.

.

.

.

..

.

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.

..

.

.

.

.

..

.

.

.

..

.

.

.

.

..

.

.

.

..

....0000....000....0000....

....0000....000....0000....

....0000....000....0000....

....0000....000....0000....

n

i

r

r

l − 1

d

hiM (n)

(in which d denotes dimM ) there is a line of zeros on the line i + n = r + 1

above the line i = l − 1, then there must be a similar line of zeros on the linei+ n = s above the line i = l − 1 for every integer s > r.

Some readers may wonder whether ‘vertical vanishing’ works just as well inthis context as ‘reverse diagonal vanishing’. It doesn’t: the following exampleprovides a counterexample to the statement ‘if Hi

R+(M)r+1 = 0 for all i ∈ N


(where R is positively graded and homogeneous and M is a finitely generatedgraded R-module), then Hi

R+(M)s = 0 for all i ∈ N and s > r’.

16.2.7 Example. We consider again the example studied in 13.5.12. ThusK is a field, d ∈ N with d ≥ 3, and A(d) is the subring of the d-th Verone-sean subring K[X,Y ](d) of the ring K[X,Y ] of polynomials over K in twoindeterminates X and Y given by A(d) := K[Xd, Xd−1Y,XY d−1, Y d], withgrading inherited from K[X,Y ](d).

Since A(d) is a 2-dimensional domain, we have HiA(d)+

(A(d)) = 0 if i = 0

or i > 2. By 13.5.12, for n ∈ Z, we have H1A(d)+

(A(d))n = 0 if and only if1 ≤ n ≤ d− 3.

Let φ : A(d) −→ K[X,Y ](d) denote the inclusion homomorphism. Weshowed in 13.5.12 that there is a homogeneous A(d)-isomorphism Cokerφ ∼=H1

A(d)+(A(d)). It therefore follows from 2.1.7(i) that there is a homogeneous

A(d)-isomorphism H2A(d)+

(A(d)) ∼= H2A(d)+

(K[X,Y ](d)). Further, argumentswe provided in 13.5.12, together with 13.5.9 and the Graded IndependenceTheorem 14.1.7, show that there are homogeneous A(d)-isomorphisms

H2A(d)+

(K[X,Y ](d)) ∼= H2A(d)+K[X,Y ](d)(K[X,Y ](d))

= H2(K[X,Y ]+)(d)(K[X,Y ](d))

∼= (H2(X,Y )(K[X,Y ]))(d,0).

It now follows from 13.5.3 and 13.5.9 that H2A(d)+

(A(d))n = 0 if and only ifn < 0.

We can now conclude, for r, s ∈ Z, that HiA(d)+

(A(d))s−i = 0 for all i ∈ Zif and only if s > d− 2. Thus A(d) is r-regular if and only if r ≥ d− 2.

Notice also that HiA(d)+

(A(d))0 = 0 for all i ∈ N, whereas, when d > 3,we have H1

A(d)+(A(d))1 = 0: thus this example shows that the statement ‘if

HiR+

(M)r+1 = 0 for all i ∈ N (where R is positively graded and homoge-neous and M is a finitely generated graded R-module), then Hi

R+(M)s = 0

for all i ∈ N and s > r’ is false.

16.2.8 Remarks. Assume that R is positively graded and homogeneous; letM be a finitely generated graded R-module. Let r ∈ Z and let l be a positiveinteger.

(i) It follows from Theorem 16.2.5 that M is r-regular at and above level lif and only if Hi

R+(M)r+1−i = 0 for all i ≥ l.

(ii) However, M is r-regular at and above level 0 (that is, M is r-regular) ifand only if Hi

R+(M)r+1−i = 0 for all i ∈ Z and H0

R+(M)s = 0 for all


s > r. This is precisely the condition used to define the condition ‘M isr-regular’ by D. Eisenbud and S. Goto [11, p. 95].

We are now ready to define Castelnuovo regularity, and also some refine-ments which have been studied by several authors, including U. Nagel [57],Nagel and P. Schenzel [58, Definition 6.1] and L. T. Hoa and C. Miyazaki [36,§2].

16.2.9 Definition. Assume that R =⊕


homogeneous, and letM be a finitely generated gradedR-module. Let l ∈ N0.The end of a graded R-module was defined in 14.1.1.

We define the (Castelnuovo–Mumford) regularity reg(M) of M by

reg(M) := sup{end(Hi

R+(M)) + i : i ∈ N0

}= sup{end(Hi

R+(M)) + i : 0 ≤ i ≤ dimM

}.

Also, the (Castelnuovo–Mumford) regularity regl(M) of M at and above level

l is defined by regl(M) := sup{end(Hi

R+(M)) + i : i ≥ l

}. Thus the reg-

ularity reg0(M) of M at and above level 0 is the regularity reg(M) of M .Furthermore, regl(M) = inf {r ∈ Z :M is r-regular at and above level l} .

Since there are only finitely many integers i for which HiR+

(M) = 0, itfollows from 16.1.5(ii) that reg(M) is either an integer or −∞. Note also thatregl(M) ≤ regl−1(M) ≤ · · · ≤ reg0(M) = reg(M).

Observe also that, by 14.1.10(ii), regl(M(t)) = regl(M)− t for all t ∈ Z.

16.2.10 �Exercise. Let the situation be as in 16.2.9.

(i) Show that regl(M) = −∞ if and only if HiR+

(M) = 0 for all i ≥ l.(ii) Let N be an R+-torsion graded submodule of M . Show that regl(M) =

regl(M/N) for all l > 0.(iii) Let p be a minimal prime of R+ + (0 :R M) and assume that l ≤

ht(p/(0 :R M)). Show that regl(M) > −∞.(iv) Assume that R0 is Artinian and local, and that M = 0. Show that

regl(M) = −∞ if and only if l > ht ((R+ + (0 :R M))/(0 :R M)).

16.2.11 �Exercise. Let R0 be a commutative Noetherian ring, let n ∈ N,and let R := R0[X1, . . . , Xn], the ring of polynomials over R0. Use Example13.5.3 to show that reg(R) = 0.

16.2.12 Proposition. Assume that R =⊕


homogeneous, and let M =⊕

n∈ZMn be a finitely generated graded R-module. Then reg1(M) = −∞ if and only if Mn = 0 for all n� 0.


Proof. (⇐) Suppose n0 ∈ N is such thatMn = 0 for all n ∈ Z with |n| ≥ n0.Then (R+)

2n0M = 0, so thatM isR+-torsion. Therefore, by 2.1.7(i), we haveHi

R+(M) = 0 for all i > 0, and so reg1(M) = −∞.

(⇒) Assume that reg1(M) = −∞. This means that HiR+

(M) = 0 for alli > 0. It therefore follows from 2.1.7(iii) and 2.1.2 that Hi

R+(M/ΓR+(M)) =

0 for all i ∈ N0. Hence M/ΓR+(M) = R+(M/ΓR+(M)), by 6.2.7 and 6.2.4.It follows from this that, since M/ΓR+(M) is finitely generated and graded,we must have M/ΓR+

(M) = 0. Hence M = ΓR+(M), and the desired result

follows from this and the fact that M is finitely generated.

16.2.13 �Exercise. Let the situation be as in Proposition 16.2.12. Show thatreg(M) = −∞ if and only if M = 0.



homogeneous. Let M =⊕

n∈ZMn be a finitely generated graded R-module,let l, l′ ∈ N0 with l ≥ l′, and let r ∈ Z.

It is clear that, if M is r-regular at and above level l′, then M is r-regular atand above level l. Give an example to show that the converse statement is notalways true.



homogeneous, let l ∈ N0, and let 0 −→ L −→ M −→ N −→ 0 be anexact sequence of finitely generated graded R-modules and homogeneous ho-momorphisms. Show that

(i) reg(L) ≤ max {reg(M), reg(N) + 1},(ii) regl+1(L) ≤ max

{regl+1(M), regl(N) + 1

},

(iii) regl(M) ≤ max{regl(L), regl(N)

}, and

(iv) regl(N) ≤ max{regl+1(L)− 1, regl(M)

}.

16.3 Degrees of generators

The following theorem is one of the main reasons for our introduction of theconcept of regularity. Among its consequences are the links, hinted at in theintroduction to this chapter, between local cohomology theory and the syzygiesof finitely generated graded modules over a polynomial ring over a field.



mogeneous, and let M =⊕

n∈ZMn be a non-zero finitely generated gradedR-module. ThenM can be generated by homogeneous elements of degrees notexceeding reg(M).

16.3 Degrees of generators 359

Proof. Our strategy in this proof has some similarities to that for our proof ofTheorem 16.2.5 above, inasmuch as we apply 16.2.3 and 16.2.4 to reduce tothe case in which R0 is local with infinite residue field.

Let N be the graded submodule of M generated by⊕

n≤reg(M)Mn: wemust show that M = N .

It suffices to show that Mp0 = Np0 for each p0 ∈ Spec(R0). We use 16.2.3:consider the positively graded, homogeneous ring R′ = R ⊗R0 (R0)p0 andthe graded R′-module M ′ = M ⊗R0

(R0)p0, which is finitely generated. By

16.2.3, reg(M ′) ≤ reg(M); also, under the canonical isomorphism betweenM ′ andMp0 , theR′-submodule ofM ′ generated by all homogeneous elementsof degrees not exceeding reg(M) is mapped onto Np0 . It is therefore enoughfor us to establish the claim in the statement of the theorem under the additionalassumption that R0 is local. We can then use 16.2.4 in a similar way to seethat it is enough for us to establish the claim under the additional assumptionthat (R0,m0) is a local ring with infinite residue field, and we shall make thisassumption in what follows.

Now d := dimM is finite, and reg(M) = −∞ since M = 0 (by 16.2.13).We argue by induction on d. Consider first the case where d = 0. ThenAssM = {m0 ⊕R+}, so that there exists t ∈ N such that (R+)

tM = 0.In this case, M = ΓR+(M) and so Hi

R+(M) = 0 for all i ∈ N, by 2.1.7(ii).

Thus reg(M) = end(M), and it is obvious that M can be generated by homo-geneous elements of degrees not exceeding end(M).

Now suppose that dimM > 0 and our desired result has been proved forall non-zero, finitely generated graded R-modules of smaller dimensions. Ofcourse, ΓR+(M), if non-zero, can be generated by homogeneous elements ofdegrees not exceeding end(ΓR+(M)); since end(ΓR+(M)) ≤ reg(M), it fol-lows from the exact sequence

0 −→ ΓR+(M) −→M −→M/ΓR+(M) −→ 0

that it is enough for us to show that M/ΓR+(M) can be generated by homo-geneous elements of degrees not exceeding reg(M). Note that

reg(M/ΓR+(M)) = reg1(M) ≤ reg(M).

Now dim(M/ΓR+(M)) ≤ dimM : if this inequality is strict, we can use theinductive assumption to achieve our aim; otherwise, for the purpose of thisinductive step, we can replace M by M/ΓR+

(M) and so (in view of 2.1.2and 2.1.1) assume that M is R+-torsion-free and that R+ contains a non-zerodivisor on M . This we do.

Note that M = R+M . Then, by Lemma 16.1.4(ii), there exists a homoge-neous element a ∈ R1 (and so of degree 1) which is a non-zerodivisor on M .


Apply 16.2.15(iv) to the exact sequence

0 −→M(−1) a−→M −→M/aM −→ 0

of graded R-modules and homogeneous homomorphisms to see that

reg(M/aM) ≤ max{reg1(M(−1))− 1, reg(M)

}= reg(M).

Now dim(M/aM) < dimM , and so it follows from the inductive hypoth-esis that M/aM can be generated by homogeneous elements of degrees notexceeding reg(M/aM). Hence M = N + aM . Now R is *local with unique*maximal graded ideal m := m0 ⊕ R+; it follows from Nakayama’s Lemmathat Mm = Nm, so that M = N by 14.1.2(ii) applied to M/N .

16.3.2 �Exercise. Let R := K[X1, . . . , Xn], the ring of polynomials in n(≥ 1) indeterminates over a field K. Let a be a proper graded ideal of R suchthat H0

R+(R/a) = Hn

R+(R/a) = 0. Use 16.2.11, 16.2.15 and 16.3.1 to show

that 0 ≤ reg1(R/a) = reg2(a)− 1.

In the following definition and subsequent exercises, we consider a geomet-ric significance of the Castelnuovo regularity.

16.3.3 Definition and Remark. Let K be an algebraically closed field, andlet r ∈ N. Let V ⊂ Pr(K) be a projective variety. Let R := K[X0, . . . , Xr],and consider R as the coordinate ring O(Ar+1(K)). Then the graded primeideal IPr(K)(V ) := IAr+1(K)(Cone(V )) of R is non-zero: we define the Cast-elnuovo–Mumford regularity reg(V ) of V by reg(V ) := reg(IPr(K)(V )).

Note that H0R+

(R) = H1R+

(R) = 0, and the homogeneous coordinate ringO(Cone(V )) of V is an integral domain. It follows from these observationsand 16.3.2 that

reg(V ) = reg(IPr(K)(V )) = reg1(IPr(K)(V )) = reg2(IPr(K)(V ))

= reg1(O(Cone(V ))) + 1 = reg(O(Cone(V ))) + 1.

Note that, by 16.3.1, the vanishing ideal IPr(K)(V ) can be generated by homo-geneous polynomials of degrees not exceeding reg(V ).

16.3.4 Exercise. Let K be an algebraically closed field, and let r ∈ N. LetV ⊂ Pr(K) be a projective variety such that dimV > 0. Let n ∈ N be suchthat reg(V ) ≤ n+ 1.

(i) Show that the n-th Veronesean V (n) ⊆ P(r+nr )−1(K) of V has

arithdepthV (n) > 1.

(ii) Show that reg(V (n)) ≤ dimV + 2.


16.3.5 Exercise. Let K be an algebraically closed field, and let d ∈ N withd > 1. Here we study again the rational normal curve in projective d-spaceN(d) ⊆ Pd(K) of 15.2.3.

Consider the Veronesean subring K[X0, X1](d) of the ring K[X0, X1] of

polynomials over K in two indeterminates X0, X1, and the polynomial ringK[Y0, . . . , Yd] in d + 1 indeterminates Y0, . . . , Yd as O(Ad+1(K)), the co-ordinate ring of affine (d + 1)-space over K. Let φ : K[Y0, . . . , Yd] −→K[X0, X1]

(d) be the K-algebra homomorphism for which

φ(Y0) = Xd0 , φ(Y1) = Xd−1

0 X1, . . . , φ(Yd−1) = X0Xd−11 , φ(Yd) = Xd

1 .

Then Kerφ = IPd(K)(N(d)).

(i) Calculate reg(N(d)), and deduce that IPd(K)(N(d)) can be generated byquadratics.

(ii) Calculate the dimension of IPd(K)(N(d))2, the component of degree 2 ofIPd(K)(N(d)), as a vector space over K.

(iii) Deduce that IPd(K)(N(d)) can be generated by the 2 × 2 minors of thematrix [

Y0 Y1 Y2 . . . Yd−2 Yd−1

Y1 Y2 Y3 . . . Yd−1 Yd

].

16.3.6 Exercise. Use the notation of 15.2.5, but in the special case in whichd = 4. Thus K is an algebraically closed field, and A(4) is the subring

K[X4, X3Y,XY 3, Y 4]

of the ring K[X,Y ] of polynomials over K in two indeterminates X and Y .Also, ψ : K[Y0, Y1, Y2, Y3] −→ A(4) is the K-algebra homomorphism forwhich ψ(Y0) = X4, ψ(Y1) = X3Y , ψ(Y2) = XY 3 and ψ(Y3) = Y 4, andKerψ is the vanishing ideal of the projective variety Σ(4) ⊆ P3(K).

(i) Calculate reg(Σ(4)), and deduce that IP3(K)(Σ(4)) can be generated byhomogeneous polynomials of degrees not exceeding 3.

(ii) For i = 2, 3, calculate the dimension of IP3(K)(Σ(4))i, the componentof degree i of IP3(K)(Σ(4)), as a vector space over K. Find a quadraticand three cubics which generate IP3(K)(Σ(4)).

The next result uses Theorem 16.3.1 to establish connections between theregularity of a finitely generated graded module M over a polynomial ringover a field and the syzygies of M : we hinted at these connections in the intro-duction to this chapter. The result is presented by Eisenbud and Goto in [11,p. 89], although they state that it is not new and that it has origins in ideas of


Castelnuovo and Mumford. We refer the reader to [7, pp. 36–37] for funda-mental facts concerning the minimal graded free resolution of M .

16.3.7 Theorem: syzygetic characterization of regularity. Let K be a fieldand let R =

⊕j∈N0

Rj := K[X1, . . . , Xn], the ring of polynomials over Kin n indeterminates (where n ∈ N). Let M be a non-zero finitely generatedgraded R-module having proj dimM = p, and let

0 −→ Fpfp−→ Fp−1 −→ . . . −→ F1

f1−→ F0 −→ 0,

be the minimal graded free resolution of M . Thus there exist b0, . . . , bp ∈ N

and, for each j = 0, . . . , p, integers a(j)i (i = 1, . . . , bj) such that a(j)1 ≥ · · · ≥a(j)bj

and Fj =⊕bj

i=1R(a(j)i ). Then

reg(M) = max{−a(j)bj

− j : j = 0, . . . , p}.

Proof. The fi (i = 1, . . . , p) are homogeneous, and there is a homogeneousR-epimorphism f0 : F0 −→M such that Ker f0 = Im f1.

In view of 16.2.11 and 16.2.9, we have

reg(F0) = reg(⊕b0

i=1R(a(0)i ))= max

{reg(R(a

(0)i )): i = 0, . . . , b0

}= max

{−a(0)i : i = 0, . . . , b0

}= −a(0)b0

.

We argue by induction on p. When p = 0, we have a homogeneous isomor-phism M ∼= F0, and the claim follows from the above equations. So suppose,inductively, that p > 0 and that the result has been proved for smaller valuesof p. Set L := Ker f0 = Im f1.

Now F0, and therefore M , can be generated by homogeneous elements ofdegrees not exceeding reg(F0) = −a(0)b0

, by 16.3.1. By the minimality, Mcannot be generated by homogeneous elements all of whose degrees are lessthan −a(0)b0

; therefore reg(M) ≥ −a(0)b0= reg(F0) by 16.3.1.

There is an exact sequence 0 −→ L⊆−→ F0 −→ M −→ 0 in the category

*C(R). Application of 16.2.15(i) to this yields that

reg(L) ≤ max{reg(F0), reg(M) + 1} = reg(M) + 1,

so that reg(L)− 1 ≤ reg(M); on the other hand, application of 16.2.15(iv) tothe same sequence yields that

reg(M) ≤ max{reg1(L)− 1, reg(F0)} ≤ max{reg(L)− 1,−a(0)b0},

and so we see that reg(M) = max{reg(L)− 1,−a(0)b0}.

The exact sequence 0 −→ Fpfp−→ Fp−1 −→ . . . −→ F1

f1−→ L −→ 0


yields the minimal graded free resolution of L, so that proj dimL = p − 1.Therefore, by the inductive hypothesis,

reg(L) = max{−a(j)bj

− (j − 1) : j = 1, . . . , p}

= max{−a(j)bj

− j : j = 1, . . . , p}+ 1.

Therefore

reg(M) = max{reg(L)− 1,−a(0)b0

}= max

{max{−a(j)bj

− j : j = 1, . . . , p},−a(0)b0

},

and the inductive step can be completed.

16.3.8 Example. Consider the situation of Theorem 16.3.7 above in the spe-cial case in which M = R/b, where b is a proper graded ideal of R. SinceR/b can be generated by one homogeneous element of degree 0, and sinceKer fj ⊆ mFj for all j = 0, . . . , p, it follows that, in this case, b0 = 1,a(0)1 = 0, and −a(j+1)

1 ≥ −a(j)1 +1 for all j = 0, . . . , p− 1. Hence −a(j)1 ≥ j

for all j = 0, . . . , p.On the other hand, by 16.3.7, we have −a(j)bj

≤ reg(M) + j for all j =

0, . . . , p. Thus, for a given integer j such that 0 ≤ j ≤ p, the −a(j)i (1 ≤ i ≤bj) are constrained to lie in the interval {r ∈ N0 : j ≤ r ≤ reg(M) + j} oflength reg(M).



homogeneous, and let M be a non-zero finitely generated graded R-module.Let a1, . . . , ah be homogeneous elements of R with deg ai = ni > 0 fori = 1, . . . , h, and suppose that a1, . . . , ah is an M -sequence. Use Exercise16.2.15 to show that

reg1(M/∑h

i=1aiM)≤ reg1(M) +

∑hi=1 ni − h ≤ reg

(M/∑h

i=1aiM),

and that both inequalities in this display are equalities if gradeM (R+) > h.

17

Hilbert polynomials

Suppose that R =⊕

n∈N0Rn is positively Z-graded and homogeneous and

such thatR0 is an Artinian ring, and letM =⊕

n∈ZMn be a non-zero finitelygenerated graded R-module of dimension d. By a classical theorem of Hilbert(see [7, Theorem 4.1.3]), there is a polynomial PM ∈ Q[X] of degree d − 1,and an n0 ∈ Z, such that PM (n) = �R0(Mn) for all n ≥ n0. The polynomialPM is necessarily uniquely determined, and known as the Hilbert polynomialof M . The classical ‘postulation’ problem asked for an explanation of the dif-ference PM (n) − �R0(Mn) for n < n0. Serre solved this in [77] using sheafcohomology. In this chapter we present, via graded local cohomology, the ideasbehind Serre’s approach; the reader will find a translation into the language ofsheaf cohomology in 20.4.16.

We study, in §17.1 below, the so-called characteristic function χM : Z −→Z of M and show that this is completely represented by a polynomial. In moredetail, it turns out that, for each i ∈ N0 and n ∈ Z, the n-th componentRiDR+(M)n of RiDR+(M) has finite length as an R0-module, and we de-note this length by diM (n); it makes sense for us to define the characteristicfunction χM : Z −→ Z of M by setting

χM (n) =∑i∈N0

(−1)idiM (n) for all n ∈ Z.

We shall see in 17.1.7 that there is a polynomial q ∈ Q[X] of degree d−1 suchthat q(n) = χM (n) for all n ∈ Z. The fact that Hi

R+(M)n = 0 for all i ≥ 0

and n � 0 (see 16.1.5) means that we can conclude that χM (n) = �R0(Mn)

for all n � 0; this not only provides a proof of Hilbert’s Theorem mentionedabove (q turns out to be the Hilbert polynomial PM ), but also yields preciseinformation about the difference PM (n)− �R0(Mn) for small n.

Also in §17.1, we use Hilbert’s Theorem and graded local duality to establishthe existence of ‘cohomological Hilbert polynomials’: for each i ∈ N0, there

Hilbert polynomials 365

is a polynomial piM ∈ Q[X] of degree less than i such that

�R0(HiR+

(M)n) = piM (n) for all n� 0.

In §17.2, we study the invariant reg2(M) (where M is as above). We showthat both DR+(M) and H1

R+(M) can be generated by homogeneous elements

of degrees not exceeding reg2(M) (provided reg2(M) is finite). This will en-able us to prove (in 20.4.13) an important result on coherent sheaves due toSerre [80]. This is a small hint about the significance of the invariant reg2.Furthermore, in the case where there is a homogeneous element a of R of de-gree 1 that is a non-zerodivisor on M , we show that, as n increases beyondreg2(M/aM)−1, the integers �R0(H

2R+

(M)n) decrease strictly to 0 and thenremain at 0. This is a generalization of a result, actually formulated in terms ofsheaf cohomology, of Mumford [54, p. 99].

To appreciate fully the significance of the invariant reg2, the reader will needto become aware of the links between local cohomology and sheaf cohomol-ogy, which will be treated in Chapter 20. In this chapter, we content ourselveswith the hints given above, and some elementary comments at the beginningof §17.2 about the significance of reg2 in the framework of defining equationsof projective varieties.

In §17.3, we generalize an important result of D. Mumford [54, p. 101],actually formulated in terms of sheaf cohomology, which shows that, for anon-zero graded ideal b of the polynomial ring K[X1, . . . , Xd] over a fieldK, the regularity reg2(b) of b at and above level 2 is bounded in terms of theHilbert polynomial Pb of b. We generalize this in 17.3.6 and prove that, for apolynomial ring R = R0[X1, . . . , Xd] over an Artinian local ring R0, for inte-gers a1, . . . , ar and a non-zero graded submodule M of the finitely generated*free graded R-module

⊕ri=1R(ai), the regularity reg2(M) is bounded in

terms of PM . The reader will find a sheaf-theoretic formulation of Mumford’sRegularity Bound in 20.4.18.

In §17.4, we apply the main result of §17.3 to give upper bounds on theinvariants reg1 and reg0 in certain circumstances. We prove that the regular-ity reg1(M) at and above level 1 is bounded in terms of PM and the numberand degrees of homogeneous generators of M . We also prove in the same sec-tion that the regularity reg0(M) = reg(M) is bounded in terms of PM , thedegrees of homogeneous generators of M and the so-called postulation num-ber pstln(M) of M , defined as the greatest integer n for which �R0

(Mn) =PM (n) if any such integers exist, and −∞ otherwise.

Once again, throughout this chapter, all graded rings and modules are tobe understood to be Z-graded, and all polynomial rings R0[X1, . . . , Xd] (overa commutative Noetherian ring R0) are to be understood to be (positively)

366 Hilbert polynomials

Z-graded so that R0 is the component of degree 0 and degXi = 1 for alli = 1, . . . , d.

17.1 The characteristic function

17.1.1 Reminders, Notation and Terminology. We shall use notation andterminology employed by Bruns and Herzog in [7, §4.1]. Consider a functionf : Z −→ Z.

(i) We say that f is of polynomial type (of degree d) if and only if thereexists a polynomial p ∈ Q[X] (of degree d) such that f(n) = p(n) forall n� 0 (that is, for all n greater than some fixed constant integer n0).(When this is the case, the polynomial p will be uniquely determined,of course, since two rational polynomials which take the same values atinfinitely many integers must coincide. We adopt the convention that thezero polynomial has degree −1.)

(ii) We define Δf : Z −→ Z by Δf(n) = f(n + 1) − f(n) for all n ∈ Z.Thus Δ is a mapping from the set of functions from Z to Z to itself. Forn ∈ N, we use Δn to denote the result of repeating Δ n times; we alsowrite Δ0f = f .

(iii) Let d ∈ N0. Recall from [7, Lemma 4.1.2] that f is of polynomial type ofdegree d if and only if there exists c ∈ Z with c = 0 such that Δdf(n) =

c for all n� 0.(iv) Recall that, for i ∈ N,(

X + i

i

):=

(X + i)(X + i− 1) . . . (X + 1)

1.2.3. . . . .i∈ Q[X],

and that(X+00

)denotes the constant polynomial 1 ∈ Q[X].

(v) Let d ∈ N. Let P ∈ Q[X] be a non-zero polynomial of degree d − 1.Recall from [7, Lemma 4.1.4] that P (n) ∈ Z for all n ∈ Z if and only ifthere exist integers e0, . . . , ed−1 such that

P (X) =d−1∑i=0

(−1)iei(X + d− i− 1

d− i− 1

).

When this is the case, we say that P is a numerical polynomial; also,the integers e0, . . . , ed−1 are uniquely determined by P , and we denotethem by e0(P ), . . . , ed−1(P ).

Also, given c := (c0, . . . , cd−1) ∈ Zd, we shall denote by pc =

17.1 The characteristic function 367

p(c0,...,cd−1) the polynomial in Q[X] given by

pc(X) =d−1∑i=0

(−1)ici(X + d− i− 1

d− i− 1

).

Note that, when d > 1, we have

pc(X)− pc(X − 1)

=

d−1∑i=0

(−1)ici((

X + d− i− 1

d− i− 1

)−((X − 1) + d− i− 1

d− i− 1

))

=

d−2∑i=0

(−1)ici(X + d− i− 2

d− i− 2

)= p(c0,...,cd−2)(X).

(vi) We shall say that f is of reverse polynomial type (of degree d) if and onlyif there exists a polynomial p ∈ Q[X] (of degree d) such that f(n) =

p(n) for all n� 0 (that is, for all n less than some fixed constant integern′0). Thus f is of reverse polynomial type of degree d if and only if thefunction g : Z −→ Z defined by g(n) = f(−n) for all n ∈ Z is ofpolynomial type of degree d in the sense of (i).

17.1.2 �Exercise. Let 0 = P ∈ Q[X] have degP = d − 1. Assume thatP takes integer values at d consecutive integers. Show that P (n) ∈ Z for alln ∈ Z.

17.1.3 �Exercise. Suppose that the function f : Z −→ Z is such that thereexists Q ∈ Q[X] for which Δf(n) = Q(n) for all n ∈ Z. By using, forvarious values of the integer n0, the fact that

f(n) = f(n0) +∑n−1

i=n0Q(i) for all n ≥ n0,

show that there exists P ∈ Q[X] such that f(n) = P (n) for all n ∈ Z.

17.1.4 Further Notation and Terminology. Assume that R =⊕

n∈N0Rn

is positively graded and homogeneous and such thatR0 is an Artinian ring, andlet M =

⊕n∈ZMn be a finitely generated graded R-module. It follows from

16.1.5(i) that theR0-moduleHiR+

(M)n has finite length, for all i ∈ N0 and alln ∈ Z: we denote this length by hiM (n). Similarly, it follows from 16.1.6(ii)that the R0-module DR+(M)n has finite length, for all n ∈ Z. Let i ∈ N.For n ∈ Z, we denote the n-th component of RiDR+

(M) by RiDR+(M)n;

by 13.5.7(iii), this R0-module is isomorphic to Hi+1R+

(M)n, and so has finitelength.

We denote �R0(RiDR+(M)n) by diM (n), for all i ∈ N0 and all n ∈ Z. Notethat diM (n) = hi+1

M (n) for i > 0, and that this is zero for i ≥ max {1, dimM}.


It therefore makes sense for us to define the characteristic function χM :

Z −→ Z of M by setting χM (n) =∑

i∈N0(−1)idiM (n) for all n ∈ Z. Note

that, for all n ∈ Z, we have d0M (n) = �R0(Mn) + h1M (n)− h0M (n) by 13.5.4,so that, for any integer d ≥ dimM ,

χM (n) =d∑

i=0

(−1)idiM (n) = d0M (n)−d∑

i=2

(−1)ihiM (n)

= �R0(Mn)−d∑

i=0

(−1)ihiM (n).

Thus it follows from 16.1.5(ii) that χM (n) = �R0(Mn) ≥ 0 for all n � 0.Note also that, by 14.1.10(ii), we have χM(t)(n) = χM (t+n) for all t, n ∈ Z.



homogeneous and such that R0 is an Artinian ring. Let

0 −→ L −→M −→ N −→ 0

be an exact sequence of finitely generated graded R-modules and homoge-neous homomorphisms. Show that, for each n ∈ Z, we have

χM (n) = χL(n) + χN (n).

17.1.6 Remark. Assume that R =⊕


mogeneous, and such that R0 is Artinian. Let m(1)0 , . . . ,m

(t)0 be the maxi-

mal ideals of R0. Let M be a finitely generated graded R-module. For eachj = 1, . . . , t, set R′(j) := R⊗R0 (R0)m(j)

0and M ′(j) :=M ⊗R0 (R0)m(j)

0.

(i) For each R0-module N , the natural R0-homomorphism ωN : N −→⊕tj=1Nm

(j)0

(for which ωN (y) = (y/1, . . . , y/1) for all y ∈ N ) is anisomorphism; therefore, when N is finitely generated, its length �R0(N)

satisfies �R0(N) =∑t

j=1 �(R0)m

(j)0

(N

m(j)0

).

(ii) Since m(1)0 +R+, . . . ,m

(t)0 +R+ are the only *maximal graded ideals of

R and, for j = 1, . . . , t, the ring R′(j) is isomorphic, as a graded ring, tothe homogeneous localization (see 14.1.1) R(

m(j)0 +R+

), it follows that

dimM = max{dimR′(j) M ′(j) : j = 1, . . . , t

}.

(iii) Fix an integer j between 1 and t. Note that, by 16.2.3, the ring R′(j) ispositively graded and homogeneous, and has 0-th component isomorphic


to the Artinian local ring (R0)m(j)0

; moreover,R+R′(j) = R

′(j)+ and there

is an (R0)m(j)0

-isomorphism(Hi

R+(M)n

)m

(j)0

∼= Hi

R′(j)+

(M ′(j))n for each i ∈ N0 and each n ∈ Z.

Also, by 16.2.2(vi), there is an (R0)m(j)0

-isomorphism(DR+(M)n

)m

(j)0

∼= DR

′(j)+

(M ′(j))n for each n ∈ Z.

(iv) It follows from parts (i) and (iii) that

regl(M) = max{regl(M ′(j)) : j = 1, . . . , t

}for all l ∈ N0,

that hiM (n) =∑t

j=1 hiM ′(j)(n) for all i ∈ N0 and n ∈ Z, and that (with

the notation of 17.1.4)

d0M (n) =

t∑j=1

d0M ′(j)(n) and χM (n) =

t∑j=1

χM ′(j)(n) for all n ∈ Z.



mogeneous, and such that R0 is Artinian; let M be a non-zero finitely gen-erated graded R-module. Then there is a (necessarily uniquely determined)polynomial PM ∈ Q[X] of degree dimM − 1 such that PM (n) = χM (n) forall n ∈ Z.

Proof. It follows from 17.1.6 (and the fact that χM (n) ≥ 0 for all n � 0)that it is sufficient for us to prove this result under the additional assumptionthat the Artinian ring R0 is local. We make this assumption in what follows,and we let m0 be the maximal ideal of R0.

We now use 16.2.4 and 16.2.2(vi) to reduce to the case where the residuefield of R0 is infinite. We therefore assume that R0/m0 is infinite for the re-mainder of this proof.

We now use induction on dimM . Consider first the case when dimM = 0.Then AssM = {m0 ⊕R+}, so that there exists t ∈ N such that (R+)

tM = 0.In this case, M = ΓR+(M) and so Hi

R+(M) = 0 for all i ∈ N, by 2.1.7(ii),

and DR+(M) = 0 by 2.2.10(i). Thus χM (n) = 0 for all n ∈ Z, and our claimis proved in this case.

Now suppose that dimM > 0 and our desired result has been proved forall non-zero, finitely generated graded R-modules of smaller dimension. SinceΓR+(M) is a graded submodule of M , it follows from 2.1.7(iii) that there is a(homogeneous) isomorphism

HiR+

(M) ∼= HiR+

(M/ΓR+(M)) for each i ∈ N,


and from 13.3.14 and 2.2.10(ii) that there is a (homogeneous) isomorphismDR+(M) ∼= DR+(M/ΓR+(M)). Hence χM = χM/ΓR+

(M). Since ΓR+(M)

is annihilated by some power of R+, it follows that

Supp(ΓR+(M)) ⊆ {m0 ⊕R+} ,

so that dimM = dim(M/ΓR+(M)). Hence, for the purpose of this induc-

tive step, we can replace M by M/ΓR+(M) and so (in view of 2.1.2, 2.1.1and 16.1.4(ii)) assume that M is R+-torsion-free and that R1 contains a non-zerodivisor a on M . This we do.

Application of 17.1.5 to the exact sequence

0 −→M(−1) a−→M −→M/aM −→ 0

of graded R-modules and homogeneous homomorphisms, together with theobservation that χM(−1)(n) = χM (n− 1) for all n ∈ Z, yields that

χM (n)− χM (n− 1) = χM/aM (n) for all n ∈ Z.

Now dimM/aM = dimM−1 (by 14.1.14), and so it follows from the induc-tive hypothesis that there is a polynomial PM/aM ∈ Q[X] of degree dimM−2such that PM/aM (n) = χM/aM (n) for all n ∈ Z. Hence, by 17.1.3, there ex-ists a polynomial PM ∈ Q[X] such that PM (n) = χM (n) for all n ∈ Z.Furthermore, if PM/aM = 0, then PM must have degree dimM − 1. How-ever, if PM/aM = 0, then the above argument shows only that degPM ≤ 0,and we must show that χM (n) = 0 for some n ∈ Z in order to complete theinductive step.

The assumption that PM/aM = 0, in conjunction with the inductive hypoth-esis, implies that dimM = 1. Hence Hi

R+(M) = 0 for all i > 1, and so

χM (n) = d0M (n) for all n ∈ Z. However, the fact that ΓR+(M) = 0 ensuresthat the map ηM : M −→ DR+(M) of 2.2.6(i) is a monomorphism, so thatDR+(M) = 0 and d0M (n) = 0 for some n ∈ Z.



mogeneous, and such that R0 is Artinian; let M =⊕

n∈ZMn be a non-zerofinitely generated graded R-module of dimension d.

It was pointed out in 17.1.4 that χM (n) = �R0(Mn) for all n � 0. ThusHilbert’s Theorem (see [7, Theorem 4.1.3]), that the function

�R0(M( • )) : Z −→ Z

is of polynomial type of degree d − 1, is a corollary of Theorem 17.1.7, andthe polynomial PM of that theorem is just the Hilbert polynomial of M of [7,4.1.5].


With the notation of 17.1.1(v), for each j = 0, . . . , d− 1, we set ej(M) :=

ej(PM ) and refer to this as the j-th Hilbert coefficient of M . Note that, in theterminology of [7, 4.1.5], the integer e0(M) is the multiplicity of M if d > 0.

To sum up, we can write

PM (X) =d−1∑i=0

(−1)iei(M)

(X + d− i− 1

d− i− 1

)and

χM (n) =d−1∑i=0

(−1)iei(M)

(n+ d− i− 1

d− i− 1

)for all n ∈ Z.



mogeneous, and such that R0 is Artinian; let M be a non-zero finitely gen-erated graded R-module of dimension d. Suppose that a ∈ R1 is a non-zerodivisor on M . Show that

(i) PM/aM (X) = PM (X)− PM (X − 1); and(ii) if d > 1, then, with the notation of 17.1.1(v),

PM/aM (X) = p(e0(M),...,ed−2(M))(X),

and ei(M/aM) = ei(M) for all i = 0, . . . , d− 2.

17.1.10 Definition and �Exercise. Assume that R =⊕

n∈N0Rn is pos-

itively graded and homogeneous, and such that R0 is Artinian; let M be afinitely generated graded R-module. The postulation number pstln(M) of Mis defined by pstln(M) = sup{n ∈ Z : �R0(Mn) = PM (n)}. Thus pstln(M)

is an integer or −∞. Show that

(i) pstln(M) ≤ reg(M) ≤ max{reg1(M), pstln(M)}, and(ii) if dimM ≤ 0, then pstln(M) = reg(M) = end(M).

17.1.11 Theorem and Definition. Assume R =⊕

n∈N0Rn is positively

graded and homogeneous, and such that R0 is Artinian; let M be a finitelygenerated graded R-module. Let i ∈ N0. Then the function hiM : Z −→ N0 isof reverse polynomial type of degree less than i (in the sense of 17.1.1(vi)); inother words, there is a polynomial piM ∈ Q[X] of degree less than i such that

�R0(HiR+

(M)n) = piM (n) for all n� 0.

The function hiM is called the i-th cohomological Hilbert function of M ,while the (uniquely determined) polynomial piM is called the i-th cohomologi-cal Hilbert polynomial of M .


Proof. By 17.1.6, it is sufficient for us to prove the result under the additionalhypothesis that R0 is local. We therefore assume for the remainder of thisproof that R0 is an Artinian local ring, with maximal ideal m0, say. With thisassumption, R is *local, and

√R+ = m0 + R+ =: m, the unique *maximal

ideal of R.Since (R0,m0) is a complete local ring, we can use Cohen’s Structure The-

orem for such rings (see [50, Theorem 29.4(ii)], for example) in conjunctionwith standard facts about the structure of positively graded homogeneous com-mutative Noetherian rings (see [7, Proposition 1.5.4]) to see that there is aGorenstein graded *local commutative Noetherian ring R′ and a surjective ho-mogeneous ring homomorphism f : R′ −→ R. Let d′ := dimR′.

We are going to use the Graded Local Duality Theorem 14.4.1. Let *Ddenote the functor *HomR( • , *E(R/m)) from *C(R) to itself. By 14.4.1,there exists a′ ∈ Z such that there is a natural transformation

ψi : HiR+−→ *D(*Extd

′−iR′ ( • , R′(a′)))

of covariant functors from *C(R) to *C(R) which is such that ψiM ′ is a (nec-

essarily homogeneous) isomorphism for all i ∈ N0 whenever M ′ is a finitelygenerated graded R-module.

Let N = *Extd′−i

R′ (M,R′(a′)), a finitely generated graded R-module. Wenow use Graded Matlis Duality 14.4.2. We use the notation ( • )∨ of 14.4.2 todenote the functor *HomR0( • , ER0(R0/m0)) from *C(R) to itself. It followsfrom 14.4.2 that, for each n ∈ Z (and with an obvious notation), there areR0-isomorphisms

HiR+

(M)n ∼= (*D(N))n ∼= (N∨)n ∼= (*HomR0(N,ER0(R0/m0)))n

= HomR0(N−n, ER0(R0/m0)),

so that hiM (n) = �R0(HomR0(N−n, ER0(R0/m0))) = �R0(N−n) by 10.2.13.Thus, in order to prove that hiM is of reverse polynomial type of degree lessthan i, it is now sufficient for us to prove that the function �R0(N( • )) : Z −→N0 is of polynomial type of degree less than i; hence, by Hilbert’s Theorem(see 17.1.8 above and [7, Theorem 4.1.3]), it is enough for us to prove thatdimN ≤ i.

To do this, let p′ ∈ SuppR′ N . Then Extd′−i

R′p′

(Mp′ , R′p′) = 0, so that, since

inj dimR′p′ R

′p′ = ht p′, we must have ht p′ ≥ d′−i. Therefore dimR′/p′ ≤ i,

so that dimRN ≤ i.

17.1.12 Exercise. Let the notation be as in 15.2.7 and 15.2.8.

(i) Compute the Hilbert polynomial, all Hilbert coefficients, the postula-

17.2 The significance of reg2 373

tion number, all cohomological Hilbert functions and the Castelnuovo–Mumford regularity for each of the three rings A, B and C.

(ii) Conclude that the vanishing ideal IP6(K)(VA) ⊂ K[X0, . . . , X6] of VAcan be generated by homogeneous polynomials of degree 2, and that thevanishing ideals IP5(K)(VB), IP5(K)(VC) ⊂ K[X0, . . . , X5] of VB andVC can be generated by homogeneous polynomials of degrees 2 and 3.(You might find 16.3.2 and 16.3.3 helpful.)

17.1.13 �Exercise. Let the situation be as in 17.1.11, and let M be a non-zero finitely generated graded R-module of dimension d. Show that the d-thcohomological Hilbert polynomial pdM of M has degree exactly d− 1.



mogeneous, and such that R0 is Artinian; let M be a non-zero finitely gener-ated graded R-module of positive dimension. This exercise involves the finite-ness dimension fR+(M) of M relative to R+ of 9.1.3, and, for r ∈ N ands ∈ Z, the notation ( • )(r,s) : *C(R) −→ *C(R(r)) of 13.5.9 for the (r, s)-thVeronesean functor.

(i) Show that fR+(M) is finite.(ii) Show that fR+

(M) = max{gradeM(r,s)((R(r))+) : r ∈ N, s ∈ Z

}.

17.2 The significance of reg2

In this section, we shall show that, if R is positively graded and homoge-neous, M is a finitely generated graded R-module, and t ∈ Z is such thatt ≥ reg2(M), then DR+(M) and H1

R+(M) can be generated by homoge-

neous elements of degrees not exceeding t. We shall also start to prepare theground for a result (to be presented in the next section) that bounds reg2(M)

in certain circumstances.As promised in the introduction to this chapter, we now give another hint

about the significance of the invariant reg2. Let K be an algebraically closedfield, let r ∈ N and let V ⊂ Pr(K) be a projective variety. Let p denotethe vanishing ideal IPr(K)(V ) of V , so that p is the (non-zero) graded primeideal IAr+1(K)(Cone(V )) of K[X0, . . . , Xr] = O(Ar+1(K)). In 16.3.3, weintroduced the Castelnuovo–Mumford regularity reg(V ) of V , and pointed outthat reg(V ) = reg(p) = reg1(p) = reg2(p) and that p can be generated byhomogeneous polynomials of degrees not exceeding reg(V ). Since p can beused to define V , in the sense that V = VPr(K)(p), the regularity reg2(p) of pat and above level 2 provides an upper bound on the degrees of homogeneous


polynomials needed to define V . This is another reason why one could beinterested in upper bounds on reg2(p).



homogeneous, and let M =⊕

n∈ZMn be a finitely generated graded R-module.

If reg2(M) = −∞, then each of DR+(M) and H1R+

(M) can be generatedby homogeneous elements of degrees not exceeding reg2(M).

If reg2(M) = −∞, then, for each t ∈ Z, both DR+(M) and H1R+

(M) canbe generated by homogeneous elements of degrees not exceeding t.

Proof. Since reg2(M) = reg2(M/ΓR+(M)) (by 16.2.10(ii)) and there arehomogeneous isomorphisms

DR+(M)∼=−→ DR+(M/ΓR+(M)) and H1

R+(M)

∼=−→ H1R+

(M/ΓR+(M))

(by 13.3.14, 2.2.10(ii) and 2.1.7(iii)), we can replace M by M/ΓR+(M) andso (in view of 2.1.2) assume that M is R+-torsion-free. We shall make thissimplification.

Since M is finitely generated, there exists h ∈ Z such that Mn = 0 for alln < h. If reg2(M) = −∞, choose t ∈ Z such that t ≤ min

{h, reg2(M)

};

if reg2(M) = −∞, let t be any integer such that t ≤ h. With the nota-tion of 16.1.1, let N := DR+(M)≥t. Note that N is a finitely generatedR-module, by 16.1.6(i). Also, by choice of t, the homogeneous monomor-phism ηM : M −→ DR+(M) satisfies Im ηM ⊆ N ; thus N/ Im ηM ⊆DR+(M)/ Im ηM ∼= H1

R+(M), and so N/ Im ηM is finitely generated and

R+-torsion. It therefore follows from 2.1.7(i) that there is a (homogeneous)isomorphism Hi

R+(M)

∼=−→ HiR+

(N) for each i > 1, so that reg2(M) =

reg2(N).Let C =

⊕n∈Z Cn be the graded R-module DR+(M)/N . Since

HiR+

(DR+(M)) = 0 for i = 0, 1

by 2.2.10(iv), it follows from the exact sequence

0 −→ N −→ DR+(M) −→ C −→ 0

(in the category *C(R)) that H0R+

(N) = 0 and there is a homogeneous iso-

morphism ΓR+(C)∼=−→ H1

R+(N). Now ΓR+(C) is a graded submodule of C,

and Cn = 0 for all n ≥ t. Therefore end(H1R+

(N)) + 1 ≤ t − 1 + 1 = t. Inthe case where reg2(M) = −∞, it follows that

reg(N) = reg2(N) = reg2(M) ≥ t,


whereas, in the case where reg2(M) = −∞, we have reg(N) ≤ t.It follows from 16.3.1 that N = DR+(M)≥t, if non-zero, can be gener-

ated by homogeneous elements of degrees not exceeding max{t, reg2(M)},so that, since H1

R+(M) is a homomorphic image of DR+(M) by a homoge-

neous homomorphism, all the claims are proved.

17.2.2 �Exercise. Let the notation and hypotheses be as in 17.2.1. Let t ∈ Zwith t ≥ reg2(M). Let p ∈ Proj(R) := * Spec(R) \ Var(R+). Recall that• (p) denotes the homogeneous localization functor with respect to p.

Show that the natural homomorphism η := ηM :M −→ DR+(M) induces

a homogeneous isomorphism η(p) : M(p)

∼=−→ DR+(M)(p) of graded R(p)-modules.

Let φ(p) : DR+(M) −→ DR+(M)(p) be the natural homogeneous ho-momorphism of graded R-modules, and consider the composition β(p) =

η−1(p) ◦ φ(p) : DR+

(M) −→M(p). As usual, the t-th component of β(p) will bedenoted by (β(p))t. Let S be a generating set for the R0-module DR+(M)t.Use 17.2.1 and the fact that R is homogeneous to show that

M(p) =∑

m∈SR(p)β(p)(m)

and (M(t)(p))0 = (M(p))t =∑

m∈S(R(p))0(β(p))t(m).

Note. Exercise 17.2.2 will be used in an application to sheaf cohomology inTheorem 20.4.13; the aim of that theorem is to establish a (generalization of a)result of Serre that certain sheaves are generated by their global sections.



homogeneous, and let M be a finitely generated graded R-module. Supposethat a ∈ R1 is a non-zerodivisor on M . Then

(i) for all integers m ≥ reg2(M/aM)− 1, the multiplication map

H2R+

(M)m−1a−→ H2

R+(M)m

is surjective; and(ii) for all integers m ≥ reg2(M/aM) such that H2

R+(M)m−1 = 0, the

multiplication map H2R+

(M)m−1a−→ H2

R+(M)m is not injective.

Proof. (i) By 14.1.10(ii), the exact sequence

0 −→Ma−→M(1) −→ (M/aM)(1) −→ 0

of graded R-modules and homogeneous homomorphisms induces an exact se-quence H2

R+(M)m−1

a−→ H2R+

(M)m −→ H2R+

(M/aM)m of R0-modules,for all m ∈ Z, and H2

R+(M/aM)m = 0 when m+ 1 ≥ reg2(M/aM).


(ii) For each m ∈ Z, let

βm : R1 ⊗R0 H1R+

(M/aM)m −→ H1R+

(M/aM)m+1

be the R0-homomorphism for which βm(r1 ⊗ z) = r1z for all r1 ∈ R1 andz ∈ H1

R+(M/aM)m, and let γm : R1 ⊗R0 H

1R+

(M)m −→ H1R+

(M)m+1 bedefined similarly.

Let m0 ∈ Z be such that H1R+

(M/aM) can be generated by homogeneouselements of degrees not exceeding m0: by Proposition 17.2.1, we can takem0 = reg2(M/aM) if reg2(M/aM) = −∞, and we can take m0 to be anarbitrary integer if reg2(M/aM) = −∞. Since R is homogeneous, it fol-lows that, for all m ≥ m0, each element y ∈ H1

R+(M/aM)m+1 can be ex-

pressed in the form∑t

i=1 aizi for suitable a1, . . . , at ∈ R1 and z1, . . . , zt ∈H1

R+(M/aM)m. In other words, βm is surjective for all m ≥ m0.

Let π :M −→M/aM be the canonical epimorphism, and, for eachm ∈ Z,let αm : H1

R+(M)m −→ H1

R+(M/aM)m be the m-th component of the

homogeneous homomorphism H1R+

(π). Observe that the diagram

R1 ⊗R0H1

R+(M)m

γm

H1R+

(M)m+1�

IdR1 ⊗αm αm+1

R1 ⊗R0 H1R+

(M/aM)mβm

H1R+

(M/aM)m+1��

commutes. It follows that, if, for somem ≥ m0, we know that αm is surjective,then αm+1 is surjective too.

Consider an m ≥ m0 and suppose that H2R+

(M)m−1a−→ H2

R+(M)m is

injective. By 14.1.10(ii), the exact sequence

0 −→M(−1) a−→M −→M/aM −→ 0


H1R+

(M)nαn−→ H1

R+(M/aM)n −→ H2

R+(M)n−1

a−→ H2R+

(M)n

of R0-modules for all n ∈ Z. This exact sequence (in the case where n = m)shows that αm is surjective, and so it follows, as explained in the immedi-ately preceding paragraph, that αm+1 is surjective and αn is surjective forall n ≥ m. It follows from the above exact sequence that H2

R+(M)n−1

a−→H2

R+(M)n is injective for all n ≥ m. It therefore follows from 16.1.5(ii) that

H2R+

(M)m−1 = 0.


We point out that, in the situation and with the notation of 17.2.3, and withthe additional assumption that R0 is Artinian, the proposition shows that, as nincreases beyond reg2(M/aM)− 1, the integers h2M (n) decrease strictly to 0

and then remain at 0.The following corollary is a consequence of Propositions 17.2.1 and 17.2.3,

and it provides the key for some crucial arguments later in this chapter.

17.2.4 Corollary. Assume that R =⊕


homogeneous, and that R0 is Artinian; let M be a finitely generated gradedR-module. Suppose that a ∈ R1 is a non-zerodivisor on M .

Let n0, q ∈ Z with q ≥ max{n0, reg

2(M/aM)− 1}

; for each n ≥ n0, set

s(n) := h2M (n0) +n∑

m=n0+1

h2M/aM (m).

Then

(i) h2M (n) ≤ s(n) for all n ≥ n0;(ii) s(n) = s(q) for all n ≥ q, and s(q) = s(q − 1) if q > n0; and

(iii) h2M (n) ≤ max {0, s(q) + q − n} for all n > q.

Proof. (i) By 14.1.10(ii), the exact sequence

0 −→Ma−→M(1) −→ (M/aM)(1) −→ 0

of graded R-modules and homogeneous homomorphisms induces, for eachm ∈ Z, an exact sequence

H2R+

(M)m−1a−→ H2

R+(M)m −→ H2

R+(M/aM)m

of R0-modules, from which we deduce that

h2M (m) = �R0(H2R+

(M)m) ≤ �R0(H2R+

(M)m−1) + �R0(H2R+

(M/aM)m)

= h2M (m− 1) + h2M/aM (m),

and the claim follows easily from this.(ii) For m > q ≥ reg2(M/aM)− 1, we have

q ≥ end(H2R+

(M/aM)) + 1,

so that H2R+

(M/aM)m = H2R+

(M/aM)q = 0. Hence

h2M/aM (m) = h2M/aM (q) = 0.

Therefore s(n) = s(q) for all n ≥ q, and s(q) = s(q − 1) if q > n0.(iii) For m ≥ q + 1 ≥ reg2(M/aM), we have, by Proposition 17.2.3,

that the multiplication map H2R+

(M)m−1a−→ H2

R+(M)m is surjective, and


is not injective unless H2R+

(M)m−1 = 0; therefore, if h2M (m − 1) = 0, thenh2M (m) ≤ h2M (m−1)−1. Thus h2M (m) ≤ max

{0, h2M (m− 1)− 1

}for all

m ≥ q + 1. Repeated use of this then shows that

h2M (n) ≤ max{0, h2M (q)− (n− q)

}for all n ≥ q + 1,

and the claim follows from part (i).

17.3 Bounds on reg2 in terms of Hilbert coefficients


n∈N0Rn is graded. Let D be a sub-

class of the class of all graded R-modules. By a numerical invariant for R-modules in D (or on D) we mean an assignment μ which, to each R-moduleM in D, assigns μ(M) ∈ Z ∪ {−∞}, and which is such that μ(M) = μ(N)

whenever M and N are modules in D for which there is a homogeneous iso-morphism M ∼= N . We say that such a numerical invariant μ is finite if andonly if μ takes only finite values.

Let μ1, . . . , μs, ρ be numerical invariants for R-modules in D, such thatμ1, . . . , μs are finite. We say that μ1, . . . , μs form a bounding system for ρ(for R-modules in D) (or on D) if and only if there is a function B : Zs −→Z such that ρ(M) ≤ B(μ1(M), . . . , μs(M)) for each R-module M in D.Furthermore, we say that μ1, . . . , μs form a minimal bounding system for ρ(for R-modules in D) precisely when they form a bounding system for ρ on Dbut no s− 1 of μ1, . . . , μs form a bounding system for ρ on D.

The main result of this section is a generalization of a ring-theoretic formu-lation of a classical result of D. Mumford on sheaves of ideals on projectivespaces. A consequence is that, over the polynomial ring S = K[X0, . . . , Xr]

where K is an algebraically closed field, e0( • ), . . . , er( • ) form a boundingsystem for reg2 for non-zero graded ideals of S.

The following exercise shows that, in general, for an Artinian local ring R0,the Hilbert coefficients do not form a bounding system for reg2 on the class ofall finitely generated graded R0[X1, . . . , Xn]-modules.

17.3.2 Exercise. Suppose that R = R0[X1, X2] is a polynomial ring in 2

indeterminates X1, X2 over an Artinian ring R0.For each t ∈ N0, set M (t) := R(t) ⊕ R(−t), and calculate e0(M (t)),

e1(M(t)), reg2(M (t)). Conclude that the numerical invariants e0( • ), e1( • )

do not form a bounding system for reg2 on the class of all finitely generatedgraded R-modules.

17.3 Bounds on reg2 in terms of Hilbert coefficients 379

17.3.3 Notation and Remark. For (n, i) ∈ Z× N0, we set

(n

i

)+:=

⎧⎨⎩(n

i

)if n ≥ i,

0 if n < i.

This notation is helpful in the following situation. Suppose that

R = R0[X1, . . . , Xd]

is a polynomial ring in d (> 0) indeterminates X1, . . . , Xd over an Artinianring R0, and let a ∈ Z. Then, for all n ∈ Z, the n-th component R(a)n of thegraded R-module R(a) has length as an R0-module given by

�R0(R(a)n) = �R0

(R0)

(n+ a+ d− 1

d− 1

)+.

17.3.4 Lemma. Suppose that R = R0[X1, . . . , Xd] is a polynomial ring ind > 1 indeterminates X1, . . . , Xd over an Artinian ring R0. Let r ∈ N and leta1, . . . , ar ∈ Z. Let M be a graded submodule of

⊕ri=1R(ai), and let f be

an integer such that f ≥ reg2(M/XdM). Then

reg2(M) ≤ �R0(R0)

r∑i=1

(f + ai + d− 3

d− 1

)+− χM (f − 2) + f.

Proof. Note that Xd is a non-zerodivisor on M , because M is a submoduleof G :=

⊕ri=1R(ai). Let i, n ∈ Z be such that i > 2 and n > f − i.

Since f ≥ reg2(M/XdM) ≥ endHi−1R+

(M/XdM) + i − 1, it follows thatHi−1

R+(M/XdM)n+1 = 0. Therefore, the exact sequence

0 −→M(−1) Xd−→M −→M/XdM −→ 0

of graded R-modules and homogeneous homomorphisms induces an exactsequence

0 −→ HiR+

(M)nXd−→ Hi

R+(M)n+1

of R0-modules. Hence HiR+

(M)n = 0 for all i > 2 and n > f − i. Thus, byDefinition 17.1.4,

χM (f − 2) = d0M (f − 2)− h2M (f − 2),

so that h2M (f − 2) = d0M (f − 2)− χM (f − 2).We are now going to use Corollary 17.2.4 with n0 = f − 2 and q = f − 1.

Observe that, with the notation of that result, h2M/XdM(m) = 0 for all m >

n0 = f − 2, so that s(n) = h2M (f − 2) for all n ≥ n0. It follows from


17.2.4 that h2M (n) ≤ max{0, h2M (f − 2) + f − 1− n

}for all n ≥ f − 1.

Hence h2M (n) = 0 for all n > h2M (f − 2) + f − 2. As we know already thathiM (n) = 0 for all i > 2 and n > f − i, it follows that

reg2(M) ≤ h2M (f − 2) + f = d0M (f − 2)− χM (f − 2) + f.

As gradeGR+ = d > 1, it follows from 6.2.7, 2.2.6(i)(c) and 13.3.14 thatηR+ : G −→ DR+(G) is a homogeneous isomorphism. It therefore followsfrom the fact that the functor DR+ is left exact that there is a homogeneousmonomorphism DR+(M) −→ G; we thus deduce, on use of 17.3.3, that (withan obvious notation)

d0M (f − 2) ≤ �R0(Gf−2) = �R0(R0)r∑

i=1

(f − 2 + ai + d− 1

d− 1

)+.

The claim now follows.

Our next theorem is the main result of this chapter. We need one preliminarylemma.

17.3.5 Lemma. Suppose that R = R0[X1, . . . , Xd] is a polynomial ring ind (> 0) indeterminates X1, . . . , Xd over an Artinian local ring (R0,m0).

Let M be a non-zero finitely generated graded R-module, and supposethat there exists a ∈ R1 which is a non-zerodivisor on M . Then there existX ′1, . . . , X

′d ∈ R1 such that X ′d is a non-zerodivisor on M , the family (X ′i)

di=1

is algebraically independent over R0, and R = R0[X′1, . . . , X

′d].

Proof. We have a = b1X1 + · · ·+ bdXd for some b1, . . . , bd ∈ R0. Now m0

is nilpotent, and so there exists j ∈ {1, . . . , d} such that bj ∈ m0. Hence bj isa unit of R0, and so, after multiplication by b−1

j , we can, and do, assume thatbj = 1. Let c := a−Xj ∈ R1.

There is a homogeneousR0-algebra homomorphism φ : R −→ R for whichφ(Xj) = Xj + c = a and φ(Xi) = Xi for all i = 1, . . . , d with i = j.Similarly, there is a homogeneous R0-algebra homomorphism ψ : R −→ R

for which ψ(Xj) = Xj − c and ψ(Xi) = Xi for all i = 1, . . . , d with i = j.Since φ(c) = ψ(c) = c, it follows that φ and ψ are inverse isomorphisms. Ifwe now take X ′j = a = Xj + c and X ′i = Xi for all i = 1, . . . , d with i = j,then (X ′i)

di=1 is algebraically independent over R0 and R = R0[X

′1, . . . , X

′d];

since X ′j is a non-zerodivisor on M , we can now reorder the X ′i (if necessary)to complete the proof.

17.3.6 Theorem. SupposeR = R0[X1, . . . , Xd] is a polynomial ring in d (>0) indeterminates X1, . . . , Xd over an Artinian local ring (R0,m0).

17.3 Bounds on reg2 in terms of Hilbert coefficients 381

Let G be a non-zero finitely generated *free graded R-module. Then

e0( • ), . . . , ed−1( • )

form a bounding system for reg2 on the class of all non-zero graded submod-ules of G.

Proof. Write G =⊕r

i=1R(ai), where a := (a1, . . . , ar) ∈ Zr, and sets := min{−ai : 1 ≤ i ≤ r}. We start by defining, for each integer h ≥ 2, anumerical function F (h)

a : Zh −→ Z. The definition will be made by inductionon h.

First, for (e0, e1) ∈ Z2, define (with the notation of 17.1.1(v))

F (2)a (e0, e1) = s+ 1− p(e0,e1)(s− 1).

Now suppose that h > 2 and that the function F (h−1)a has already been defined.

Then, for e := (e0, . . . , eh−1) ∈ Zh and with e′ := (e0, . . . , eh−2), we writef = F

(h−1)a (e′) and set (again with the notation of 17.1.1(v))

F (h)a (e) := �R0(R0)

r∑i=1

(f + ai + h− 3

h− 1

)+− pe (f − 2) + f.

Consider first the case where d = 1, and let M be a non-zero graded sub-module of G. Then dimM = 1, so that Hi

R+(M) = 0 for all i > 1 and

reg2(M) = −∞. We therefore assume henceforth in this proof that d > 1.As in the proof of Theorem 17.1.7, we can use the ideas of Example 16.2.4 to

show that it is enough for us to prove the theorem under the additional assump-tion thatR0/m0 is infinite, and we shall make this assumption in what follows.We shall prove the result by induction on d. We shall make frequent use of thefact that, as each associated prime p of a free R-module has dimR/p = d,every non-zero submodule of a free R-module has dimension d.

Consider now the case where d = 2, and let M be a non-zero graded sub-module of G. Then dimM/X2M < 2, and so reg2(M/X2M) = −∞. UseLemma 17.3.4 with the choice f = s+1: since f+ai+d−3 = s+ai+d−2 ≤d− 2 < d− 1 for each i = 1, . . . , r, we obtain that

reg2(M) ≤ �R0(R0)r∑

i=1

(s+ ai + d− 2

d− 1

)+− χM (s− 1) + s+ 1

= s+ 1− PM (s− 1) = s+ 1− p(e0(M),e1(M))(s− 1)

= F (2)a (e0(M), e1(M)).

Therefore e0( • ), e1( • ) form a bounding system for reg2 on the class of allnon-zero graded submodules of G.


Now suppose d > 2 and we have proved, for R := R0[X1, . . . , Xd−1]

and for every non-zero graded submodule N of⊕r

i=1R(ai), that reg2(N)

≤ F(d−1)a (e0(N), . . . , ed−2(N)).

Now let M be a non-zero graded submodule of G. As gradeGR+ = d > 1,it follows from 6.2.7, 2.2.6(i)(c) and 13.3.14 that ηR+ : G −→ DR+(G)

is a homogeneous isomorphism. It therefore follows from the fact that thefunctor DR+

is left exact that there is a homogeneous monomorphism ε :

DR+(M) −→ G: let L := ε(DR+(M)), a d-dimensional graded submodule

of G. Since there is a homogeneous isomorphism L∼=−→ DR+(M), it fol-

lows from 13.3.14 and 2.2.10(iii) that there is a homogeneous isomorphismDR+(L)

∼=−→ DR+(M), and from 13.4.3, 13.4.4 and 2.2.10(v) that there are

homogeneous isomorphisms HiR+

(L)∼=−→ Hi

R+(M) for all i > 1. Hence

χL = χM , so that PL = PM and ei(L) = ei(M) for all i = 0, . . . , d− 1; alsoreg2(L) = reg2(M). Now H1

R+(L) ∼= H1

R+(DR+(M)) = 0 by 2.2.10(iv),

so that, since H0R+

(G) = 0, we have H0R+

(G/L) = 0. It is therefore suf-ficient for us to complete this inductive step under the additional assumptionthat H0

R+(G/M) = 0, and so we shall make this assumption in the remainder

of this proof. In view of 2.1.1 and 16.1.4(ii), this means thatR1 contains a non-zerodivisor on G/M , and it follows from Lemma 17.3.5 that we can assumewithout loss of generality that Xd is a non-zerodivisor on G/M . This we do.

Set R := R0[X1, . . . , Xd−1], G := G/XdG, and M := M/XdM . We canviewG andM asR-modules by means of the inclusion homomorphismR −→R, and the natural mapM −→ G is anR-monomorphism becauseXd is a non-zerodivisor on G/M . Note also that there is a homogeneous R-isomorphismG

∼=−→⊕r

i=1R(ai). It therefore follows from the inductive hypothesis that

reg2(M) ≤ F (d−1)a (e0(M� R), . . . , ed−2(M� R)).

A straightforward adaptation of 4.2.2 to the graded case, with use of theGraded Independence Theorem 14.1.7 instead of the Independence Theorem4.2.1, will show that there are homogeneous R-isomorphisms

HiR+

(M� R)∼=−→ Hi

R+(M)� R for all i ∈ N0,

and a similar argument based on 14.1.4 will show that there is a homogeneousR-isomorphism

DR+(M� R)

∼=−→ DR+(M)� R.

It follows that reg2(M� R) = reg2(M/XdM) and, in view of 17.1.9, that

ei(M� R) = ei(M/XdM) = ei(M) for all i = 0, . . . , d− 2.

17.4 Bounds on reg1 and reg0 383

Set e := (e0(M), . . . , ed−1(M)) ∈ Zd and e′ := (e0(M), . . . , ed−2(M)).Thus reg2(M/XdM) ≤ F

(d−1)a (e′). Recall that χM (n) = PM (n) = pe(n)

for all n ∈ Z. We can now deduce from Lemma 17.3.4 with the choice f =

F(d−1)a (e′) that

reg2(M) ≤ �R0(R0)

r∑i=1

(f + ai + d− 3

d− 1

)+− χM (f − 2) + f

= �R0(R0)

r∑i=1

(f + ai + d− 3

d− 1

)+− pe(f − 2) + f

= F (d)a (e0(M), . . . , ed−1(M)).

This completes the inductive step.

Theorem 17.3.6 is based on work of D. Mumford: see [54, p. 101]. Inour notation (of 17.3.6), Mumford’s work is concerned with the case whereG = R; thus Mumford showed that e0( • ), . . . , ed−1( • ) form a boundingsystem for reg2 on the class of all non-zero graded ideals of R. In fact, Mum-ford showed that there exists a rational polynomial q in d indeterminates suchthat, for every non-zero graded ideal b of R, reg2(b) is bounded above byq(e0(b), . . . , ed−1(b)).

17.4 Bounds on reg1 and reg0

In this section we show that the bounding result for reg2 obtained in the lastsection leads to bounding results for reg1 and reg0 in which the Hilbert coeffi-cients play an important role.



mogeneous, and such that R0 is Artinian and local, with maximal ideal m0;assume also that R is generated as R0-algebra by d homogeneous elements ofdegree 1, where d > 1. Let r, t ∈ N and a := (a1, . . . , ar) ∈ Zr be fixed. LetD denote the class of all graded R-modules M of dimension t which can bewritten in the form M =

∑rj=1Rmj with mj ∈M−aj for all j ∈ {1, . . . , r}.

Then e0( • ), . . . , et−1( • ) form a bounding system for reg1 on D.

Proof. We begin by defining a function that we shall use to bound reg1. LetE denote the class of non-zero graded submodules of G =

⊕ri=1R(ai). By

17.3.6, there is a function B : Zd −→ Z such that

reg2(L) ≤ B(e0(L), . . . , ed−1(L))


for each R-module L in E . Let c := (c0, . . . , ct−1) ∈ Zt and let pc be thepolynomial of degree t− 1 in Q[X] given by

pc(X) =t−1∑i=0

(−1)ici(X + t− i− 1

t− i− 1

).

(See 17.1.1(v).) Also, define p(a)c ∈ Q[X] by

p(a)c (X) :=r∑

j=1

�R0(R0)

(X + aj + d− 1

d− 1

)− pc(X).

With the notation of 17.1.1(v), if deg(p(a)c ) = d− 1, set

C(d)a (c) := max{B(e0(p

(a)c ), . . . , ed−1(p

(a)c ))− 1,−a1, . . . ,−ar},

but let C(d)a (c) := max{−a1, . . . ,−ar} if deg(p(a)c ) < d− 1.

The hypotheses ensure that there is a surjective homogeneous homomor-phism φ : S := R0[X1, . . . , Xd] −→ R of graded rings. On use of the GradedIndependence Theorem 14.1.7, we see that we can replace R by S; thus weassume that R = R0[X1, . . . , Xd]. Let M be a module in D; then there is anexact sequence 0 −→ N −→ G −→ M −→ 0 in the category *C(R). Writee := (e0(M), . . . , et−1(M)). The Hilbert polynomials PN and PM satisfy

PN (X) = PG(X)− PM (X) =

r∑j=1

�R0(R0)

(X + aj + d− 1

d− 1

)− pe(X)

= p(a)e (X).

If N = 0, then deg(p(a)e ) = −1 < d− 1, and

reg1(M) = reg1(⊕r

j=1R(aj)) ≤ reg(⊕r

j=1R(aj))

= max{−a1, . . . ,−ar} = C(d)a (e).

Now consider the case where N = 0. Then dimN = d, and so deg(p(a)e ) =

d − 1. Notice that N lies in the class E . Therefore, in view of 17.3.6 and16.2.15(iv), we have

reg1(M) ≤ max{reg2(N)− 1, reg(⊕r

j=1R(aj))}≤ max{B(e0(N), . . . , ed−1(N))− 1, reg(

⊕rj=1R(aj))}

= max{B(e0(p(a)e ), . . . , ed−1(p

(a)e ))− 1,−a1, . . . ,−ar}

= C(d)a (e).


Observe that

ei(p(a)e )

=

⎧⎨⎩ei(⊕r

j=1R(aj))

if 0 ≤ i ≤ d− t− 1,

ei

(⊕rj=1R(aj)

)− (−1)d−tei−(d−t)(M) if d− t ≤ i ≤ d− 1,

and that each such ei(⊕r

j=1R(aj))

depends only on �R0(R0), a1, . . . , ar and

d. Thus e0( • ), . . . , et−1( • ) form a bounding system for reg1 on D.

17.4.2 Corollary. Let R =⊕

n∈N0Rn be as in 17.4.1. Let W =

⊕n∈ZWn

be a finitely generated graded R-module and let 0 = P ∈ Q[x]. Then there isan integer G such that, for each homogeneous R-homomorphism f : W −→M of finitely generated gradedR-modules that is surjective in all large degreesand is such that PM = P , we have reg1(M) ≤ G.

Proof Choose r ∈ N and a := (a1, . . . , ar) ∈ Zr such that W =∑r

j=1Rwj

with wj ∈ W−aj for all j ∈ {1, . . . , r}. Let t := degP + 1, and defineD as in 17.4.1, that is, as the class of all graded R-modules N of dimensiont which can be written in the form N =

∑rj=1Ryj with yj ∈ N−aj for

all j ∈ {1, . . . , r}. By 17.4.1, there is a function C : Zt −→ Z such thatreg1(N) ≤ C(e0(N), . . . , et−1(N)) for each R-module N in D.

Let f : W −→ M be a homomorphism as described in the statement,and set W := W/Ker f . Then W =

∑rj=1Rwj with wj ∈ W−aj for all

j ∈ {1, . . . , r}. Moreover, f induces an exact sequence

0 −→W −→M −→ C −→ 0

(in the category *C(R)), where Cn = 0 for all except finitely many n ∈ Z. Inparticular, this means that PW = PM = P and reg1(C) = −∞. Therefore,by 16.2.15(iii) and 17.4.1, we have

reg1(M) ≤ reg1(W ) ≤ C(e0(P ), . . . , edegP (P )) =: G.

Note, in particular, that Corollary 17.4.2 tells us (with the notation of thatcorollary) that, for all graded homomorphic images M , with specified Hilbertpolynomial P , of a fixed finitely generated graded R-module W , the invariantreg1 is bounded in terms of P .

Next, we bound the regularity in terms of the Hilbert coefficients, the postu-lation number and the degrees of generators.

17.4.3 Corollary. Let R =⊕

n∈N0Rn be as in 17.4.1. Let g, π, t ∈ Z with

t > 0.


Let G denote the class of all finitely generated graded R-modules of dimen-sion t that can be generated by homogeneous elements of degrees not exceed-ing g and whose postulation number does not exceed π. Then the invariantse0( • ), . . . , et−1( • ) form a bounding system for reg on G.

Proof Let g := max{g, π + 1}, and, for a graded module M ∈ G, considerthe graded submodule M :=

⊕n≥gMn of M . Observe that PM = PM = pe,

where e := (e0(M), . . . , et−1(M)). Observe also that �R0(Mg) = PM (g) =

pe(g), and that M is generated by pe(g) > 0 homogeneous elements of degreeg. This time write a := (−g, . . . ,−g) ∈ Zpe(g), and define D′ to be the classof all graded R-modules N of dimension t that can be generated by pe(g)

homogeneous elements all of degree g. By 17.4.1, there is a function C ′ :

Zt −→ Z such that reg1(M) ≤ C ′(e0(M), . . . , et−1(M)) for each R-moduleM in D′.

Note that there is an exact sequence 0 −→ M −→ M −→ C −→ 0 (in thecategory *C(R)) in which C is finitely generated and R+-torsion. Therefore,by 16.2.15(iii), reg1(M) ≤ reg1(M). Since M ∈ D′, we have reg1(M) ≤C ′(e0(M), . . . , et−1(M)). However, it follows from 17.1.10(i) that reg(M) ≤max{reg1(M), pstln(M)}. Putting these inequalities together, we have

reg(M) ≤ max{C ′(e0(M), . . . , et−1(M)), π},

and this is enough to complete the proof.

17.4.4 Exercise. Let R = K[X1, X2] be a polynomial ring in two indeter-minates X1, X2 over a field K. By the generating degree gendeg(M) of anon-zero finitely generated graded R-module M we mean the smallest integerg such thatM is generated by homogeneous elements of degrees not exceedingg; thus

gendeg(M) = inf{g ∈ Z :M = R

∑n≤gMn

}.

For each h ∈ Z, let hK be the graded R-module (see 14.1.13) such that, for alln ∈ Z,

(hK)n =

{K if n = h,

0 if n = h.

(i) For each t ∈ N, letM (t) := Xt1(R/X2R)⊕ 0K⊕ 1K⊕· · ·⊕ t−1K, and

determine end(ΓR+(M(t)), end(H1

R+(M (t)), PM(t) and pstln(M (t)).

Conclude that the multiplicity e0(•) and the postulation number pstln(•)do not form a bounding system of invariants for either reg1 or reg on theclass of all non-zero finitely generated graded R-modules.


(ii) Use the graded R-modules N (t) := R ⊕ (R/Rt+) (t ∈ N) to show that

e0(•), e1(•) and gendeg(•) do not form a bounding system of invariantsfor reg on the class D of all finitely generated graded R-modules ofdimension 2.

(iii) For each t ∈ N, let L(t) :=⊕t−1

h=0(hK)h+1

⊕R≥t, where R≥t is as

defined in 16.1.1 and (hK)h+1 denotes the direct sum of h+1 copies ofhK. Use the L(t) (t ∈ N) to show that e0( • ), e1( • ) and pstln( • ) do notform a bounding system of invariants for reg1 on the class D of part (ii).

17.4.5 Definition and Exercise. Assume that R =⊕


graded and that the ideal a is graded. The saturation of a is defined as asat :=⋃n∈N(a :R (R+)

n). We say that a is saturated precisely when a = asat. Showthat

(i) asat is a graded ideal of R that contains a;(ii) asat/a = ΓR+(R/a);

(iii) asat is the largest graded ideal of R which coincides with a in all largedegrees;

(iv) (asat)sat = asat; and(v) if ΓR+(R) = H1

R+(R) = 0, then there is a homogeneous isomorphism

asat/a ∼= H1R+

(a).

17.4.6 Exercise. Let R = R0[X1, . . . , Xd] be the polynomial ring over thecommutative Noetherian ring R0 in d > 1 indeterminates. Assume that a isgraded.

Show that a is saturated if and only if H1R+

(a) = 0, and that reg(a) =

reg2(a) if a is saturated.Now assume in addition that the base ring R0 is local and Artinian. For

P ∈ Q[T ], let AP := {b is a graded ideal of R : Pb = P}. Show that themaximal members of AP are precisely the saturated ideals in AP . Show alsothat the Hilbert coefficients e0( • ), . . . , ed−1( • ) form a bounding system forreg on the class of all non-zero saturated graded ideals of R.

18

Applications to reductions of ideals

Graded local cohomology theory has played a substantial role in the study ofRees rings and associated graded rings of proper ideals in local rings. We donot have enough space in this book to include all we would like about theapplications of local cohomology to this area, and so we have decided to selecta small portion of the theory which gives some idea of the flavour. The partwe have chosen to present in this chapter concerns links between the theory ofreductions of ideals in local rings and the concept of Castelnuovo regularity,discussed in Chapter 16. The highlight will be a theorem of L. T. Hoa whichasserts that, if b is a proper ideal in a local ring having infinite residue field,then there exist t0 ∈ N and c ∈ N0 such that, for all t > t0 and every minimalreduction a of bt, the reduction number ra(bt) of bt with respect to a is equalto c. This statement of Hoa’s Theorem is satisfyingly simple, and makes nomention of local cohomology, and yet Hoa’s proof, which we present towardsthe end of this chapter, makes significant use of graded local cohomology.

Throughout this chapter, all graded rings and modules are to be understoodto be Z-graded, and all polynomial rings R[X1, . . . , Xt] (and R[T ]) over Rare to be understood to be (positively) Z-graded so that each indeterminate hasdegree 1 and deg a = 0 for all a ∈ R \ {0}.

18.1 Reductions and integral closures

Reductions of ideals of local rings were first considered by D. G. Northcottand D. Rees in [63]. Nowadays, the concept is recognized as being of majorimportance in commutative algebra. The original paper [63] of Northcott andRees was written under unnecessarily restrictive hypotheses, and so we beginthis chapter with a rapid development of the links between the two concepts of

18.1 Reductions and integral closures 389

reduction and integral closure of ideals over a general commutative Noetherianring. Throughout this chapter, b will denote a second ideal of R.

18.1.1 Definitions. (See D. G. Northcott and D. Rees [63].)

(i) We say that a is a reduction of b precisely when a ⊆ b and there existss ∈ N0 such that abs = bs+1; then the least such s is denoted by ra(b)and called the reduction number of b with respect to a. Note that, if a isa reduction of b, then ambj = bm+j for all m ∈ N and j ≥ ra(b).

(ii) We say that a is a minimal reduction of b if and only if a is a reductionof b and there is no reduction c of b with c ⊂ a.

(iii) We say that r ∈ R is integrally dependent on a if and only if there existn ∈ N and c1, . . . , cn ∈ R with ci ∈ ai for i = 1, . . . , n such that

rn + c1rn−1 + · · ·+ cn−1r + cn = 0.

18.1.2 �Exercise. Assume that a is a reduction of b.

(i) Show that√a =

√b.

(ii) Let c be a third ideal of R such that b is a reduction of c. Show that a isa reduction of c.

18.1.3 �Exercise. Show that, if a is a reduction of b and also a reduction ofanother ideal b′ of R, then a is a reduction of b+ b′.

18.1.4 Notation and �Exercise. Let {a1, . . . , ah} be a generating set for a,and let T be an indeterminate. We use R[aT, T−1] to denote the subring

R[a1T, . . . , ahT, T−1]

of R[T, T−1] = R[T ]T , and refer to this as the extended Rees ring of a.(Note that R[a1T, . . . , ahT, T−1] is independent of the choice of finite gener-ating set {a1, . . . , ah} for a.) Note that R[aT, T−1] inherits a Z-grading fromR[T, T−1] = R[T ]T .

Let r ∈ R. Show that r is integrally dependent on a if and only if the elementrT of R[T, T−1] is integral over R[aT, T−1].

18.1.5 Proposition. Assume that a ⊆ b and let r ∈ R.

(i) The element r is integrally dependent on a if and only if a is a reductionof a+Rr.

(ii) The ideal a is a reduction of b if and only if each element of b is inte-grally dependent on a.

390 Applications to reductions of ideals

Proof. First we suppose that a is a reduction of b, so that there exists s ∈ Nsuch that ambs = bm+s for all m ∈ N. Hence, provided we interpret ai as Rwhen the integer i is negative, bn ⊆ an−s for all n ∈ N0. Therefore, within thering R[T, T−1], we have R[bT, T−1] ⊆ T sR[aT, T−1], a finitely generatedmodule over the Noetherian ring R[aT, T−1]. It follows that R[bT, T−1] isintegral over its subring R[aT, T−1], and so, by 18.1.4, each element of b isintegrally dependent on a.

(i) It follows from the above paragraph that, if a is a reduction of a + Rr,then r is integrally dependent on a. Conversely, if r is integrally dependent ona, then there exist n ∈ N and c1, . . . , cn ∈ R with ci ∈ ai for i = 1, . . . , n

such that rn + c1rn−1 + · · ·+ cn−1r + cn = 0. Then

(a+Rr)n = a(a+Rr)n−1 +Rrn = a(a+Rr)n−1

and a is a reduction of a+Rr.(ii) Suppose each element of b is integrally dependent on a. Let {b1, . . . , bt}

be a generating set for b. By part (i), for each i = 1, . . . , t, the ideal a is areduction of a+Rbi. Hence a is a reduction of a+

∑ti=1Rbi = b, by 18.1.3.

The converse statement has already been proved in the first paragraph of thisproof.

18.1.6 Corollary and Definition. By 18.1.3, the set I of all ideals of R thathave a as a reduction has a unique maximal member, a say: a is the union ofthe members of I. By 18.1.5, this ideal a is precisely the set of all elements ofR which are integrally dependent on a (and so the latter set is an ideal of R):we refer to a as the integral closure of a. �

18.1.7 Exercise. Show that

(i) the integral closure a is not a reduction of any ideal of R which properlycontains it, and

(ii) a = b if and only if a and b are both reductions of a+ b.

We now reproduce the classical argument of Northcott and Rees whichshows that, when R is local, every reduction of b contains a minimal reductionof b.

18.1.8 Lemma. Assume that (R,m) is local and that a ⊆ b + am. Thena ⊆ b.

Proof. Since b + a = b + am, we have (b + a)/b = m ((b+ a)/b), and sothe result follows from Nakayama’s Lemma.

18.1 Reductions and integral closures 391

18.1.9 Lemma (Northcott and Rees [63, §2, Lemma 2]). Assume that (R,m)

is local and that a ⊆ b. Then a is a reduction of b if and only if a + bm is areduction of b.

Proof. It is clear that, if a is a reduction of b, then a+ bm is a reduction of b.Now assume that a+ bm is a reduction of b. Then there exists n ∈ N such that(a+ bm)bn = bn+1, that is, bn+1 = abn + bn+1m. It therefore follows fromLemma 18.1.8 that bn+1 = abn.

18.1.10 Lemma. Assume that (R,m) is local and that a is a reduction of b.Let a1, . . . , at ∈ a be such that their natural images in (a + bm)/bm form abasis for this R/m-space, and set a′ =

∑ti=1Rai. Then

(i) a′ is a reduction of b contained in a and a′ ∩ bm = a′m; and(ii) if a is a minimal reduction of b, then a = a′ and a ∩ bm = am.

Proof. (i) Note that a′ ⊆ a ⊆ b and a′+bm = a+bm. Since a is a reductionof b, it follows from Lemma 18.1.9 that a + bm = a′ + bm is a reduction ofb, so that a′ is a reduction of b by the same lemma. Also, if r1, . . . , rt ∈ R

are such that∑t

i=1 riai ∈ bm, then, by choice of a1, . . . , at, we must haver1, . . . , rt ∈ m.

(ii) Now suppose that a is a minimal reduction of b. Then a′ = a by part (i),so that a ∩ bm = am, again by part (i).

18.1.11 �Exercise. Assume (R,m) is local and that a is a minimal reductionof b. Let d be an ideal of R such that a ⊆ d ⊆ b. Show that every minimalgenerating set for a can be extended to a minimal generating set for d.

18.1.12 Theorem (Northcott and Rees [63, §2, Theorem 1]). Assume that(R,m) is local and that a is a reduction of b. Then a contains a minimalreduction of b.

Proof. Let Σ be the set of all ideals of R of the form d + bm, where d is areduction of b and is contained in a. Note that a + bm ∈ Σ. Since b/bm is afinite-dimensional vector space over R/m, the set Σ has a minimal member,and so there exists a reduction c′ of b contained in a such that c′ + bm is aminimal member of Σ. Let c1, . . . , ct ∈ c′ be such that their natural images in(c′ + bm)/bm form a basis for this R/m-space, and set c =

∑ti=1Rci. Now

c is a reduction of b contained in c′, and c ∩ bm = cm by 18.1.10(i); we shallprove that c is a minimal reduction of b.

Suppose that c0 is a reduction of b with c0 ⊆ c. Then c0 + bm ∈ Σ andc0 + bm ⊆ c′ + bm. Hence, by the choice of c′, we must have

c0 + bm = c′ + bm = c+ bm.


It follows that c ⊆ c0 + bm; therefore c ⊆ c0 + (bm ∩ c) ⊆ c0 + cm by theimmediately preceding paragraph in this proof. It now follows from 18.1.8 thatc ⊆ c0, so that c = c0.

There is an important connection between minimal reductions of ideals inlocal rings and analytical independence.

18.1.13 Definition. Assume that (R,m) is local. Then v1, . . . , vt ∈ b aresaid to be analytically independent in b if and only if, whenever h ∈ N andf ∈ R[X1, . . . , Xt] (the ring of polynomials over R in t indeterminates) is ahomogeneous polynomial of degree h such that f(v1, . . . , vt) ∈ bhm, then allthe coefficients of f lie in m.

Note that, if v1, . . . , vt ∈ b are analytically independent in b, and c :=∑ti=1Rvi, then ch ∩ bhm = chm for all h ∈ N.

18.1.14 Lemma (Northcott and Rees [63, §4, Lemma 1]). Assume (R,m)

is local and b is proper, and let v1, . . . , vt ∈ b be analytically independentin b. Then v1, . . . , vt form a minimal generating set for

∑ti=1Rvi, and, fur-

thermore, if {w1, . . . , wt} is a minimal generating set for∑t

i=1Rvi, thenw1, . . . , wt are analytically independent in b.

Proof. Set c :=∑t

i=1Rvi. If r1, . . . , rt ∈ R are such that∑t

i=1 rivi ∈ cm,then, since c ⊆ b, it follows that r1, . . . , rt ∈ m. Hence v1, . . . , vt form aminimal generating set for c.

Next, let h ∈ N and f ∈ R[X1, . . . , Xt] be a homogeneous polynomial ofdegree h such that f(w1, . . . , wt) ∈ bhm. There exist rij ∈ R (1 ≤ i, j ≤ t)

such that wi =∑t

j=1 rijvj for i = 1, . . . , t, and so

f(∑t

j=1r1jvj , . . . ,∑t

j=1rtjvj

)∈ bhm.

Since v1, . . . , vt ∈ b are analytically independent in b, it follows that all thecoefficients of the homogeneous polynomial

f(∑t

j=1 r1jXj , . . . ,∑t

j=1 rtjXj

)lie in m.

Denote the natural image in k := R/m (respectively k[X1, . . . , Xt]) of r ∈R (respectively q ∈ R[X1, . . . , Xt]) by r (respectively q). Now [rij ] is aninvertible t × t matrix over k, and so there exists a t × t matrix [sij ] over Rsuch that [rij ][sij ] = It, the t× t identity matrix. Since

f(∑t

j=1 r1jXj , . . . ,∑t

j=1 rtjXj

)= 0

18.2 The analytic spread 393

in k[X1, . . . , Xt], it follows that

f(∑t

j=1r1j

(∑tl=1sjlXl

), . . . ,∑t

j=1rtj

(∑tl=1sjlXl

))= 0,

and so f(X1, . . . , Xt) = 0. Thus all the coefficients of f lie in m.

18.1.15 Proposition (Northcott and Rees [63, §4, Lemma 2]). Assume that(R,m) is local and that k := R/m is infinite. Suppose that b is proper and a

is a minimal reduction of b, and let v1, . . . , vt form a minimal generating setfor a. Then v1, . . . , vt are analytically independent in b.

Proof. Let h ∈ N and f ∈ R[X1, . . . , Xt] be a homogeneous polynomial ofdegree h such that f(v1, . . . , vt) ∈ bhm. Suppose that the coefficient u of Xh

1

is a unit of R. Then vh1 ∈ bhm+ v2ah−1 + · · ·+ vta

h−1, and so

ah = Rvh1 + (Rv2 + · · ·+Rvt)ah−1 ⊆ bhm+ (Rv2 + · · ·+Rvt)a

h−1.

Set a′ := Rv2 + · · · + Rvt. Now there exists n ∈ N such that abn = bn+1.Hence bh+n = ahbn ⊆ bh+nm + a′ah−1bn = bh+nm + a′bh+n−1. Hencebh+n ⊆ a′bh+n−1 by 18.1.8, so that bh+n = a′bh+n−1 and a′ is a reductionof b. This contradicts the fact that a is a minimal reduction of b, since a′ ⊂ a.Hence u ∈ m.

Next let [rij ] be a t × t matrix over R such that all the entries in the firstcolumn, that is r11, r21, . . . , rt1, are units of R and det[rij ] is also a unit.Then there is a minimal generating set {w1, . . . , wt} for a such that vi =∑t

j=1 rijwj for all i = 1, . . . , t. Since

f(∑t

j=1r1jwj , . . . ,∑t

j=1rtjwj

)∈ bhm,

it follows from the first paragraph of this proof that f(r11, . . . , rt1), which isthe coefficient of Xh

1 in the form f(∑t

j=1 r1jXj , . . . ,∑t

j=1 rtjXj

), lies in

m. Thus, if we denote the natural image in k = R/m of r ∈ R by r, and thenatural image in k[X1, . . . , Xt] of q ∈ R[X1, . . . , Xt] by q, then

f(r11, . . . , rt1) = 0.

Therefore f(α1, . . . , αt) = 0 for all choices of (α1, . . . , αt) ∈ (k \ {0})t;hence, since k is infinite, f = 0.

18.2 The analytic spread

We now consider the graded rings which will provide the framework for con-nections between reductions of ideals and graded local cohomology.


18.2.1 Notation and Definition. Suppose that b is proper and a is a reductionof b.

(i) By 18.1.6 and 18.1.4, the extended Rees ringR[bT, T−1] is integral overits subring R[aT, T−1]. The associated graded ring

⊕i∈N0

bi/bi+1 willbe denoted by G(b) =

⊕i∈N0

G(b)i. Since there is a homogeneous iso-

morphism of graded rings R[bT, T−1]/T−1R[bT, T−1]∼=−→ G(b) (and

a similar one for G(a)), it follows that G(b) is integral over the naturalimage of G(a).

(ii) Now suppose, in addition, that (R,m) is local. The extension mG(b) ofm to G(b) under the natural ring homomorphism is a graded ideal: infact, mG(b) =

⊕i∈N0

mbi/bi+1. The analytic spread spr(b) of b isdefined by

spr(b) := dim (G(b)/mG(b)) .

Note that G(b)/mG(b) is (homogeneously) isomorphic to the gradedring⊕

i∈N0bi/mbi, in which (ri+mbi)(r′j +mbj) = rir

′j +mbi+j for

(i, j ∈ N0 and) ri ∈ bi, r′j ∈ bj . Some authors refer to the graded ringG(b)/mG(b) as the fibre cone of b.

(iii) Note that, since a is a reduction of b, it follows that G(b)/mG(b) isintegral over the natural image of G(a)/mG(a). Observe also that, if acan be generated by elements which are analytically independent in b,then ai ∩ bim = aim for all i ∈ N, so that the natural homogeneous ringhomomorphism G(a)/mG(a) −→ G(b)/mG(b) is injective.

18.2.2 Lemma. Suppose (R,m) is local and b is proper. Then spr(b) =

spr(bt) for all t ∈ N.

Proof. By Definition 18.2.1(ii),

spr(b) := dim (G(b)/mG(b)) and spr(bt) := dim(G(bt)/mG(bt)

).

Now there are homogeneous ring isomorphisms

G(b)/mG(b)∼=−→⊕i∈N0

bi/mbi and G(bt)/mG(bt)∼=−→⊕i∈N0

bti/mbti.

It is clear from these that G(bt)/mG(bt) is homogeneously isomorphic to thet-th Veronesean subring (see 13.5.9) of G(b)/mG(b). The claim now followsfrom the fact that a commutative Noetherian graded ring has the same dimen-sion as its t-th Veronesean subring because the former is integral over thelatter.

The observations in 18.2.1(iii) have some very important consequences.

18.2 The analytic spread 395

18.2.3 Remark. Suppose that (R,m) is local, that a ⊆ b ⊂ R and thata is generated by v1, . . . , vt which are analytically independent in b. Set k :=

R/m. Then v1, . . . , vt are, of course, analytically independent in a, and it is im-mediate from Definition 18.1.13 that there is a homogeneous isomorphism φ :

k[X1, . . . , Xt]∼=−→⊕

i∈N0ai/mai of graded k-algebras (where X1, . . . , Xt

are independent indeterminates) such that φ(Xi) = vi+ma for all i = 1, . . . , t.Since G(a)/mG(a) ∼=

⊕i∈N0

ai/mai, we have dim(G(a)/mG(a)) = t.

18.2.4 Theorem (Northcott and Rees [63, §4, Theorems 1 and 2]). Assumethat (R,m) is local and that k := R/m is infinite. Suppose that b is proper andthat a is a reduction of b, and let t := dimk(a/ma), the number of elements ineach minimal generating set for a. Let {v1, . . . , vt} be a minimal generatingset for a. Then

(i) spr(b) ≤ t;(ii) a is a minimal reduction of b if and only if spr(b) = t;

(iii) hence, at least spr(b) elements are needed to generate a, and, if a canbe generated by spr(b) elements, then it is a minimal reduction of b;

(iv) a is a minimal reduction of b if and only if v1, . . . , vt are analyticallyindependent in b.

Proof. Note that a contains a minimal reduction c of b, by Theorem 18.1.12;let w1, . . . , ws form a minimal generating set for c. Then w1, . . . , ws are ana-lytically independent in b by 18.1.15. By 18.2.1(iii), we can view the gradedring G(b)/mG(b) as an integral extension ring of G(c)/mG(c), and so

dim(G(c)/mG(c)) = dim(G(b)/mG(b)) = spr(b).

However, it is immediate from 18.2.3 that dim(G(c)/mG(c)) = s, and so s =spr(b).

(i) By 18.1.11, every minimal generating set for c can be extended to a min-imal generating set for a. Hence spr(b) = s ≤ t.

(ii) If a is a minimal reduction of b, then a = c and t = s = spr(b).Now suppose that t = spr(b). By 18.1.11, every minimal generating set for

c can be extended to a minimal generating set for a, and so spr(b) = s ≤ t =

spr(b). Therefore s = t and w1, . . . , ws actually generate a. It follows thata = c, and a is a minimal reduction of b.

(iii) This is a restatement of most of parts (i) and (ii).(iv) If a is a minimal reduction of b, then v1, . . . , vt are analytically inde-

pendent in b by 18.1.15. Conversely, if v1, . . . , vt are analytically indepen-dent in b, then t = dim(G(a)/mG(a)) by 18.2.3, and G(b)/mG(b) can beviewed as an integral extension ring of G(a)/mG(a), by 18.2.1(iii); therefore


t = dim(G(b)/mG(b)) = spr(b), so that we can deduce from part (ii) that ais a minimal reduction of b.

The next Lemma 18.2.5 is included to help the reader solve Exercise 18.2.6;that exercise presents an important result of Northcott and Rees.

18.2.5 Lemma. Suppose that (R,m) is local and that v1, . . . , vt are analyti-cally independent in b. Then t ≤ spr(b).

Proof. Set k := R/m and c :=∑t

i=1Rvi. Since ci∩bim = cim for all i ∈ N,the natural homogeneous ring homomorphism G(c)/mG(c) −→ G(b)/mG(b)is injective. Therefore, by 18.2.3, and with an obvious notation,

dimk (G(b)/mG(b))i ≥ dimk (G(c)/mG(c))i =(i+ t− 1

t− 1

)for all i ∈ N0.

Hence, with the notation of 17.1.8, degPG(b)/mG(b) ≥ t − 1, and thereforet ≤ spr(b) by 17.1.7.

18.2.6 Exercise (Northcott and Rees [63, §4, Theorem 3]). Assume (R,m) islocal and that k := R/m is infinite. Suppose that b is proper. Show that spr(b)is the maximum number of elements of b which are analytically independentin b.

18.2.7 Exercise. Assume (R,m) is local and let v1, . . . , vt ∈ R. Recall from[50, pp. 106–107] that v1, . . . , vt are analytically independent if and only if,whenever f ∈ R[X1, . . . , Xt] is a homogeneous polynomial such that

f(v1, . . . , vt) = 0,

then all the coefficients of f lie in m.Show that v1, . . . , vt are analytically independent if and only if they are

analytically independent in the ideal∑t

i=1Rvi which they generate.

18.2.8 Exercise (Northcott and Rees [63, §4, Lemma 4, Theorems 4, 5, and§6, Theorem 1]). Assume that (R,m) is local and that k := R/m is infinite.Suppose that b is proper. We say that b is basic precisely when it has no reduc-tion other than itself. Let v1, . . . , vt form a minimal generating set for b.

(i) Show that ht b ≤ spr(b) ≤ dimk(b/mb) = t.(ii) Show that the following statements are equivalent:

(a) b is basic;(b) v1, . . . , vt are analytically independent (see Exercise 18.2.7);(c) spr(b) = dimk(b/mb).

18.3 Links with Castelnuovo regularity 397

(iii) Show that, if b can be generated by t elements and has height t, thenb is basic and the members of each minimal generating set for b areanalytically independent.

(iv) Deduce that the members of a system of parameters for R are analyti-cally independent.

(v) Let q be an m-primary ideal of R. Show that spr(q) = dimR. (Here isa hint: �R(qn/qn+1) = �R(R/q

n+1) − �R(R/qn) for all n ∈ N, where

‘�’ denotes length.) Deduce that each minimal reduction of q is an idealgenerated by a system of parameters of R.

18.3 Links with Castelnuovo regularity

Local cohomology has still not made an appearance in this chapter! We aresoon going to make, in the case where (R,m) is local, some calculations withlocal cohomology over the associated graded ring G(b) =

⊕n∈N0

G(b)n of aproper ideal b of R. We note that G(b) is a positively graded, homogeneouscommutative Noetherian ring: the theory in the next part of this chapter con-cerns such rings and will eventually be applied to G(b).



homogeneous. We say that an ideal A of R is a *reduction of R+ if and onlyif A is graded, can be generated by homogeneous elements of R of degree 1,and is a reduction of R+.



mogeneous, and let A =⊕

n∈N An be a graded ideal ofR generated by homo-geneous elements of degree 1. Then A is a *reduction ofR+ if and only if thereexists m ∈ N0 such that Rm+1 = Am+1; when this is the case, Ri+1 = Ai+1

for all i ≥ m, and the least i ∈ N0 such that Ri+1 = Ai+1 is equal to thereduction number rA(R+) (see 18.1.1), so that rA(R+) = end(R/A).

Proof. Suppose that R = R0[x1, . . . , xt], where x1, . . . , xt ∈ R1, and thatA is generated by a1, . . . , ah ∈ A1. We use the notation R≥i, for i ∈ N, of16.1.1.

Let i ∈ N0. Then a typical element of Ri is a sum of finitely many elementsof the form r0x

j11 . . . xjtt , where r0 ∈ R0 and j1, . . . , jt ∈ N0 are such that∑t

k=1 jk = i, and a typical element of Ai+1 is a sum of finitely many elementsof the form alri, where l ∈ {1, . . . , h} and ri ∈ Ri. It follows that, if Ri+1 =

Ai+1, then Ri+2 = Ai+2, that (R+)i = R≥i, and that A(R+)

i = A≥i+1. Allthe claims now follow easily.



n∈N0Rn is positively graded and that

(R0,m0) is local; set k := R0/m0. Let A =⊕

n∈N An be a graded ideal of Rgenerated by homogeneous elements of degree 1. Then the minimum numberof homogeneous elements (of degree 1) needed to generate A is

dimk(A1/m0A1).

18.3.4 Lemma. Suppose that (R,m) is local, and that a ⊆ b ⊂ R. LetG(a) denote the ideal of the associated graded ring G(b) =

⊕i∈N0

G(b)i ofb generated by {a+ b2 : a ∈ a} ⊆ G(b)1.

Then a is a reduction of b if and only if G(a) is a *reduction of G(b)+.Furthermore, when this is the case, ra(b) = rG(a)(G(b)+).

Proof. Let i ∈ N0. Then the (i+ 1)-th component G(a)i+1 of G(a) is givenby G(a)i+1 = (abi + bi+2)/bi+2. By 18.1.8 and the fact that b is proper,abi = bi+1 if and only if abi + bi+2 = bi+1, that is, if and only if G(a)i+1 =

G(b)i+1. The claims therefore follow from 18.3.2.

18.3.5 Lemma (L. T. Hoa [35, Lemma 2.3]). Assume that (R,m) is localand that k := R/m is infinite. Suppose that b is proper. Then spr(b) =

ara(G(b)+), and their common value is the greatest integer i such that

HiG(b)+(G(b)) = 0.

Proof. By 18.1.12 and 18.2.4, there exists a minimal reduction a of b, and thiscan be generated by s := spr(b) elements. Therefore, by 18.3.4, and with thenotation of that Lemma, G(a) is a *reduction of G(b)+. Hence, by 18.1.2(i),we have

√G(a) =

√G(b)+, and so ara(G(b)+) ≤ s. Let M be an arbitrary

G(b)-module. By 3.3.3, we have HiG(b)+(M) = 0 for every i > ara(G(b)+),

and so, in particular, for every i > s.Consequently, the epimorphism G(b) −→ G(b)/mG(b) induces an epimor-

phism HsG(b)+(G(b)) −→ Hs

G(b)+(G(b)/mG(b)), and so it is enough for us toshow that Hs

G(b)+(G(b)/mG(b)) = 0 in order to complete the proof. This wedo.

The extension of G(b)+ to the *local graded ring G(b)/mG(b) is the unique*maximal ideal, and is, in fact, also maximal; it is therefore of height s, by14.1.14. It therefore follows from Theorem 6.1.4 and the Graded IndependenceTheorem 14.1.7 that Hs

G(b)+(G(b)/mG(b)) = 0.

18.3.6 �Exercise. Let the situation be as in Lemma 18.3.5, and let a be aminimal reduction of b (such an a certainly exists, by 18.1.12). Show thatthe graded ideal G(a) of G(b) defined in 18.3.4 can be generated by spr(b)

homogeneous elements of degree 1, and not by fewer.


In order to make further progress, we wish to use the fact that, when R =⊕n∈N0

Rn is positively graded, homogeneous, and such that (R0,m0) is localwith infinite residue field, and the non-zero graded ideal A =

⊕n∈N An of R

is a *reduction of R+, it is possible to generate A by the members of an R+-filter-regular sequence of length dimR0/m0

(A1/m0A1). We therefore explainthe concept of R+-filter-regular sequence.

18.3.7 Definition. Assume that R is positively graded, and let M be a non-zero finitely generated graded R-module. Let f1, . . . , fh be a sequence ofhomogeneous elements of R. We say that f1, . . . , fh is an R+-filter-regularsequence with respect to M if and only if, for all i = 1, . . . , h, we havefi ∈⋃

P∈Ass(M/(f1,...,fi−1)M)\Var(R+) P.

We collect some elementary properties of R+-filter-regular sequences to-gether in the next exercise.

18.3.8 �Exercise. Assume that R is positively graded, and let M be a non-zero finitely generated graded R-module and f1, . . . , fh be a sequence of ho-mogeneous elements of R. Show that the following statements are equivalent:

(i) f1, . . . , fh is an R+-filter-regular sequence with respect to M ;(ii) for all P ∈ Spec(R)\Var(R+), the sequence f1/1, . . . , fh/1 of natural

images in RP is a poor MP-sequence;(iii) ((f1, . . . , fi−1)M :M fi)/(f1, . . . , fi−1)M is R+-torsion for all i =

1, . . . , h;(iv) end ((f1, . . . , fi−1)M :M fi)/(f1, . . . , fi−1)M) < ∞ for i = 1, . . . , h

(the end of a graded R-module was defined in 14.1.1);(v) fi is a non-zerodivisor on M/(f1, . . . , fi−1)M for each i = 1, . . . , h,

where L, for an R-module L, denotes L/ΓR+(L).

18.3.9 Exercise. Let the situation be as in 18.3.8, and assume that f1, . . . , fhis an R+-filter-regular sequence with respect to M . Let deg fi = ni for i =1, . . . , h. Show that

reg1(M/∑h

i=1fiM)≤ reg1(M) +

∑hi=1 ni − h.

We now turn to the construction of R+-filter-regular sequences.


n∈N0Rn is positively graded,

that (R0,m0) is local with infinite residue field k := R0/m0, and that thenon-zero graded ideal A =

⊕n∈N An of R is a *reduction of R+. Let t :=

dimk(A1/m0A1). Then there exist f1, . . . , ft ∈ A1 such that A =∑t

i=1Rfiand f1, . . . , ft is an R+-filter-regular sequence with respect to R.


Proof. Suppose, inductively, that i ∈ N0 with i < t and that an R+-filter-regular sequence (with respect to R) f1, . . . , fi of elements of A1 has beenconstructed such that the natural images of f1, . . . , fi in A1/m0A1 are linearlyindependent in this k-space. This is certainly the case when i = 0. Set f :=∑i

j=1Rfj . Note that m0A + f = A and that√A =√R+ by 18.1.2(i). We

can use 16.1.3 to see that there exists

fi+1 ∈ A1 \

⎛⎝(m0A+ f)⋃⎛⎝ ⋃

P∈Ass(R/f)\Var(R+)

P

⎞⎠⎞⎠ .Then the extended sequence f1, . . . , fi, fi+1 is R+-filter-regular (with respecttoR), and the choice of fi+1 ensures that the natural images of f1, . . . , fi, fi+1

in A1/m0A1 are linearly independent. This completes the inductive step.In this way, we can construct an R+-filter-regular sequence (with respect

to R) f1, . . . , ft of elements of A1 whose natural images in A1/m0A1 forma basis for this k-space. Since A can be generated by elements of degree 1, itfollows that f1, . . . , ft generate A.

We now show that R+-filter-regular sequences lend themselves to satisfac-tory calculations with various Castelnuovo regularities. We remind the readerthat, when R is positively graded and homogeneous, the regularity regl(M) ofM at and above level l (for l ∈ N0) of a finitely generated graded R-moduleM was defined in 16.2.9.



homogeneous, let M be a non-zero finitely generated graded R-module, andlet f1, . . . , fh ∈ R1 be an R+-filter-regular sequence with respect to M . Thenregl(M) ≤ regl−h(M/(f1, . . . , fh)M) for all l ≥ h and

regh(M) ≤ reg(M/(f1, . . . , fh)M) ≤ reg(M).

Proof. We prove this by induction on h, there being nothing to prove whenh = 0. Suppose, inductively, that h > 0, and that both statements in the claimhave been proved for smaller values of h. Set f :=

∑h−1i=1 Rfi. By our inductive

hypothesis, regl(M) ≤ regl−h+1(M/fM) for all l ≥ h−1 and regh−1(M) ≤reg(M/fM) ≤ reg(M).

Since (fM :M fh)/fM isR+-torsion, we haveHjR+

((fM :M fh)/fM) = 0

for all j ∈ N, by 2.1.7(i); therefore, the canonical epimorphism π :M/fM −→M/(fM :M fh) induces homogeneous isomorphisms

HjR+

(π) : HjR+

(M/fM)∼=−→ Hj

R+(M/(fM :M fh)) for all j ∈ N.

Hence regl(M/fM) = regl(M/(fM :M fh)) for all l ∈ N.


Multiplication by fh leads to an exact sequence

0 −→ (M/(fM :M fh))(−1) −→M/fM −→M/(f+Rfh)M −→ 0

of graded R-modules and homogeneous homomorphisms. By 16.2.15(iv) andthe inductive hypothesis,

reg(M/(f+Rfh)M)

≤ max{reg1((M/(fM :M fh))(−1))− 1, reg(M/fM)

}= max

{reg1(M/fM), reg(M/fM)

}= reg(M/fM)

≤ reg(M).

We can also apply 16.2.15(ii) to deduce that, for l ≥ h,

regl−h+1(M/fM)

= regl−h+1((M/(fM :M fh))(−1))− 1

≤ max{regl−h+1(M/fM)− 1, regl−h(M/(f+Rfh)M)

},

so that regl−h+1(M/fM) ≤ regl−h(M/(f + Rfh)M). We can therefore usethe inductive hypothesis to see that

regl(M) ≤ regl−h+1(M/fM) ≤ regl−h(M/(f+Rfh)M) for all l ≥ h.

In particular, regh(M) ≤ reg(M/(f + Rfh)M), and so this completes theinductive step.

18.3.12 Theorem. (See N. V. Trung [86, Proposition 3.2].) Assume (R,m)

is local and that k := R/m is infinite. Suppose that b is proper, and set s :=

spr(b). Let a be a minimal reduction of b. Then

regs(G(b)) ≤ ra(b) ≤ reg(G(b)).

Proof. By 18.3.4, and with the notation of that lemma, G(a) is a *reductionof G(b)+, so that G(b)/G(a) is G(b)+-torsion and (by 18.3.2)

ra(b) = rG(a)(G(b)+) = end(G(b)/G(a)).

Note that, by 2.1.7(i), we have HiG(b)+(G(b)/G(a)) = 0 for all i ∈ N, and so

ra(b) = end(G(b)/G(a)) = reg(G(b)/G(a)).

Next, we observe from 18.3.6 that G(a) can be generated by s := spr(b)

homogeneous elements of degree 1, and not by fewer. Therefore, by 18.3.3and 18.3.10, there exists in G(a) a G(b)+-filter-regular sequence (with respect


to G(b)) f1, . . . , fs of elements of degree 1 which generate this ideal. Conse-quently, by Proposition 18.3.11, we have

regs(G(b)) ≤ reg(G(b)/G(a)) ≤ reg(G(b))

and this completes the proof because we have already shown that ra(b) =

reg(G(b)/G(a)).

We are now going to use Trung’s Theorem 18.3.12 to derive a theorem dueto L. T. Hoa, which shows that, in the situation of Trung’s Theorem, for t ∈ Nsufficiently large, the reduction number of bt with respect to a minimal reduc-tion of bt is independent of the choice of minimal reduction and independentof t. We need one preparatory result, also due to Hoa.

18.3.13 Proposition (L. T. Hoa [35, Lemma 2.4]). Assume that (R,m) islocal and that k := R/m is infinite. Suppose that b is proper, and assume thats := spr(b) ≥ 1. For q ∈ Q, we use !q" to denote max{i ∈ Z : i ≤ q}; weinterpret !q" as −∞ when q = −∞. Let t ∈ N. Then

(i) end(HjG(bt)+

(G(bt)))≤⌊end(HjG(b)+(G(b))

)/t⌋

for all j ∈ N0;

(ii) end(HsG(bt)+

(G(bt)))=⌊end(HsG(b)+(G(b))

)/t⌋

.

Proof. By 18.2.2 and 18.3.5, we have

end(HjG(bt)+

(G(bt)))= end(HjG(b)+(G(b))

)= −∞ for all j > s.

The inequality in part (i) is therefore certainly true when j > s.Now consider the case where 0 ≤ j ≤ s, and let i ∈ N0 with i ≤ t. The

extension biG(bt) of bi to G(bt) under the composition R → R/bt → G(bt)of canonical ring homomorphisms is a graded ideal of G(bt) with grading givenby

biG(bt) =⊕n∈N0

btn+i/bt(n+1).

Note that btG(bt) = 0. Our strategy is to consider the chain of submodules

G(bt) = b0G(bt) ⊇ · · · ⊇ brG(bt) ⊇ br+1G(bt) ⊇ · · · ⊇ btG(bt) = 0,

and to obtain information about the ‘subquotients’ in the chain by use ofVeronesean functors (see 13.5.9).

There is a natural homogeneous surjective ring homomorphism θ : G(bt)→(G(b))(t), and so any graded G(b)(t)-module can be regarded as a gradedG(bt)-module by means of θ. This applies, in particular, to

(G(b))(t,i) =⊕n∈N0

btn+i/btn+i+1.


It follows that, for i < t, there is an exact sequence

0 −→ bi+1G(bt) −→ biG(bt) −→ (G(b))(t,i) −→ 0

of graded G(bt)-modules and homogeneous homomorphisms.Let r ∈ Z with r >

⌊end(HjG(b)+(G(b))

)/t⌋

. By 13.5.9(v) and the

Graded Independence Theorem 14.1.7, there are homogeneous G(bt)-isomor-phisms

HjG(bt)+

((G(b))(t,i))∼=−→ Hj

G(b)(t)+

((G(b))(t,i))∼=−→(HjG(b)+(G(b))

)(t,i).

Hence

HjG(bt)+

((G(b))(t,i))r ∼= HjG(b)+(G(b))rt+i = 0,

since rt + i > end(HjG(b)+(G(b))). Therefore the short exact sequence dis-

played in the last paragraph induces an R/bt-epimorphism

HjG(bt)+

(bi+1G(bt))r −→ HjG(bt)+

(biG(bt))r.

Since btG(bt) = 0, we can therefore deduce by descending induction thatHjG(bt)+

(biG(bt))r = 0 for i = t − 1, . . . , 1, 0. Hence HjG(bt)+

(G(bt))r = 0.

Hence end(HjG(bt)+

(G(bt)))≤⌊end(HjG(b)+(G(b))

)/t⌋

.

Now consider the case where j = s. Let u :=⌊end(HsG(b)+(G(b))

)/t⌋

.Thus end(Hs

G(b)+(G(b))) = ut + i for some integer i with 0 ≤ i ≤ t − 1.Recall from 18.3.5 that s is the greatest integer i such that Hi

G(b)+(G(b)) = 0.It therefore follows from 16.2.5 that Hs

G(b)+(G(b))n = 0 for all n ≤ ut + i,and so, in particular, for n = ut. Recall also, from 18.2.2 and 18.3.5, thats = ara(G(bt)+). It therefore follows from the exact sequence

0 −→ bG(bt) −→ G(bt) −→ (G(b))(t,0) −→ 0

of graded G(bt)-modules and homogeneous homomorphisms that there is anexact sequence of R/bt-modules

HsG(bt)+

(G(bt))u −→ HsG(b)+(G(b))ut −→ 0.

Hence HsG(bt)+

(G(bt))u = 0, and end(HsG(bt)+

(G(bt)))≥ u. This, in con-

junction with the result of part (i), completes the proof.

We are now ready to present Hoa’s proof of his theorem, mentioned in theintroduction to this chapter, about the asymptotic behaviour, with respect toreduction numbers, of powers of a proper ideal of a local ring having infiniteresidue field. We remark again that there is no mention of local cohomology in


the statement of the theorem, but the powerful tool of graded local cohomologyplays a major role in the proof.

18.3.14 Theorem (L. T. Hoa [35, Theorem 2.1]). Assume that (R,m) is localand that R/m is infinite. Suppose that b is proper. Then there exist t0 ∈ N andc ∈ N0 such that, for all t > t0 and every minimal reduction a of bt, we havera(b

t) = c.

Proof. Let s := spr(b). If s = 0, then b is nilpotent, and the result is obviousbecause the only minimal reduction of the zero ideal of R is the zero idealitself. We therefore assume that s ≥ 1 for the remainder of this proof. Sett0 := max{| end(Hi

G(b)+(G(b)))| : i ∈ N0 and HiG(b)+(G(b)) = 0}. Note

that t0 ∈ N0, by 18.3.5.Suppose that t ∈ N with t > t0. By Proposition 18.3.13, we have

end(HiG(bt)+

(G(bt)))≤ 0 for i = 0, . . . , s− 1

and

end(HsG(bt)+

(G(bt)))=

⎧⎨⎩0 if end(HsG(b)+(G(b))

)≥ 0,

−1 if end(HsG(b)+(G(b))

)< 0.

Of course, HiG(bt)+

(G(bt)) = 0 for all i > s, by 18.2.2 and 18.3.5. Hence

regs(G(bt)) = reg(G(bt)) =

⎧⎨⎩s if end(HsG(b)+(G(b))

)≥ 0,

s− 1 if end(HsG(b)+(G(b))

)< 0.

The result now follows from Trung’s Theorem 18.3.12.

18.3.15 Exercise. Assume that (R,m) is local and that R/m is infinite; letdimR = d. Suppose that q is an m-primary ideal of R such that G(q) isCohen–Macaulay. Show that each minimal reduction q′ of q has rq′(q) =

end(HdG(q)+(G(q))

)+ d.

As was mentioned in the introduction to this chapter, we have only been ableto present a small portion of the body of work linking graded local cohomologyand reductions of ideals. An interested reader might like to consult, in additionto papers already cited in this chapter, [48] by T. Marley, [87] by N. V. Trung,[31] by M. Herrmann, E. Hyry and T. Korb, [9] by C. D’Cruz, V. Kodiyalamand J. K. Verma, [12] by J. Elias, and some of the papers cited by these authors.

19

Connectivity in algebraic varieties

The study of the topological connectivity of algebraic sets is a fundamentalsubject in algebraic geometry. Local cohomology is a powerful tool in thisfield. In this chapter we shall use this tool to prove some results on connectiv-ity which are of basic significance. Our main result will be the ConnectednessBound for Complete Local Rings, a refinement of Grothendieck’s Connect-edness Theorem. We shall apply this result to projective varieties in order toobtain a refined version of the Bertini–Gothendieck Connectivity Theorem.Another central result of this chapter will be the Intersection Inequality forConnectedness Dimensions of Affine Algebraic Cones. As an application itwill furnish a refined version of the Connectedness Theorem for Projective Va-rieties due to W. Barth, to W. Fulton and J. Hansen, and to G. Faltings. The finalgoal of the chapter will be a ring-theoretic version of Zariski’s Main Theoremon the Connectivity of Fibres of Blowing-up.

The crucial appearances of local cohomology in this chapter are just in twoproofs, but the resulting far-reaching consequences in algebraic geometry il-lustrate again the power of local cohomology as a tool in the subject. We shalluse little more from local cohomology than the Mayer–Vietoris sequence 3.2.3and its graded version 14.1.5, the Lichtenbaum–Hartshorne Vanishing Theo-rem 8.2.1 and the graded version 14.1.16, and the vanishing result of 3.3.3.See the proofs of Proposition 19.2.8 and Lemma 19.7.2. The use of these tech-niques in this context originally goes back to Hartshorne [29] and has beenpushed further by J. Rung (see [5]).

Throughout this chapter, all graded rings and modules are to be understoodto be Z-graded, and all polynomial rings K[X1, . . . , Xd] (over a field K) areto be understood to be (positively) Z-graded so that K is the component ofdegree 0 and degXi = 1 for all i = 1, . . . , d.

406 Connectivity in algebraic varieties

19.1 The connectedness dimension

To begin, we have to introduce a measure for the connectivity of an algebraicset, or, more generally, of a Noetherian topological space. We start with somereminders of topological concepts which are fundamental to our work in thischapter.

19.1.1 Reminders. Here we are concerned with a general topological space.

(i) Recall that a non-empty topological space is said to be disconnectedprecisely when it can be expressed as the disjoint union of two properopen (or closed) subsets. Otherwise the space is said to be connected.We adopt the convention whereby the empty set is considered to be dis-connected.

(ii) Recall also that a topological space T is said to be quasi-compact pre-cisely when every open covering of T has a finite subcovering, thatis, if and only if, whenever (Uα)α∈Λ is a family of open subsets of Tsuch that T =

⋃α∈Λ Uα, then there is a finite subset Φ of Λ such that

T =⋃

α∈Φ Uα.(iii) Recall that a non-empty topological space T is said to be irreducible

precisely when T is not the union of two proper closed subsets, that is,if and only if every pair of non-empty open subsets of T has non-emptyintersection.

19.1.2 �Exercise. Let T be a non-empty topological space.

(i) Show that the following statements are equivalent:

(a) T is irreducible;(b) every non-empty open subset of T is dense in T ;(c) every non-empty open subset of T is connected.

(ii) Let S be an irreducible subset of T (that is, a subset of T which is anirreducible space in the topology induced from T ), and let (Ci)1≤i≤n

be a finite covering of S by closed subsets of T (so that C1, . . . , Cn areclosed subsets of T such that S ⊆

⋃ni=1 Ci). Show that S ⊆ Ci for some

i with 1 ≤ i ≤ n.(iii) Show that, if T is irreducible, then every non-empty open subset of T is

irreducible.(iv) Let (Ui)1≤i≤n be a finite open covering of T , with Ui = ∅ for all i =

1, . . . , n. Prove that T is irreducible if and only if Ui is irreducible forall i = 1, . . . , n and Ui ∩ Uj = ∅ for all i, j = 1, . . . , n.

(v) Let S be a subset of T . Show that S is irreducible if and only if its closureS is irreducible.

19.1 The connectedness dimension 407

(vi) Let T ′ be a second topological space and let f : T → T ′ be a continuousmap. Show that, if T is irreducible, then so too is f(T ).

19.1.3 �Exercise and Definition. Use Zorn’s Lemma to show that a non-empty topological space T has maximal irreducible subsets.

The maximal irreducible subsets of T are called its irreducible components.Show that the irreducible components of T are closed and that they cover

T ; show also that every irreducible subset of T is contained in an irreduciblecomponent of T .

19.1.4 �Exercise. Let K be an algebraically closed field, and let r ∈ N.

(i) Show that the irreducible components of an affine algebraic cone inAr(K) (see 15.1.2(i)) are again affine algebraic cones.

(ii) Let W be a non-empty closed subset of Pr(K); suppose that the distinctirreducible components of W are W1, . . . ,Wn. Show that Cone(W )

(see 15.2.1(iii)) has Cone(W1), . . . ,Cone(Wn) as its (distinct) irreduci-ble components.

19.1.5 Definition. Let T be a topological space. We say that T is a Noethe-rian topological space precisely when it satisfies the following equivalent con-ditions.

(i) Whenever (Ci)i∈N is a family of closed subsets of T such that

C1 ⊇ C2 ⊇ · · · ⊇ Ci ⊇ Ci+1 ⊇ · · · ,

then there exists k ∈ N such that Ck = Ck+i for all i ∈ N.(ii) Every non-empty set of closed subsets of T contains a minimal element

with respect to inclusion.

19.1.6 �Exercise. Let T be a topological space.

(i) Show that T is Noetherian if and only if every open subset of T is quasi-compact.

(ii) Show that the spectrum of a commutative Noetherian ring, furnishedwith its Zariski topology, is a Noetherian topological space.

(iii) Show that a quasi-affine variety over an algebraically closed field (see2.3.1) is a Noetherian topological space.

19.1.7 Lemma. Let T be a non-empty Noetherian topological space. Then


T has only finitely many irreducible components. Also, if T1, . . . , Tn are thedistinct irreducible components of T , then

Tj ⊆n⋃

i=1i�=j

Ti for all j = 1, . . . , n.

Proof. Suppose that T has infinitely many irreducible components. Let Sbe the set of non-empty closed subsets of T which have infinitely many irre-ducible components. Since T is Noetherian, S has a minimal member: let Cbe one such. Then C itself cannot be irreducible, so that C can be written asC = C1 ∪ C2 for some proper closed subsets C1 and C2 of C. Note that C1

and C2 are closed in T , and so, by the minimality of C, each of C1 and C2 hasonly finitely many irreducible components. But, by 19.1.2(ii), each irreduciblesubset of C must be contained in C1 or C2, and so each irreducible componentof C must be an irreducible component of C1 or C2 (by 19.1.3). Hence therecan only be finitely many irreducible components of C, a contradiction.

The final claim follows easily from another use of 19.1.2(ii) since, by 19.1.3,each Ti (1 ≤ i ≤ n) is closed in T .

19.1.8 Definition. Let T be a non-empty Noetherian topological space. Thedimension of T , denoted by dimT , is defined as the supremum of the lengthsn of all strictly descending chains Z0 ⊃ Z1 ⊃ · · · ⊃ Zn of closed irreduciblesubsets of T if this supremum exists, and∞ otherwise. Thus, by 19.1.3, dimT

is a non-negative integer or∞.The dimension of the empty space is defined to be −1.Note that, for our Noetherian ring R, we have dim(Spec(R)) = dimR;

thus, in view of [56, Appendix, Example 1], the dimension of a Noetheriantopological space can be∞.

19.1.9 Definition. Let T be a Noetherian topological space. The connected-ness dimension c(T ) of T is defined to be the minimum of the dimensions ofthose closed subsets Z of T for which T \ Z is disconnected. (Observe thatT \ T is certainly disconnected!) Thus

c(T ) := min {dimZ : Z ⊆ T, Z is closed and T \ Z is disconnected} .

For our Noetherian ring R, we write c(R) := c(Spec(R)).

19.1.10 Examples. Let T be a Noetherian topological space.

(i) Note that T is disconnected, that is, T \ ∅ is disconnected, if and only ifc(T ) = −1. We can thus conclude that c(T ) is negative if and only if Tis disconnected.

19.1 The connectedness dimension 409

(ii) It follows from part (i) that, if (R,m) is local, then c(R) ≥ 0.(iii) Suppose that T is irreducible. Then if Z is a proper closed subset of T ,

it is impossible for T \ Z to be disconnected (by 19.1.2(i)); on the otherhand, T \ T is disconnected. Thus c(T ) = dimT in this case.

(iv) It follows from part (iii) that, if R is an integral domain, then c(R) =

dimR. Thus, in view of [56, Appendix, Example 1], the connectednessdimension of a Noetherian topological space can be∞.

19.1.11 Example. Assume that (R,m) is local, and has exactly two minimalprime ideals, p and q. Then c(R) = 0 if and only if dimR/(p+ q) = 0.

Proof. Note that the hypotheses ensure that dimR > 0.(⇐) We have Spec(R) = Var(p) ∪Var(q). Since m is the only prime ideal

of R which contains both p and q, it follows that

Var(p) ∩ (Spec(R) \ {m}) and Var(q) ∩ (Spec(R) \ {m})

are two non-empty disjoint closed subsets of Spec(R) \ {m} which cover thisspace. Therefore, bearing in mind 19.1.10(ii), we see that c(R) = 0.

(⇒) Assume that c(R) = 0. Thus there is a closed subset Z of Spec(R) forwhich dimZ = 0 and Spec(R) \ Z is disconnected. We must have Z = {m}.Set T := Spec(R)\{m}. Thus there exist ideals a, b ofR such that T ∩Var(a)and T ∩Var(b) are non-empty disjoint subsets of T which cover T .

Then p ⊇ a or p ⊇ b; also q ⊇ a or q ⊇ b. For the sake of argument, let usassume that p ⊇ a.

Then q ⊇ a, since otherwise Spec(R) = Var(a) and T ∩ Var(b) = ∅.Therefore q ⊇ b, and it follows that (T ∩Var(p))∩ (T ∩Var(q)) = ∅, so thatdimR/(p+ q) = 0.

19.1.12 Exercise. Let f , g be non-constant and irreducible polynomials inC[X,Y ] which are not associates of each other. Let T := VA2(fg). Show that

c(T ) =

{−1 if VA2(f) ∩ VA2(g) = ∅,0 otherwise.

19.1.13 Notation and �Exercise. Assume that (R,m) is local. The topologi-cal space Spec(R)\{m}, with the topology induced from the Zariski topologyon Spec(R), is called the punctured spectrum of R, and denoted by Spec◦ (R).Show that c(Spec◦ (R)) = c(R)− 1.

19.1.14 Notation. For r ∈ N, denote by S(r) the set of all ordered pairs(A,B) of non-empty subsets of {1, . . . , r} for which A ∪B = {1, . . . , r}.


19.1.15 Lemma. Let T be a non-empty Noetherian topological space with(distinct) irreducible components T1, . . . , Tr. With the notation of 19.1.14, wehave c(T ) = min

{dim((⋃

i∈ATi)∩(⋃

j∈BTj))

: (A,B) ∈ S(r)}.

Proof. We write c = c(T ) and m for the minimum that occurs on the right-hand side of the equation in the above statement.

We first show that c ≥ m. To achieve this, let Z be a closed subset of Twith dimZ = c and such that T \ Z is disconnected. If Ti ⊆ Z for somei ∈ {1, . . . , r}, then c = dimZ ≥ dimTi ≥ m. Thus we can, and do, assumethat Ti ∩ (T \ Z) = ∅ for all i ∈ {1, . . . , r}. As T \ Z is disconnected we canwrite T \ Z = U1 ∪ U2, where U1 and U2 are non-empty open sets in T suchthat U1 ∩ U2 = ∅. Set

A := {i ∈ {1, . . . , r} : Ti ∩ U1 = ∅} ,

B := {j ∈ {1, . . . , r} : Tj ∩ U2 = ∅} .

Then, the pair (A,B) ∈ S(r). Moreover,A andB are disjoint, as otherwise forany index i ∈ A∩B the irreducible space Ti would contain the two non-emptyand disjoint open subsets Ti ∩U1 and Ti ∩U2. Thus

(⋃i∈ATi)∩(⋃

j∈BTj)

has no point in common with U1 ∪ U2, so that it is contained in Z and hasdimension not exceeding c. This proves that c ≥ m.

To prove the inequality m ≥ c, let (A,B) ∈ S(r) be a pair such that Z :=(⋃i∈ATi

)∩(⋃

j∈BTj

)is of dimension m. If r = 1, we have m = dimT ≥

c, as required. Assume therefore that r > 1. Then, by the minimality in thedefinition of m, we can, and do, assume that A and B are disjoint. Then theopen sets U1 = T \

⋃i∈ATi and U2 = T \

⋃j∈BTj are non-empty. Note that

U1 ∩ U2 = ∅, since(⋃

i∈A Ti)∪(⋃

j∈B Tj)= T . Hence T \ Z = U1 ∪ U2

is disconnected. Consequently c ≤ dimZ = m.

19.1.16 �Exercise. Let V be an affine variety over the algebraically closedfield K. Let b be an ideal ofO(V ), and let V (b) denote the closed subset of Vdetermined by b. Show that c(V (b)) = c(O(V )/b).

19.2 Complete local rings and connectivity

We now introduce another invariant of Noetherian spaces.

19.2.1 Definition. Let T be a non-empty Noetherian topological space. Thesubdimension sdimT of T is defined as the minimum of the dimensions of

19.2 Complete local rings and connectivity 411

the irreducible components of T . For our Noetherian ring R, we write sdimR

instead of sdim(SpecR).

Notice the following easy fact.

19.2.2 Lemma. Let T be a non-empty Noetherian topological space of finitedimension. Then c(T ) ≤ sdimT . Moreover, equality holds here if and only ifT is irreducible.

Proof. Let T1, . . . , Tr be the irreducible components of T . Then

sdimT = min {dimTi : i = 1, . . . , r} ;

also, we have, by 19.1.7, for each j = 1, . . . , r, that

Tj ∩

⎛⎜⎜⎝ r⋃i=1i�=j

Ti

⎞⎟⎟⎠ ⊂ Tj ,

so that, since T has finite dimension,

dim

⎛⎜⎜⎝Tj ∩⎛⎜⎜⎝ r⋃

i=1i�=j

Ti

⎞⎟⎟⎠⎞⎟⎟⎠ < dimTj .

The claims now follow easily from Lemma 19.1.15.

19.2.3 Exercise. Calculate c(T ) and sdimT for the following choices of theNoetherian topological space T :

(i) T := VA3(X21 −X1, X

22 −X2);

(ii) T := VA4(X1X3, X1X4, X2X3, X2X4);(iii) T := VA3((X1 − 1)(X2

1 +X22 +X2

3 − 1)).

19.2.4 �Exercise. Assume that (R,m) is local and that dimR > 0. Showthat the subdimension of the punctured spectrum Spec◦ (R) of R (see 19.1.13)is given by sdim(Spec◦ (R)) = sdimR− 1.

19.2.5 Remark. We shall frequently consider connectedness dimensions andsubdimensions of spectra of Noetherian rings. Therefore it will be helpfulto translate the previous lemmas into ring-theoretic terms. So, let p1, . . . , prbe the distinct minimal prime ideals of R. Then, again using the notation of19.1.14, it follows from 19.1.15 that

c(R) = min{dim(R/((⋂

i∈Api)+(⋂

j∈Bpj)))

: (A,B) ∈ S(r)}.


Also, 19.2.2 shows that sdimR = min {dimR/pi : i = 1, . . . , r} ≥ c(R),with equality if and only if r = 1.

19.2.6 Exercise. Suppose that (R,m) is local, catenary and S2, and that Rhas more than one minimal prime ideal; set d := dimR. Use Proposition12.2.8 to show that c(R) = d− 1.

Our applications of local cohomology to connectedness dimensions will in-volve use of the concepts of arithmetic rank and cohomological dimension ofan ideal, introduced in 3.3.2 and 3.3.4. Recall that cohd(a) ≤ ara(a). Thereader should be aware of the elementary properties of arithmetic rank de-scribed in the following exercise.

19.2.7 �Exercise. Assume that the ideal a is proper.

(i) Show that ht a ≤ ara(a).(ii) Let R′ be a second commutative Noetherian ring and let f : R −→ R′

be a ring homomorphism. Prove that ara(aR′) ≤ ara(a).

The next result will play a crucial role in our approach to connectivity. Itsproof uses most of the main ingredients from local cohomology theory thatwe shall need in this chapter. The result relates, in certain circumstances, thecohomological dimension cohd(a ∩ b) of the intersection of two ideals a andb in a complete local domain R with the dimensions of R and R/(a+ b).

19.2.8 Proposition. Assume that (R,m) is a complete local domain. Let bbe a second ideal of R, and assume that a and b are both proper and thatmin {dimR/a, dimR/b} > dimR/(a+ b). Then

cohd(a ∩ b) ≥ dimR− dimR/(a+ b)− 1,

so that ara(a ∩ b) ≥ dimR− dimR/(a+ b)− 1.

Proof. Set d := dimR and δ := dimR/(a + b). We proceed by inductionon δ. First, let δ = 0. Then, we have to show that cohd(a∩ b) ≥ d− 1. By theMayer–Vietoris sequence 3.2.3, there is an exact sequence

Hd−1a∩b (R) −→ Hd

a+b(R) −→ Hda (R)⊕Hd

b(R).

It follows from the local Lichtenbaum–Hartshorne Vanishing Theorem 8.2.1that Hd

a (R) = Hdb(R) = 0. As δ = 0, the ideal a + b is m-primary, and so

Hda+b(R) = 0 by 1.2.3 and 6.1.4. Altogether we obtain that Hd−1

a∩b (R) = 0,and so cohd(a ∩ b) ≥ d− 1. This proves the claim when δ = 0.

So, let δ > 0 and make the obvious inductive assumption. Set cohd(a∩b) =:

r. As a+ b is not m-primary we can find an element y ∈ m which lies outside


all the minimal prime ideals of a, b and a + b. We write a′ = a + Ry andb′ = b+Ry and note that dimR/(a′ + b′) = δ − 1,

dimR/a′ = dimR/a− 1 > δ − 1 and dimR/b′ = dimR/b− 1 > δ − 1.

As√a′ ∩ b′ =

√(a+Ry) ∩ (b+Ry) =

√(a ∩ b) +Ry,

we have cohd(a′ ∩ b′) ≤ cohd(a∩ b) + 1 = r+ 1 by 8.1.3. Therefore, by theinductive hypothesis, we have r + 1 ≥ d− (δ − 1)− 1; hence r ≥ d− δ − 1.This completes the inductive step, and the proof.

As an application of this we can now prove the following.

19.2.9 Lemma. Assume that (R,m) is a complete local ring. Let b be asecond ideal of R, and assume that a and b are both proper and that

min {dimR/a, dimR/b} > dimR/(a+ b).

Then

dimR/(a+ b) ≥ min {c(R), sdimR− 1} − cohd(a ∩ b),

so that dimR/(a+ b) ≥ min {c(R), sdimR− 1} − ara(a ∩ b).

Proof. Set δ := dimR/(a+b). Let p1, . . . , pn be the distinct minimal primeideals of R.

First, we treat the case where, for all i ∈ {1, . . . , n}, either

dimR/(a+ pi) ≤ δ or dimR/(b+ pi) ≤ δ.

After an appropriate reordering of the pi, there will be an s ∈ N0 such thats ≤ n and dimR/(a + pi) ≤ δ for 1 ≤ i ≤ s and dimR/(b + pj) ≤ δ fors+ 1 ≤ j ≤ n. As max {dimR/(a+ pk) : 1 ≤ k ≤ n} = dimR/a > δ, wesee that s < n. As max {dimR/(b+ pk) : 1 ≤ k ≤ n} = dimR/b > δ, wesee that 1 ≤ s.

Now, let p be a minimal prime ideal of the ideal

c := (p1 ∩ · · · ∩ ps) + (ps+1 ∩ · · · ∩ pn)

such that dimR/p = dimR/c. By 19.2.5, we have dimR/p ≥ c(R). More-over we can choose indices i and j with 1 ≤ i ≤ s < j ≤ n and such thatpi, pj ⊆ p. It follows that

δ ≥ dimR/(a+ pi) ≥ dimR/(a+ p)


and δ ≥ dimR/(b+ pj) ≥ dimR/(b+ p); hence

δ ≥ dimR/((a+ p) ∩ (b+ p)) = dimR/((a ∩ b) + p).

As R/p is catenary (see [50, Theorem 29.4(ii)]), we can write

dimR/((a ∩ b) + p) = dimR/p− ht((a ∩ b) + p)/p.

As dimR/p ≥ c(R) and

ht((a ∩ b) + p)/p ≤ cohd(((a ∩ b) + p)/p) ≤ cohd(a ∩ b)

(by 6.1.6 and 4.2.3), we thus obtain that δ ≥ c(R)− cohd(a ∩ b).Therefore, it remains for us to treat the case in which there exists i ∈

{1, . . . , n} such that

dimR/(a+ pi) > δ and dimR/(b+ pi) > δ.

As dimR/(a+ b+ pi) =: δ′ ≤ δ, it follows from 19.2.8 that

δ ≥ δ′ ≥ dimR/pi − cohd (((a+ pi) ∩ (b+ pi))/pi)− 1.

Observing that dimR/pi ≥ sdimR and (by 4.2.3)

cohd (((a+ pi) ∩ (b+ pi))/pi) = cohd (((a ∩ b) + pi)/pi) ≤ cohd(a ∩ b),

we thus deduce that δ ≥ sdimR− 1− cohd(a ∩ b).

Note. In the First Edition of this book, only the statements involving arith-metic rank appeared in the results corresponding to 19.2.8 and 19.2.9. We arevery grateful to M. Varbaro for pointing out to us that those statements can bestrengthened by replacement of ‘arithmetic rank’ by ‘cohomological dimen-sion’. We have left to the interested reader the formulation of similar strength-enings of some subsequent results in this chapter, such as 19.2.10, 19.2.11 and19.2.12.

We are now in a position to prove the first main result of this chapter, namelythe Connectedness Bound for Complete Local Rings.

19.2.10 Connectedness Bound for Complete Local Rings. Suppose that(R,m) is a complete local ring, and let a be proper. Then

c(R/a) ≥ min {c(R), sdimR− 1} − ara(a).

Proof. Without loss of generality we can, and do, assume that a =√a. Let

p1, . . . , pn be the distinct minimal prime ideals of a, and set c := c(R/a).If n = 1, we have a = p1 and c = dimR/p1 by 19.2.5. Choose a minimal


prime ideal p of R with p ⊆ p1, and observe that ht p1/p ≤ cohd(a) by 4.2.3and 6.1.6. Since R is catenary, we deduce that

c = dimR/p− ht p1/p ≥ sdimR− cohd(a),

from which our claim follows (since cohd(a) ≤ ara(a)).Consider now the case where n > 1. By 19.2.5, there exist two non-empty

subsets A, B of {1, . . . , n} for which A ∪B = {1, . . . , n} and

c = dim(R/((⋂

i∈Api)+(⋂

j∈Bpj)))

;

moreover, we can, and do, assume that A and B are disjoint. Set r :=⋂

i∈Apiand s :=

⋂j∈Bpj ; then dimR/r > c and dimR/s > c (by the final comment

of 19.2.5), and r ∩ s = a. We can now use 19.2.9 to complete the proof.

19.2.11 Corollary. Let (R,m) and a be as in 19.2.10. Then

c(R/a) ≥ c(R)− ara(a)− 1.

If R has more than one minimal prime ideal, then the inequality is strict.

Proof. By 19.2.5, we have c(R) ≤ sdimR, with strict inequality if R hasmore than one minimal prime ideal. The claim therefore follows from 19.2.10.

As another application of 19.2.10 we now prove Grothendieck’s Connected-ness Theorem.

19.2.12 Grothendieck’s Connectedness Theorem. (See [26, Expose XIII,Theoreme 2.1].) Assume that (R,m) is a complete local ring, and let a beproper. Let k ∈ N0 be such that c(R) ≥ k and sdimR ≥ k+1. Then c(R/a) ≥k − ara(a).

Proof. By 19.2.10, we have

c(R/a) ≥ min {c(R), sdimR− 1} − ara(a) ≥ k − ara(a).

19.2.13 Remark. Let the situation be as in 19.2.12. In Grothendieck’s origi-nal version of that result, connectivity and subdimension are considered on thepunctured spectrum Spec◦ (R) of R (see 19.1.13). However, since

c(Spec◦ (R)) = c(R)− 1 and sdim(Spec◦ (R)) = sdimR− 1

(by 19.1.13 and 19.2.4), Grothendieck’s version can be recovered from ours.


19.3 Some local dimensions

Up to now, we have obtained a certain understanding of the connectivity in thespectrum of a complete local Noetherian ring. In order to apply this knowledgeto the non-complete case, we prove the following lemma. The reader is warnedthat our proof of part (iv) of the lemma uses L. J. Ratliff’s Theorem [67] (seealso [50, Theorem 31.7]) that a local ring (R,m) is universally catenary onlyif, for every p ∈ Spec(R) and for every minimal prime P of the ideal pR ofR, we have dim R/P = dimR/p.

19.3.1 Lemma. Assume that (R,m) is local. The following hold:

(i) c(R) ≥ c(R);(ii) if pR ∈ Spec(R) for all minimal prime ideals p ofR, then equality holds

in (i);(iii) sdimR ≥ sdim R; and(iv) if R is universally catenary, then equality holds in (iii).

Proof. Let P be the set of minimal prime ideals of R and Q be the set ofminimal prime ideals of R. Note that, by [50, Theorem 7.3(i)] for example,each p ∈ P is the contraction to R of some member of Q. Also, Q ∩ R ∈ Pfor all Q ∈ Q (by [50, Theorem 15.1(ii)], for example).

(i) By 19.2.5, we can find two non-empty subsets P1 and P2 of P such thatP1 ∪ P2 = P and, if a1 :=

⋂p∈P1

p and a2 :=⋂

p∈P2p, then

c := c(R) = dimR/(a1 + a2).

Let Qi := {Q ∈ Q : Q ∩R ∈ Pi} for i = 1, 2; then Q1 and Q2 are non-empty and such that Q1 ∪ Q2 = Q. Let Bi =

⋂Q∈Qi

Q (i = 1, 2). Then, by19.2.5, we have c(R) ≤ dim R/(B1 +B2). Since aiR ⊆ Bi for i = 1, 2, itfollows that c(R) ≤ dim R/(B1 +B2) ≤ dim R/(a1R+ a2R) = c.

(ii) Suppose that pR ∈ Spec(R) for all p ∈ P . By 19.2.5, we can findtwo non-empty subsets Q3 and Q4 of Q such that Q3 ∪ Q4 = Q and, ifBi =⋂

Q∈QiQ for i = 3, 4, then c := c(R) = dim R/(B3 +B4).

Set Pi := {Q ∩R : Q ∈ Qi} and ai =⋂

p∈Pip for i = 3, 4. Since each

p ∈ P is the contraction to R of some minimal prime ideal of R, we haveP3 ∪ P4 = P . In particular, we have (Q ∩ R)R = Q for all Q ∈ Q, and sowe can use [50, Theorem 7.4(ii)] to see that aiR = Bi for i = 3, 4. Hence, by19.2.5,

c = dim R/(B3 +B4) = dim R/(a3R+ a4R) = dimR/(a3 + a4) ≥ c(R).

The claim follows from this and part (i).

19.3 Some local dimensions 417

(iii) Let p ∈ P be such that dimR/p = sdimR. Now there exists Q ∈ Qsuch that Q ∩R = p. Then

sdim R ≤ dim R/Q ≤ dim R/pR = dimR/p = sdimR.

(iv) Assume now that R is universally catenary. Let Q ∈ Q be such thatsdim R = dim R/Q. Now p := Q ∩ R ∈ P; by Ratliff’s Theorem [50,Theorem 31.7], we have dim R/Q = dimR/p. Hence

sdim R = dim R/Q = dimR/p ≥ sdimR.

The claim follows from this and part (iii).

19.3.2 Definitions. Let T be a Noetherian topological space, and let p ∈ T .The local dimension of T at p, denoted by dimp T , is defined as the supre-

mum of the lengths of all strictly descending chains Z0 ⊃ Z1 ⊃ · · · ⊃ Zn

of closed irreducible subsets of T which all contain p if this supremum exists,and∞ otherwise. Thus, by 19.1.3, dimp T is a non-negative integer or∞.

Note that, for p ∈ Spec(R), we have dimp Spec(R) = dimRp.Let V be an affine variety over the algebraically closed field K. Since, in

an integral domain R′ which is a finitely generated K-algebra, every maximalideal has height equal to dimR′ (see, for example, [81, 14.33]), it follows thatdimq V = dimOV,q = dimO(V ) = dimV for all q ∈ V .

If T1, . . . , Tr are the irreducible components of T which contain p, we writeT p := T1 ∪ · · · ∪ Tr, and call this subspace T p of T the p-component of T .

The local connectedness dimension of T at p, denoted by cp(T ), is definedto be the minimum of the local dimensions at p of those closed subsets Z of thep-component T p of T which contain p and for which T p \ Z is disconnected.Thus

cp(T ) := min {dimp Z : Z ⊆ T p, p ∈ Z, Z is closed

and T p \ Z is disconnected} .

The local subdimension sdimp T of T at p is defined as the minimum of thelocal dimensions at p of the irreducible components of T which contain p. (Ofcourse, p does belong to at least one irreducible component of T , by 19.1.3.)

19.3.3 �Exercise. Let T be a Noetherian topological space and let p ∈ T .

(i) Show that dimp Z = dimp(Tp ∩Z) and sdimp Z = sdimp(T

p ∩Z) foreach closed subset Z of T for which p ∈ Z.

(ii) Let T1, . . . , Tr be the (distinct) irreducible components of T that containp. Show that, with the notation of 19.1.14,

cp(T ) = min{dimp

((⋃i∈ATi)∩(⋃

j∈BTj))

: (A,B) ∈ S(r)}.


(Use the ideas of the proof of Theorem 19.1.15.)

The next exercise provides justification for the appearance of the word‘local’ in the definitions in 19.3.2.

19.3.4 �Exercise. Let T be a Noetherian topological space and let p ∈ T ; letU be an open subset of T which contains p. Show that

(i) Up = T p ∩ U ;(ii) dimp T = dimp U ;

(iii) sdimp T = sdimp U ; and(iv) cp(T ) = cp(U).

(Use Exercise 19.1.2(i),(iii),(v) to establish the existence of a bijection be-tween the set of irreducible closed subsets of T containing p and the set ofirreducible closed subsets of U containing p; then use Exercise 19.3.3(ii).)

19.3.5 Exercise. Let p ∈ V := VA3(X1X2X3). Calculate cp(V )

(i) when p = (0, 0, 0);(ii) when exactly two of the co-ordinates of p are 0; and

(iii) in all other cases.

19.3.6 �Exercise. Let p ∈ Spec(R). Show that cp(Spec(R)) = c(Rp) andsdimp(Spec(R)) = sdim(Rp).

We wish to study the connectivity of varieties over an algebraically closedfield. We remind the reader about some elementary facts concerning such va-rieties.

19.3.7 Reminders. Let K be an algebraically closed field, and let r ∈ N. LetV ⊆ Pr(K) be a quasi-projective variety.

(i) Regard the polynomial ring K[X0, X1, . . . , Xr] as the coordinate ringO(Ar+1(K)), and let i ∈ {0, . . . , r}. Let UiPr(K) denote the opensubset of Pr(K) given by

UiPr(K) = Pr(K) \ VPr(K)(Xi)

= {(c0 : · · · : ci : · · · : cr) ∈ Pr(K) : ci = 0} .

There is an isomorphism of varieties σi : Ar(K)∼=−→ UiPr(K) given

by σi((a1, . . . , ar)) = (a1 : · · · : ai : 1 : ai+1 : · · · : ar) for all(a1, . . . , ar) ∈ Ar(K). Note that Pr(K) =

⋃rj=0 UjPr(K); it follows

that the quasi-projective variety V ⊆ Pr(K) has a finite covering byopen sets each of which is quasi-affine (in the sense that it is isomorphic


to a quasi-affine variety over K). As any quasi-affine variety has a finitecovering by open sets which are affine, we thus see that quasi-projectivevarieties also have finite coverings by affine open sets.

The above isomorphisms show that affine varieties, and also quasi-affine varieties, are quasi-projective: the reader should remember that,for us, the word ‘variety’, as introduced in 15.2.1(ii), is synonymous with‘quasi-projective variety’ (and does not mean the same as ‘(abstract) va-riety’ in the sense of Hartshorne [30, p. 105]).

(ii) Let p ∈ V . By part (i), there is an open subset U of V such that p ∈ Uand U is an affine variety. We can, and do, identify the local ring OV,p

of p on V with OU,p = O(U)IU (p) (see 6.4.1). Let C be a closed subsetof V such that p ∈ C. The vanishing ideal IU (C ∩ U) was defined in15.1.2. Recall that the local vanishing ideal IV,p(C) of C at p is the(radical) ideal of OV,p consisting of all germs of regular functions f ∈O(W ) defined on some open neighbourhood W of p in V and such thatf(W ∩ C) = 0; hence

IV,p(C) = IU,p(C ∩ U) = IU (C ∩ U)O(U)IU (p) = IU (C ∩ U)OU,p.

19.3.8 �Exercise. Let K be an algebraically closed field.

(i) Let V be an affine variety over K. Let b be a proper ideal of O(V ), andlet V (b) denote the closed subset of V determined by b; let p ∈ V (b).Show that cp(V (b)) = c(OV,p/bOV,p) and

sdimp V (b) = sdim(OV,p/bOV,p)

= min {dimO(V )/q : q is a minimal prime of b

such that p ∈ V (q)} .

(ii) Let V ′ be a variety over K. Let W be a non-empty closed subset of V ′,and let p ∈ W . Show that, with the notation of 19.3.7(ii), sdimpW =

sdim(OV ′,p/IV ′,p(W )) and cp(W ) = c(OV ′,p/IV ′,p(W )). (Use part(i) and 19.3.4.)

19.3.9 Definition. Let V be a variety over the algebraically closed field K.Let W be a non-empty closed subset of V , and let p ∈ W . We define theformal connectedness dimension of W at p, denoted cp(W ), by

cp(W ) = c(OV,p/IV,p(W )OV,p

)= c ((OV,p/IV,p(W ))) .

Here again, IV,p(W ) is the local vanishing ideal of W at p: see 19.3.7(ii).


19.3.10 Remark. Let V , W and p be as in 19.3.9. Then it follows from19.3.1(i) and 19.3.8(ii) that cp(W ) ≥ cp(W ). Moreover, if V is affine and b isany ideal of O(V ) for which W = V (b), then cp(W ) = c ((OV,p/bOV,p)).19.3.11 Exercise. Let

V1 := VA2(X3 − Y 2) and V2 := VA2(X2 +X3 − Y 2);

also let

V := VA4(X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ),

as in Example 2.3.7.

(i) Calculate c(0,0)(Vi) and c(0,0)(Vi) for i = 1, 2.(ii) Show that c(0,0,0,0)(V ) = 0. (You might find 8.2.13(iv) and 19.1.11

helpful.)

19.3.12 Definition. Let V be a variety over the algebraically closed field K.Let W and Z be closed subsets of V with Z ⊆ W , and let p ∈ Z. Note thatIV,p(W ) ⊆ IV,p(Z).

We define the (local) arithmetic rank of Z at p with respect to W , denotedaraW,p(Z), by

araW,p(Z) = ara (IV,p(Z)/IV,p(W )) .

(Here, IV,p(Z)/IV,p(W ) is considered as an ideal of OV,p/IV,p(W ).)

19.3.13 Definition. Let V be a variety over the algebraically closed field K.Let W be a non-empty closed subset of V , and let p ∈ W . We extend theterminology of 8.2.15 and say that W is analytically reducible at p preciselywhen OV,p/IV,p(W )OV,p has more than one minimal prime ideal. Otherwise,W is said to be analytically irreducible at p. Note that OV,p/IV,p(W )OV,p

∼=(OV,p/IV,p(W )), and by [50, Theorem 32.2(i) and p. 259, Remark 1], W isanalytically irreducible at p if and only if (OV,p/IV,p(W )) is a domain.

19.3.14 Remark. Note that, in the situation of 19.3.9, 19.3.12 and 19.3.13,the invariants cp(W ) and araW,p(Z), and the notion (for W ) of analytical re-ducibility at p, do not depend on the ambient variety V , and, indeed, remainunchanged if V , W and Z are replaced, respectively, by U , W ∩U and Z ∩U ,where U is any open subset of V containing p.

We can now deduce the following from 19.2.10 and 19.2.11.

19.3.15 Proposition. Let V be a variety over the algebraically closed fieldK. Let W and Z be closed subsets of V with Z ⊆W , and let p ∈ Z. Then


(i) cp(Z) ≥ min {cp(W ), sdimpW − 1} − araW,p(Z);(ii) cp(Z) ≥ cp(W )− araW,p(Z)− 1; and

(iii) if W is analytically reducible at p, then the inequality in part (ii) isstrict.

Proof. Let R := OV,p/IV,p(W ) and a := IV,p(Z)/IV,p(W ). Then

cp(W ) = c(R) and cp(Z) = c(OV,p

/IV,p(Z)OV,p

)= c(R/aR).

Next, araW,p(Z) = ara(a) (see 19.3.12), and so it follows from 19.2.7(ii) thataraW,p(Z) ≥ ara(aR). Also, sdimpW = sdimR = sdim R, by 19.3.8 and19.3.1(iv).

We can now apply the Connectedness Bound for Complete Local Rings19.2.10 to prove part (i), and 19.2.11 to prove parts (ii) and (iii).

19.3.16 Exercise. Let V be the affine variety in A4 of Example 2.3.7 givenby

V := VA4(X1X4 −X2X3, X21X3 +X1X2 −X2

2 , X33 +X3X4 −X2

4 ).

Use 19.3.11(ii) and 19.3.15 to show that araA4,(0,0,0,0)(V ) ≥ 3.

It seems natural to ask whether, if the formal connectedness dimensions inthe inequality of 19.3.15(ii) are replaced by the corresponding local connect-edness dimensions, the resulting statement is still true. We shall now providean example which shows that this is not always the case.

19.3.17 Example. Let R be the subring of R′ := C[X1, X2, X3] = O(A3)

given by R := C[X1, X2, X1X3, X2X3, X23 − 1, X3(X

23 − 1)].

Let Y1, . . . , Y6 be independent indeterminates over C, and let

f : O(A6) = C[Y1, Y2, Y3, Y4, Y5, Y6] −→ R

be the C-algebra homomorphism such that f(Y1) = X1, f(Y2) = X2, f(Y3)= X1X3, f(Y4) = X2X3, f(Y5) = X2

3 − 1 and f(Y6) = X3(X23 − 1).

Then p := Ker f is a prime ideal of O(A6) (since R is an integral domain);let V := VA6(p) denote the affine variety determined by p, so that there is anatural isomorphism of C-algebras O(V ) ∼= R.

The inclusion mappingR→ R′ = O(A3), which makesR′ integral over itssubring R, therefore gives rise to a finite morphism of varieties α : A3 → V

such that

α((c1, c2, c3)) = (c1, c2, c1c3, c2c3, c23 − 1, c3(c

23 − 1))

for all (c1, c2, c3) ∈ A3. Let p = (0, 0, 1), q = (0, 0,−1) ∈ A3, and let


0 denote (0, 0, 0, 0, 0, 0) ∈ A6. Now it is straightforward to check that α� :A3 \ {p, q} −→ V \ {0} is an isomorphism of (quasi-affine) varieties, withinverse β : V \ {0} −→ A3 \ {p, q} given by

β((d1, d2, d3, d4, d5, d6)) =

⎧⎪⎪⎨⎪⎪⎩(d1, d2, d3/d1) if d1 = 0,

(d1, d2, d4/d2) if d2 = 0,

(d1, d2, d6/d5) if d5 = 0

(for all (d1, d2, d3, d4, d5, d6) ∈ V \ {0}).Let W := V ∩ VA6(Y5), a closed subset of V such that 0 ∈ W . Note that

araV,0(W ) = ara(IV (W )OV,0) = 1.Let E1 := VA3(X3 − 1) and E2 := VA3(X3 + 1). Since α is finite, it is a

closed map, and so α(E1) and α(E2) are closed subsets of W and we havedim(α(Ei)) = dimEi = 2 for i = 1, 2.

Since α−1(W ) = VA3(X23 − 1) = E1 ∪ E2, it follows that W can be

expressed as W = α(E1) ∪ α(E2), where α(E1) and α(E2) are closed ir-reducible subsets of dimension 2. Therefore α(E1) and α(E2) must be theirreducible components of W . Hence, by 19.3.3(ii), we have

c0(W ) = dim0(α(E1) ∩ α(E2)) = dim0{0} = 0.

On the other hand, since V is irreducible, c0(V ) = dim0 V = 3. We thereforehave the strict inequality

c0(W ) = 0 < 1 = 3− 1− 1 = c0(V )− araV,0(W )− 1.

Thus, if the formal connectedness dimensions in the inequality of 19.3.15(ii)are replaced by the corresponding local connectedness dimensions, the result-ing statement is not always true.

19.4 Connectivity of affine algebraic cones

Connectedness dimensions of affine algebraic cones behave particularly satis-factorily, and the next two lemmas provide the key to this good behaviour. Weshall use the notation of 15.1.2 for affine algebraic cones.

19.4.1 Lemma. Let K be an algebraically closed field, let r ∈ N, and letC ⊆ Ar(K) be an affine algebraic cone. Then C is irreducible (that is, C is avariety) if and only if C is analytically irreducible at 0.

Proof. (⇒) Assume that C is irreducible. Then, by 15.1.2(ii), the ring O(C)

19.4 Connectivity of affine algebraic cones 423

is a positively graded *local homogeneous domain with O(C)0 = K; fur-thermore, O(C)+ is the unique *maximal graded ideal of O(C), and is ac-tually maximal and equal to IC(0). Hence OC,0

∼= O(C)O(C)+ . Therefore,

by 14.1.15(iv),(v), the completion OC,0 is a domain, and so C is analyticallyirreducible at 0.(⇐) Assume that C is reducible, so that C has more than one irreducible

component. By 19.1.4(i), these irreducible components all contain 0. Thismeans that the local vanishing ideal IAr(K),0(C) of C at 0 has more than oneminimal prime. HenceOC,0 is not a domain, and neither is its completion.

19.4.2 Lemma. Let V be an affine variety over the algebraically closed fieldK, let W ⊆ V be a closed subset, let p ∈ W , and assume that all the irre-ducible components of W which contain p are analytically irreducible at p.Then cp(W ) = cp(W ).

Proof. Let b := IV (W ) and R := OV,p/bOV,p. By 19.3.8(ii) and 19.3.10,it is enough for us to show that c(R) = c(R); therefore, by 19.3.1(ii), it isenough for us to show that, for each minimal prime ideal p of R, we havepR ∈ Spec(R), and this is what we shall do. Now p = qOV,p/bOV,p for someminimal prime ideal q of b such that q ⊆ IV (p). But then Z := V (q), theclosed subset of V determined by q, is an irreducible component of W whichcontains p. By hypothesis, Z is analytically irreducible at p, and so, by 19.3.13,the ring (OV,p/IV,p(Z)) = (OV,p/qOV,p) is a domain. Hence (R/p) is adomain, so that pR ∈ Spec(R).

The next lemma establishes a very useful fact about the connectedness di-mensions of an affine algebraic cone C: the formal connectedness dimensionof C at the origin and the local connectedness dimension of C at the origin areequal, and they are both equal to the connectedness dimension c(C).

19.4.3 Proposition. Let K be an algebraically closed field, let r ∈ N, andlet C ⊆ Ar(K) be an affine algebraic cone. As in 15.1.2, we use 0 to denotethe origin of Ar(K). Then

(i) dimC = dim0 C,(ii) sdimC = sdim0 C, and

(iii) c(C) = c0(C) = c0(C).

Proof. By 19.1.4(i), all the irreducible components ofC are themselves affinealgebraic cones in Ar(K), and so contain 0. Now the dimension of an ir-reducible affine algebraic cone C ′ satisfies dimC ′ = dim0 C

′ (by 19.3.2).The claims in parts (i) and (ii) now follow immediately, while the equalityc(C) = c0(C) follows from these considerations, 19.1.15 and 19.3.3(ii).


Finally, Lemma 19.4.1 shows that all the irreducible components of C are ana-lytically irreducible at 0, and so Lemma 19.4.2 shows that c0(C) = c0(C).

We can now deduce the following corollary from 19.3.15.

19.4.4 Corollary. Let K be an algebraically closed field, let r ∈ N, letD,E ⊆ Ar(K) be affine algebraic cones such that E ⊆ D. Then

(i) c(E) ≥ min {c(D), sdimD − 1} − araD,0(E);(ii) c(E) ≥ c(D)− araD,0(E)− 1; and

(iii) if D is reducible, then the inequality in part (ii) is strict.

Proof. In view of Proposition 19.4.3, statements (i) and (ii) follow immedi-ately from the corresponding statements of 19.3.15 (used with V = Ar(K),W = D and Z = E). Statement (iii) follows from 19.3.15(iii) and Lemma19.4.1.

19.5 Connectivity of projective varieties

In view of the close relationship between affine algebraic cones and projectivealgebraic sets (see 15.2.1(iii)), we can exploit 19.4.4 to study the connectivityof closed sets in projective varieties. We intend to do this, but first we need afew preliminaries.

19.5.1 Lemma. Let K be an algebraically closed field, let r ∈ N, let W ⊆Pr(K) be a non-empty closed subset of Pr(K), and consider the affine coneCone(W ) ⊆ Ar+1(K) over W , as in 15.2.1(iii). Then

(i) dimW = dim(Cone(W ))− 1 = dim0(Cone(W ))− 1;(ii) sdimW = sdim(Cone(W ))− 1 = sdim0(Cone(W ))− 1; and

(iii) c(W ) = c(Cone(W ))− 1 = c0(Cone(W ))− 1 = c0(Cone(W ))− 1.

Proof. Let the distinct irreducible components of W be W1, . . . ,Wn. By19.1.4(ii), Cone(W1), . . . ,Cone(Wn) are the irreducible components (againdistinct) of Cone(W ). Furthermore, it follows from 15.2.1(vii) that

dimWi = dim(Cone(Wi))− 1 for i = 1, . . . , n.

In view of 19.4.3, the claims in statements (i) and (ii) are now immediate.(iii) With the notation of 19.1.14, let (A,B) ∈ S(r). By 15.2.1(v), we have

Cone((⋃

i∈AWi

)∩(⋃

j∈B Wj

))=(⋃

i∈A Cone(Wi))∩(⋃

j∈B Cone(Wj)).

19.5 Connectivity of projective varieties 425

Since Cone(Wi) (i = 1, . . . , n) are the irreducible components of Cone(W ),it now follows from part (i) and 19.1.15 that c(W ) = c(Cone(W ))−1. Finally,we can use 19.4.3 to complete the proof.

19.5.2 Definition. Let K be an algebraically closed field and let r ∈ N. LetW and Z be non-empty closed subsets of Pr(K) with Z ⊆ W . Note thatIAr+1(K)(Cone(W )) ⊆ IAr+1(K)(Cone(Z)), and that

IAr+1(K)(Cone(Z))/IAr+1(K)(Cone(W ))

is an ideal of O(Ar+1(K))/IAr+1(K)(Cone(W )).We define the arithmetic rank of Z with respect to W , denoted araW (Z),

by

araW (Z) = ara(IAr+1(K)(Cone(Z))/IAr+1(K)(Cone(W ))

).

It should be noted that, by 19.2.7(ii) and 19.3.7(ii), we have

araW (Z) ≥ araCone(W ),0(Cone(Z)).

We are now able to state and prove a form of the Bertini–GrothendieckConnectivity Theorem.

19.5.3 The Bertini–Grothendieck Connectivity Theorem. (See [26, Expo-se XIII, Corollaire 2.3].) Let K be an algebraically closed field and let r ∈ N.Let W and Z be non-empty closed subsets of Pr(K) with Z ⊆W . Then

(i) c(Z) ≥ min {c(W ), sdimW − 1} − araW (Z);(ii) c(Z) ≥ c(W )− araW (Z)− 1; and

(iii) if W is reducible, then the inequality in part (ii) is strict.

Proof. By Lemma 19.5.1, we have sdimW = sdim(Cone(W ))− 1,

c(W ) = c(Cone(W ))− 1 and c(Z) = c(Cone(Z))− 1.

Moreover, araW (Z) ≥ araCone(W ),0(Cone(Z)) by 19.5.2, while 15.2.1(v)shows that Cone(W ) is reducible if W is reducible. Therefore, all three state-ments follow from the corresponding statements of Corollary 19.4.4.

Let K be an algebraically closed field and let r ∈ N. Recall that a hyper-surface in Pr(K) is a closed set VPr(K)(f) defined by a single homogeneouspolynomial f ∈ K[X0, X1, . . . , Xr] of positive degree. We can now deducethe following corollary from the Bertini–Grothendieck Connectivity Theorem19.5.3.


19.5.4 Corollary. Let K be an algebraically closed field, let r ∈ N, let Wbe a non-empty closed subset of Pr(K), and let H1, . . . , Ht ⊆ Pr(K) behypersurfaces. Then

(i) c(W ∩H1 ∩ · · · ∩Ht) ≥ min {c(W ), sdimW − 1} − t;(ii) c(W ∩H1 ∩ · · · ∩Ht) ≥ c(W )− t− 1; and

(iii) if W is reducible, then the inequality in part (ii) is strict.

Proof. For i = 1, . . . , t, there is a homogeneous polynomial (of positive de-gree) fi ∈ K[X0, X1, . . . , Xr] such that Hi = VPr(K)(fi). Set

Z :=W ∩H1 ∩ · · · ∩Ht = VPr(K)

(IPr(K)(W ) + (f1, . . . , ft)

),

so that IPr(K)(Z) =√IPr(K)(W ) + (f1, . . . , ft). This equation shows that

araW (Z) = ara(IPr(K)(Z)/IPr(K)(W )

)≤ t. The claims now follow from

application of 19.5.3.

19.5.5 Exercise. Let K be an algebraically closed field and let r ∈ N.

(i) Prove the ‘classical’ form of Bertini’s Connectivity Theorem, that, ifV ⊆ Pr(K) is a projective variety such that dimV > 1 andH ⊆ Pr(K)

is a hypersurface, then V ∩H is connected.(ii) Provide an example which shows that if the irreducibility of V is dropped

from the statement in part (i) above, then the resulting statement is nolonger always true.

(iii) Provide an example of an affine variety V ⊆ Ar(K) with dimV > 1

such that V ∩H is disconnected for a hyperplane H ⊆ Ar(K).

19.6 Connectivity of intersections

Our next aim is the study of the connectivity of the intersection of two affine al-gebraic cones. For this, we recall, in 19.6.1 and 19.6.2 below, some elementaryfacts about products of affine algebraic sets.

19.6.1 Reminder and Remark. Let K be an algebraically closed field andlet r, s ∈ N. We consider polynomial rings K[X1, . . . , Xr] = O(Ar(K)),K[Y1, . . . , Ys] = O(As(K)) and

K[X1, . . . , Xr;Y1, . . . , Ys] = O(Ar+s(K)).

Let a be an ideal of K[X1, . . . , Xr] and b be an ideal of K[Y1, . . . , Ys], andset V := VAr(K)(a) and W := VAs(K)(b). Recall that the product of V and

19.6 Connectivity of intersections 427

W is just the Cartesian product V ×W ⊆ Ar+s(K); it is an affine algebraicset because

V ×W= VAr+s(K)(aK[X1, . . . , Xr;Y1, . . . , Ys] + bK[X1, . . . , Xr;Y1, . . . , Ys]).

(i) Recall from Hartshorne [30, Chapter I, Exercise 3.15] that, when V ⊆Ar(K) and W ⊆ As(K) are irreducible, then V ×W ⊆ Ar+s(K) isagain irreducible and, moreover, dim(V ×W ) = dimV + dimW .

(ii) Now suppose r = s. Let Δ(r) be the diagonal

{(c, c) : c ∈ Ar(K)} ⊆ A2r(K).

Note that Δ(r) = VA2r(K)(X1 − Y1, . . . , Xr − Yr), so that Δ(r) is ir-reducible, and also that there is the diagonal isomorphism of varietiesδ(r) : Ar(K) −→ Δ(r) for which δ(r)(c) = (c, c) for all c ∈ Ar(K). IfV,W ⊆ Ar(K) are closed subsets of Ar(K), then

δ(r)(V ∩W ) = (V ×W ) ∩Δ(r),

and δ(r) gives rise to a homeomorphism

δ(r)� : V ∩W ≈−→ (V ×W ) ∩Δ(r).

19.6.2 �Exercise. Let K be an algebraically closed field, let r, s ∈ N, andlet V ⊆ Ar(K) and W ⊆ As(K) be non-empty closed sets. Consider theirproduct V ×W ⊆ Ar+s(K), as in 19.6.1.

(i) Show that, if V and W are affine algebraic cones, then V ×W is againan affine algebraic cone.

(ii) Let V1, . . . , Vp (respectively W1, . . . ,Wq) be the distinct irreduciblecomponents of V (respectively W ). Show that the products

Vi ×Wj (i = 1, . . . , p, j = 1, . . . , q)

are the (distinct) irreducible components of V ×W .(iii) Show that dim(V × W ) = dimV + dimW and sdim(V × W ) =

sdimV + sdimW .

19.6.3 Lemma. Let K be an algebraically closed field, let r, s ∈ N, and letV ⊆ Ar(K) and W ⊆ As(K) be non-empty closed sets. Then

(i) c(V ×W ) ≥ min {sdimV + c(W ), sdimW + c(V )};(ii) c(V ×W ) ≥ c(V ) + c(W ); and

(iii) if V and W are both reducible, then the inequality in part (ii) is strict.


Proof. Let V1, . . . , Vp be the distinct irreducible components of V and letW1, . . . ,Wq be the distinct irreducible components of W . By 19.6.2(ii), theproducts Vi ×Wj (i = 1, . . . , p, j = 1, . . . , q) are the (distinct) irreduciblecomponents of V ×W . By 19.1.15, there are two non-empty subsets A,B ⊆{1, . . . , p} × {1, . . . , q} such that A ∪B = {1, . . . , p} × {1, . . . , q} and

c(V ×W ) = dim((⋃

(i,j)∈AVi ×Wj

)∩(⋃

(k,l)∈BVk ×Wl

)).

The argument now splits into two cases. Suppose first that there exists i0 ∈{1, . . . , p} such that there are indices j, l ∈ {1, . . . , q} for which (i0, j) ∈ A

and (i0, l) ∈ B. Then

A := {j ∈ N : (i0, j) ∈ A} and B := {l ∈ N : (i0, l) ∈ B}

are non-empty sets such that A ∪B = {1, . . . , q}. Set

Z :=(⋃

j∈AWj

)∩(⋃

l∈BWl

).

By 19.1.15, dimZ ≥ c(W ). Let Y be an irreducible component of Z suchthat dimY = dimZ. Then there exist j ∈ A and l ∈ B such that Y ⊆ Wj

and Y ⊆ Wl. Therefore Vi0 × Y ⊆ (Vi0 ×Wj) ∩ (Vi0 ×Wl), and so (see19.6.2(iii))

c(V ×W ) ≥ dim ((Vi0 ×Wj) ∩ (Vi0 ×Wl))

≥ dim(Vi0 × Y ) = dimVi0 + dimY ≥ sdimV + c(W ).

We now deal with the remaining case, when there is no index i0 with theproperties described above. Choose i ∈ {1, . . . , p} such that (i, j′) ∈ A forsome j′ ∈ {1, . . . , q}. Then, for all j ∈ {1, . . . , q}, we must have (i, j) ∈ B,so that (i, j) ∈ A. But there is also a pair (k, j0) ∈ B; thus (i, j0) ∈ A and(k, j0) ∈ B, and we can use the argument of the previous paragraph, with theroles of V and W interchanged, to deduce that c(V ×W ) ≥ sdimW + c(V ).This proves statement (i).

Parts (ii) and (iii) are now immediate from part (i) and 19.2.2.

19.6.4 Exercise. Let V ′ := VA2(XY ) and W ′ := VA2(X) (with the notationof 2.3.1).

(i) Calculate sdim(V ′ × V ′) and c(V ′ × V ′).(ii) Calculate sdim(V ′ ×W ′) and c(V ′ ×W ′).

(iii) Show that it is possible for the inequality in 19.6.3(ii) to be an equalitywhen just one of V and W is reducible.


19.6.5 Proposition: the Intersection Inequality for the Connectedness

Dimensions of Affine Algebraic Cones. Let K be an algebraically closedfield, let r ∈ N, and let C,D ⊆ Ar(K) be affine algebraic cones.

(i) We have

c(C ∩D)

≥ min {sdimC + sdimD − 1, sdimC + c(D), sdimD + c(C)} − r.

(ii) Furthermore,

c(C ∩D) ≥ min {sdimC + c(D), sdimD + c(C)} − r − 1,

with strict inequality if C or D is reducible.(iii) Consequently, c(C ∩D) ≥ c(C) + c(D)− r− 1+ ε, where ε = 0, 1 or

2 according as none, one or both of C and D are reducible.

Proof. By 19.6.2(i), the product C × D ⊆ A2r(K) is an affine algebraiccone, and by 19.6.2(iii) we have sdim(C ×D) = sdimC + sdimD. Also, by19.6.3, we have c(C ×D) ≥ min {sdimC + c(D), sdimD + c(C)} .

As in 19.6.1, write K[X1, . . . , Xr;Y1, . . . , Yr] = O(A2r(K)), and notethat the diagonal Δ(r) ⊆ A2r(K) is an affine algebraic cone in A2r(K). SinceIA2r(K)(Δ

(r)) = (X1 − Y1, . . . , Xr − Yr), we have

IA2r(K)((C ×D) ∩Δ(r)) =√IA2r(K)(C ×D) + (X1 − Y1, . . . , Xr − Yr).

It therefore follows from 19.2.7(ii) that

araC×D,0((C ×D) ∩Δ(r))

≤ ara(IA2r(K)((C ×D) ∩Δ(r))/IA2r(K)(C ×D)

)≤ r.

If we now apply 19.4.4(i) to the two affine algebraic cones (C ×D) ∩Δ(r) ⊆C ×D in A2r(K), we obtain

c((C ×D) ∩Δ(r))

≥ min {sdimC + sdimD − 1, sdimC + c(D), sdimD + c(C)} − r.

In view of the homeomorphism provided by 19.6.1(ii), we have

c((C ×D) ∩Δ(r)) = c(C ∩D),

and so the claim in (i) is proved.If C orD is reducible, then so too is C×D (by 19.6.2(ii)); we can therefore

deduce part (ii) from 19.4.4(ii),(iii).


In view of 19.2.2, part (iii) follows from part (ii) if at least one of C, Dis irreducible, and so we deal now with the case where both C and D arereducible. Then, by 19.2.2, we have sdimC ≥ c(C) + 1 and also sdimD ≥c(D) + 1, so that

sdimC+c(D) ≥ c(C)+c(D)+1 and sdimD+c(C) ≥ c(C)+c(D)+1.

We use these inequalities in conjunction with part (ii) to see that

c(C ∩D) ≥ min {sdimC + c(D), sdimD + c(C)} − r − 1 + 1

≥ c(C) + c(D) + 1− r − 1 + 1 = c(C) + c(D)− r − 1 + 2.

We now apply Proposition 19.6.5 to affine cones over projective algebraicsets to deduce the following theorem.

19.6.6 Theorem: the Intersection Inequality for the Connectedness

Dimensions of Projective Algebraic Sets. Let K be an algebraically closedfield, let r ∈ N, and let V,W ⊆ Pr(K) be non-empty closed sets.

(i) We have

c(V ∩W ) + r

≥ min {sdimV + sdimW − 1, sdimV + c(W ), sdimW + c(V )} .

(ii) Furthermore,

c(V ∩W ) + r ≥ min {sdimV + c(W ), sdimW + c(V )} − 1,

with strict inequality if V or W is reducible.(iii) Consequently, c(V ∩W ) + r ≥ c(V ) + c(W ) − 1 + ε, where ε = 0, 1

or 2 according as none, one or both of V and W are reducible.

Proof. Let C := Cone(V ) ⊆ Ar+1(K) and D := Cone(W ) ⊆ Ar+1(K)

be the affine cones in Ar+1(K) over V and W respectively. Observe thatC ∩ D = Cone(V ) ∩ Cone(W ) = Cone(V ∩ W ). By Lemma 19.5.1, wehave sdimC = sdimV + 1, sdimD = sdimW + 1,

c(C) = c(V ) + 1, c(D) = c(W ) + 1, c(C ∩D) = c(V ∩W ) + 1.

Moreover, it follows from 15.2.1(v) that C is reducible if and only if V is,and D is reducible if and only if W is. All three parts of the theorem nowfollow from the corresponding parts of Proposition 19.6.5 applied to the affinealgebraic cones C,D ⊆ Ar+1(K).

Part of Theorem 19.6.6(iii) amounts to the following formulation of the Con-nectivity Theorem due to W. Fulton and J. Hansen and to G. Faltings.


19.6.7 Corollary: the Connectivity Theorem of Fulton–Hansen and

Faltings. (See [20] and [15].) LetK be an algebraically closed field, let r ∈ N,and let V,W ⊆ Pr(K) be projective varieties. Then

c(V ∩W ) ≥ dimV + dimW − r − 1.

Proof. By 19.1.10(iii), we have c(V ) = dimV and c(W ) = dimW , and sothe claim follows from 19.6.6(iii).

19.6.8 Corollary (W. Fulton and J. Hansen [20, Corollary 1]). Let K be analgebraically closed field, let r ∈ N, and let V,W ⊆ Pr(K) be projectivevarieties such that dimV + dimW > r. Then V ∩W is connected.

Proof. By Corollary 19.6.7, we have c(V ∩W ) ≥ 0, so that V ∩W is con-nected by 19.1.10(i).

19.6.9 Example. Interpret the polynomial ring C[X1, X2, X3] as O(A3), asin 2.3.1. In A3, we consider the two surfaces

V◦:= VA3(X1 −X2X3) and W

◦:= VA3(X1 +X2

2 −X2X3 − 1).

Then

V◦ ∩W◦ = VA3(X1 −X2X3, X1 +X2

2 −X2X3 − 1)

= VA3(X1 −X2X3, X22 − 1)

= VA3(X1 −X2X3, X2 + 1) ∪ VA3(X1 −X2X3, X2 − 1)

= VA3(X1 +X3, X2 + 1) ∪ VA3(X1 −X3, X2 − 1),

so that V◦ ∩W◦ is the union of the two lines L1

◦= VA3(X1 +X3, X2 + 1) and

L2◦= VA3(X1 −X3, X2 − 1) and is disconnected. On the other hand,

dimV◦+ dimW

◦= 2 + 2 > 3

and V◦

and W◦

are both irreducible (as their defining polynomials are). There-fore, the analogue of 19.6.8 for affine varieties is not always true.

Now interpret the polynomial ring C[X0, X1, X2, X3] as O(A4) and con-sider the projective varieties V,W ⊆ P3 defined by

V := VP3(X0X1 −X2X3) and W := VP3(X0X1 +X22 −X2X3 −X2

0 ).

(Note that bothX0X1−X2X3 andX0X1+X22−X2X3−X2

0 are irreduciblepolynomials. One can think of V andW as the projective closures of V

◦andW

◦

respectively with respect to the isomorphism of varieties σ0 : A3∼=−→ U0P3

of 19.3.7(i): see [30, Chapter I, Exercise 2.9].) Since dimV = dimW = 2,Corollary 19.6.8 tells us that V ∩W is connected.


Indeed, we have

V ∩W= VP3(X0X1 −X2X3, X0X1 +X2

2 −X2X3 −X20 )

= VP3(X1 +X3, X2 +X0) ∪ VP3(X1 −X3, X2 −X0) ∪ VP3(X0, X2)

= L1 ∪ L2 ∪ L,

where L1 := VP3(X1+X3, X2+X0) and L2 := VP3(X1−X3, X2−X0) arethe projective closures of L1

◦and L2

◦respectively with respect to σ0, and L is

the ‘line at infinity’ VP3(X0, X2) which intersects L1 at p := (0 : 1 : 0 : −1)and L2 at q := (0 : 1 : 0 : 1).

19.6.10 Exercise. Let V◦:= VA3(X3

1 +X21 −X2

2 ), W◦:= VA3(X2), and

V := VP3(X31 +X0X

21 −X0X

22 ), W := VP3(X2).

Determine V◦ ∩W◦ and V ∩W .

19.7 The projective spectrum and connectedness

Let us take stock. Among the results we have presented so far in this chapter arethe Connectedness Bound for Complete Local Rings 19.2.10, Grothendieck’sConnectedness Theorem 19.2.12, the Bertini–Grothendieck Connectivity The-orem 19.5.3, the Intersection Inequality for the Connectedness Dimensions ofProjective Algebraic Sets 19.6.6, and the Fulton–Hansen Connectivity Theo-rem 19.6.7. All of these are important results about connectivity, but none ofthem mentions local cohomology in its statement. However, our proofs aboveof these results depend on one proposition, crucial for our approach, that doesuse local cohomology, namely Proposition 19.2.8. The key argument in ourproof of that proposition concerned part of an (exact) Mayer–Vietoris sequence

Hd−1a∩b (R) −→ Hd

a+b(R) −→ Hda (R)⊕Hd

b(R),

where (R,m) is a d-dimensional complete local domain and a and b are non-zero proper ideals of R whose sum is m-primary. An exact sequence like theone displayed above was considered in [5, p. 484], and so we shall refer tothe above sequence as Rung’s display. We used it in conjunction with the lo-cal Lichtenbaum–Hartshorne Vanishing Theorem 8.2.1 and the non-vanishingresult of 6.1.4. From these few arguments from local cohomology, we havederived far-reaching geometric consequences which do not involve local coho-mology in their statements!

19.7 The projective spectrum and connectedness 433

Our final part of this chapter is concerned with a graded analogue of Rung’sdisplay, which we shall use in conjunction with the Graded Lichtenbaum–Hartshorne Vanishing Theorem 14.1.16 and the non-vanishing result of Ex-ercise 16.1.10(ii) in order to prove a ring-theoretic version of Zariski’s MainTheorem on the Connectivity of Fibres of Blowing-up. To prepare for this, weremind the reader, in 19.7.1 below, of some facts concerning the projectivespectrum of a positively graded commutative Noetherian ring.

19.7.1 Reminder and �Exercise. Assume that R =⊕


graded.

(i) Show that, for any two prime ideals p, q ∈ * Spec(R) with p ⊂ q andht q/p > 1, there exists a prime ideal s ∈ *Spec(R) with p ⊂ s ⊂ q.(Recall that ⊂ denotes strict inclusion.)

Recall that the projective spectrum of R, denoted by Proj(R), is the set*Spec(R)\Var(R+) of all graded prime ideals of R which do not contain theirrelevant ideal R+ (see 15.1.1). The Zariski topology on Proj(R) is definedas the topology induced by the Zariski topology on Spec(R). Note that, since,for an ideal b ofR, we have * Spec(R)∩Var(b) = * Spec(R)∩Var(b) whereb is the ideal of R generated by all the homogeneous components of all theelements of b, it follows that the set of closed sets for the Zariski topology onProj(R) is {Proj(R) ∩Var(c) : c is a graded ideal of R}.

Let π : Proj(R) −→ Spec(R0) be the natural map, defined by π(p) =

p ∩ R0 for all p ∈ Proj(R). In the case when (R0,m0) is local, we refer toπ−1(m0) as the special fibre of π.

(ii) Show that * Spec(R), with the topology induced by the Zariski topologyon Spec(R), is a Noetherian topological space.

(iii) Show that Proj(R) is a Noetherian topological space.(iv) Show that the set of closed irreducible subsets of Proj(R) is

{Proj(R) ∩Var(p) : p ∈ Proj(R)}.

(v) Let b be a graded ideal of R. Show that

Proj(R) ∩Var(b) = Proj(R) ∩Var(b ∩R+),

and deduce that each closed set in Proj(R) can be defined by finitelymany homogeneous elements of positive degree in R.

(vi) Show that π−1(Var(b0)) = Proj(R) ∩ Var(b0R) for each ideal b0 ofR0, and deduce that π : Proj(R) −→ Spec(R0) is continuous.


(vii) Let b be a graded ideal of R. Show that

π(Proj(R) ∩Var(b)) = Var(⋃

n∈N(b :R (R+)n) ∩R0

).

(Here are some hints: show that it is sufficient to prove that, for p ∈Proj(R), we have π(Proj(R)∩Var(p)) = Var(p∩R0); then show that,in the special case in which (R0,m0) is local, R+ ⊆

√m0R+ p when

p ∈ Proj(R).)Deduce that π is closed and that π(Proj(R)) = Var

(R0 ∩ ΓR+(R)

).

(viii) Assume that (R0,m0) is local. Show that, for a graded ideal b of R,we have m0 ∈ π(Proj(R) ∩ Var(b)) if and only if R+ ⊆

√m0R+ b.

Show that π−1(m0) ∩W = ∅ for every non-empty closed subset W ofProj(R).

Deduce that π−1(m0) is connected if and only if π−1(Z) is connectedfor each non-empty closed subset Z of Spec(R0).

Our next lemma uses the promised graded analogue of Rung’s display.


n∈N0Rn is positively graded and that

(R0,m0) is local and complete. Then the following statements are equivalent:

(i) Proj(R) is connected;(ii) the special fibre π−1(m0) (under the natural map π : Proj(R) −→

Spec(R0) of 19.7.1) is connected.

Proof. (ii)⇒ (i) This is immediate from Exercise 19.7.1(viii).(i) ⇒ (ii) Since Proj(R) = ∅, we have π−1(m0) = ∅, by 19.7.1(viii), and

R+ = 0. If dimR0 = 0, then π−1(m0) = Proj(R). Therefore we can, anddo, assume that dimR0 > 0.

Suppose that π−1(m0) = Proj(R) ∩ Var(m0R) is disconnected; we shallobtain a contradiction. Then there exist two non-empty closed subsets Z1, Z2

of Proj(R) such that Z1 ∩ Z2 = ∅ and Z1 ∪ Z2 = π−1(m0). Let T1, . . . , Trbe the distinct irreducible components of Proj(R), and set

A := {i ∈ {1, . . . , r} : Ti∩Z1 = ∅}, B := {j ∈ {1, . . . , r} : Tj∩Z2 = ∅}.

Clearly each of A and B is non-empty, since Z1 = ∅ = Z2. By 19.7.1(viii),it follows that, for each k ∈ {1, . . . , r}, we have Tk ∩ (Z1 ∪ Z2) = ∅. Thisshows that A ∪B = {1, . . . , r}.

We show next that A ∩ B = ∅. Suppose, on the contrary, that A ∩ B = ∅,and seek a contradiction. Then W :=

(⋃i∈ATi

)∩(⋃

j∈BTj

)would be non-

empty because Proj(R) is connected; also, we would have

π−1(m0) ∩W = (Z1 ∪ Z2) ∩W = (Z1 ∩W ) ∪ (Z2 ∩W ) = ∅,


contrary to 19.7.1(viii). Hence there exists k ∈ A ∩B.Then Zi ∩ Tk = ∅ for i = 1, 2 and so π−1(m0) ∩ Tk is disconnected. By

19.7.1(iv), there exists p ∈ Proj(R) for which Tk = Proj(R) ∩ Var(p). LetR =⊕

n∈N0Rn = R/p, graded in the natural way, let m0 be the maximal

ideal of R0, and let π : Proj(R) −→ Spec(R0) be the natural map. Thenatural homeomorphism between Tk and Proj(R) maps π−1(m0) ∩ Tk ontoπ−1(m0), and so this latter special fibre is disconnected. Hence, in our searchfor a contradiction, we can, and do, assume that the graded ring R is a domain.

There exist graded ideals b, c of R such that Proj(R) ∩ Var(b) = ∅,Proj(R) ∩Var(c) = ∅, Proj(R) ∩Var(b) ∩Var(c) = ∅, and

(Proj(R) ∩Var(b)) ∪ (Proj(R) ∩Var(c)) = π−1(m0).

Since Proj(R) ∩Var(b) ⊆ π−1(m0), we have

Proj(R) ∩Var(b) = Proj(R) ∩Var(b+m0R),

and a similar comment applies to c; we therefore assume that b ⊇ m0R andc ⊇ m0R. We can now deduce that

√b+ c = m0R+R+ and

√b ∩ c =

√m0R.

Set d := dimR. We can now use 1.2.3 in conjunction with the Graded Mayer–Vietoris sequence 14.1.5 to see that there is an exact sequence of graded R-modules and homogeneous homomorphisms

Hd−1m0R

(R) −→ Hdm0R+R+

(R) −→ Hdb(R)⊕Hd

c (R).

As Proj(R) ∩Var(b) = ∅ and Proj(R) ∩Var(c) = ∅, we have dimR/b > 0

and dimR/c > 0. Therefore, by the Graded Lichtenbaum–Hartshorne Vanish-ing Theorem 14.1.16, we have Hd

b(R) = Hdc (R) = 0; also, Lemma 14.1.12

shows that Hd−1m0R

(R)n ∼= Hd−1m0

(Rn) for all n ∈ Z, and this is 0 for n < 0.Hence Hd

m0R+R+(R)n = 0 for all n < 0, contrary to 16.1.10(ii).

19.7.3 Theorem: the Connectedness Criterion for the Special Fibre. As-sume that R =

⊕n∈N0

Rn is positively graded and that (R0,m0) is local. Let

R0 denote the completion ofR0. As in 16.2.2, set R := R⊗R0 R0; after an ob-vious identification, we can consider R as a graded ring

⊕n∈N0

(Rn⊗R0 R0).Let π : Proj(R) −→ Spec(R0) be the natural map.

The following statements are equivalent:

(i) Proj(R) is connected;(ii) the special fibre π−1(m0) is connected;

(iii) for each non-empty closed subset Z of Spec(R0), the set π−1(Z) is con-nected.


Proof. The equivalence of (ii) and (iii) is the subject of part of Exercise19.7.1(viii).

(i) ⇔ (ii) Let π : Proj(R) −→ Spec(R0) be the natural map. Let m0 =

m0R0, the maximal ideal of R0. Since there are natural homeomorphismsπ−1(m0) ≈ Proj(R/m0R) and π−1(m0) ≈ Proj(R/m0R), and since thereis a homogeneous ring isomorphism R/m0R

∼=−→ R/m0R, there is a home-omorphism π−1(m0) ≈ π−1(m0). We can therefore apply Lemma 19.7.2 tocomplete the proof.

We are now going to apply the Connectedness Criterion for the Special Fibre19.7.3 to the (ordinary) Rees ring of an ideal of a commutative Noetherian ring.

19.7.4 Notation. We use R(a) to denote the ordinary Rees ring⊕

n∈N0an

of a. If, as in 18.1.4, we let {a1, . . . , ah} be a generating set for a and T be anindeterminate, then there is a homogeneous isomorphism of gradedR-algebrasR(a)

∼=−→ R[a1T, . . . , ahT ] =: R[aT ].We remark here that R(a) is also called the blowing-up ring of a; this ter-

minology has its roots in the fact that Proj (R(a)) is the topological spaceunderlying the scheme obtained by blowing up Spec(R) with respect to a.

19.7.5 Corollary. Let p ∈ Spec(R) be such that Rp has only one minimalprime ideal and the ideal aRp of Rp is not nilpotent. Let R(a) denote theordinary Rees ring of a (see 19.7.4), and let πa : Proj(R(a)) −→ Spec(R) bethe natural map. Then the fibre π−1

a (p) of p under πa is connected.

Proof. We can use localization at p to see that it is enough for us to provethe claim under the assumption that (R,m) is local and p = m; we makethis assumption in what follows. Now R is flat over R; also, R(a) can beviewed as a subring of R(aR) and there is a homogeneous isomorphism ofR(a)-algebras R(a) ⊗R R

∼=−→ R(aR). Therefore Proj(R(a) ⊗R R

)and

Proj(R(aR)

)are homeomorphic. By Theorem 19.7.3, it is sufficient for us to

prove that Proj(R(aR)) is connected.In order to prove this, we let p be the unique minimal prime ideal of R,

and P be the ideal of R(aR) given by P :=⊕

n∈N0(anR ∩ p); in fact P ∈

*Spec(R(aR)). Since p =√0, there exists t ∈ N such that p t = 0; hence

Pt = 0, so that P is the unique minimal prime ofR(aR).It follows from the faithful flatness of R → R that aR ⊆

√0 = p, and so

anR ∩ p ⊂ anR for all n ∈ N. Hence

P ∈ * Spec(R(aR)) \Var(R(aR)+) = Proj(R(aR)).

As P is the unique minimal prime of R(aR), we also have Proj(R(aR)) =


Proj(R(aR)) ∩ Var(P). Hence, by 19.7.1(iv), Proj(R(aR)) is irreducible,and so it is connected by 19.1.2(i).

We are now able to deduce from 19.7.5 the last main connectedness resultof this chapter.

19.7.6 Corollary: ring-theoretic version of Zariski’s Main Theorem on

the Connectivity of Fibres of Blowing-up. Assume that R is a domain andthat a = 0. Let p ∈ Spec(R) be such that Rp is also a domain. Let πa :

Proj(R(a)) −→ Spec(R) be the natural map. Then the fibre π−1a (p) is con-

nected. �

19.7.7 Exercise. Assume that R is a domain and that a = 0. Define πa :

Proj(R(a)) −→ Spec(R) as in 19.7.5. Let Z ⊆ Spec(R) be a connectedclosed subset of Spec(R) such that Rm is an integral domain for each maximalideal m of R which belongs to Z. Prove that π−1

a (Z) is connected.

19.7.8 Exercise. Assume that (R,m) is a local domain and that a is non-zeroand proper. We use G(a) to denote the associated graded ring

⊕i∈N0

ai/ai+1:see 18.2.1.

(i) Show that, if R is an integral domain, then Proj(G(a)) is connected.(ii) Show that Proj(G(m)) is connected if and only if Proj(R(mR)) is con-

nected.

19.7.9 Exercise. Let V denote the Cartesian curve VA2(X3+X2−Y 2), andW denote the cuspidal curve VA2(X3 − Y 2).

(i) Let (R,m) be the local ring OV,0 of the origin 0 on V . Show that R hastwo minimal primes, and that Proj(G(m)) is a discrete topological spacewith just two points.

(ii) Let (R′,m′) be the local ring OW,0 of the origin 0 on W . Show that R′

is a domain and that Proj(G(m′)) is a singleton set.

20

Links with sheaf cohomology

In this last chapter we shall develop the links between local cohomology andthe cohomology of quasi-coherent sheaves over certain Noetherian schemes.Here we shall assume for the first time that the reader has some basic know-ledge about schemes and sheaves: our reference for these topics is Hartshorne’sbook [30]. The central idea in this chapter is to extend our earlier relations in2.3.2 and 15.1.3 between ideal transforms and rings of regular functions onvarieties to quasi-coherent sheaves over certain Noetherian schemes. We shallbe very concerned with a generalization of the ‘Deligne Isomorphism’ (see [30,Chapter III, Exercise 3.7, p. 217]) which links the group of sections (over anopen subset) of an induced sheaf on an affine scheme with an ideal transform.More precisely, let M denote the sheaf induced by an R-module M on theaffine scheme Spec(R), and let U = Spec(R)\Var(a), where a ⊂ R; then thegroup of sections Γ(U, M) is isomorphic to the ideal transform Da(M). Weshall use standard techniques involving negative strongly connected sequencesof functors to extend this Deligne Isomorphism, and our generalization of it,to produce the Deligne Correspondence 20.3.11. This correspondence providesconnections between higher cohomology groups of induced sheaves on the onehand, and local cohomology modules on the other.

We shall also examine the case when R is graded in some detail. Here thecentral result for us is the Serre–Grothendieck Correspondence 20.3.15, whichwe shall also derive from the Deligne Isomorphism by standard ‘connected se-quence’ arguments. In this introduction, we mention only some consequencesof the Serre–Grothendieck Correspondence for projective schemes. Consider,therefore, the special case in which R =

⊕n∈N0

Rn is positively Z-gradedand homogeneous; set T := Proj(R) = *Spec(R) \ Var(R+) and considerthe projective scheme (T,OT ) defined by R (see [30, Chapter II, §2, p. 76]).If F is a coherent sheaf of OT -modules, there is a finitely generated gradedR-moduleN such that F is isomorphic to the sheaf ofOT -modules associated

20.1 The Deligne Isomorphism 439

to N on Proj(R). The classical form of the Serre–Grothendieck Correspon-dence yields, for each i ∈ N0 and n ∈ Z, an R0-isomorphism between the i-thcohomology group Hi(T,F(n)) of the twisted sheaf F(n) andRiDR+(N)n,so that, for i > 0, we have Hi(T,F(n)) ∼= Hi+1

R+(N)n. These results enable

us to deduce quickly, from algebraic results about local cohomology estab-lished earlier in the book, significant results about the cohomology of coherentsheaves of OT -modules.

In the final two sections, we use this approach to present proofs of somefundamental theorems, and extensions thereof, from projective algebraic ge-ometry, including Serre’s Finiteness Theorem for the cohomology of coherentsheaves over projective schemes (see 20.4.8), Serre’s Criterion for the globalgeneration of coherent sheaves over projective schemes (see 20.4.13), the ex-istence of a Hilbert polynomial for a coherent sheaf over a projective schemeover an Artinian base ring (see 20.4.16), Mumford’s Regularity Bound forcoherent sheaves of ideals over a projective space (see 20.4.18), the Severi–Enriques–Zariski–Serre Vanishing Theorem for the cohomology of coherentsheaves over projective schemes (see 20.4.23), Serre’s Cohomological Crite-rion for local freeness of coherent sheaves over regular projective schemes(see 20.5.6), Horrocks’ Splitting Criterion for coherent sheaves over projectivespaces (see 20.5.8), and Grothendieck’s Splitting Theorem for coherent locallyfree sheaves over the projective line (see 20.5.9).

20.1 The Deligne Isomorphism

The basic result in this chapter is Theorem 20.1.14, a generalized version ofthe Deligne Isomorphism. This result is not formulated explicitly in sheaf-theoretic terms, but rather in terms of certain local families of fractions. Wenow start to develop the notions of ‘S-topology’ and ‘S-local family of frac-tions’ which we shall use in our formulation and proof of Theorem 20.1.14.

20.1.1 Notation and Terminology. Throughout this chapter, we shall use Sto denote a non-empty subset of R which is closed under multiplication. Itshould be noted that we do not assume that 1 ∈ S.

(i) We denote by AS the set {(S′) : ∅ = S′ ⊆ S} of all ideals of Rgenerated by elements of S. As R is Noetherian, each ideal in AS canbe generated by finitely many elements of S. Note that, if (ai)i∈I is afamily of ideals in AS , then

∑i∈I ai ∈ AS , and if b1, . . . , br ∈ AS ,

then∏r

j=1 bj ∈ AS too.

440 Links with sheaf cohomology

(ii) We shall say that a subset T of Spec(R) is essential with respect to Sprecisely when

(a) T ⊆ Spec(R) \Var((S));(b) there exists b ∈ AS such that T ⊆ Var(b); and(c)

√c =⋂

p∈Var((S))∪(T∩Var(c)) p for all c ∈ AS .

Observe that condition (a) is automatically satisfied if S contains aunit, that condition (b) is automatically satisfied if S contains 0, andthat, for every subset T of Spec(R), we certainly have

√c ⊆

⋂p∈Var((S))∪(T∩Var(c))

p for all c ∈ AS .

Note also that, if S = {0}, then the empty set is the unique subset ofSpec(R) which is essential with respect to S. In general, if the empty setis essential with respect to S, then

√c =√

(S) for all c ∈ AS , so that,in particular, ht(S) ≤ 1 if S contains a non-unit; if, in addition, 0 ∈ S,then√(S) =

√0.

Note also that condition (a) implies that, for each p ∈ T , there existssp ∈ S \ p.

(iii) Let T be a subset of Spec(R) which is essential with respect to S. It isan easy consequence of the last sentence in part (i) that

{T ∩Var(b) : b ∈ AS}

is the set of closed sets in a topology on T : we refer to this topologyas the S-topology on T . We denote by U (S)

T the set of open sets in theS-topology on T : thus U (S)

T = {T \ Var(b) : b ∈ AS}. For p ∈ T , wedenote by U (S)

T,p the set {U ∈ U (S)T : p ∈ U} of all open neighbourhoods

of p in the S-topology on T .Since every non-empty set of closed subsets of T (in the S-topology)

has a minimal member with respect to inclusion, the S-topology makesT into a Noetherian topological space, and hence (see 19.1.6(i)) everyopen subset of T is quasi-compact.

We now state the assumptions that will be in force throughout this chapter.

20.1.2 Standard hypotheses. Throughout this chapter, S will denote a non-empty subset of R which is closed under multiplication, AS will denote the set{(S′) : ∅ = S′ ⊆ S} of all ideals of R generated by non-empty subsets ofS, T will denote a subset of Spec(R) which is essential with respect to S (see20.1.1(ii)), a will denote an ideal in AS , and M will denote an R-module.

20.1.3 Examples. (The hypotheses of 20.1.2 apply.)


(i) If S = R, then AS is the set of all ideals of R, and T := Spec(R)

is essential with respect to S. The S-topology on T is just the ordinaryZariski topology.

(ii) Assume that G is a finitely generated torsion-free Abelian group andR =⊕

g∈GRg is G-graded, and take S :=⋃

g∈GRg to be the set ofhomogeneous elements of R. Then AS is the set of all graded ideals ofR, and T := *Spec(R) is essential with respect to S. It is easy to seethat the S-topology on T is again the ordinary Zariski topology.


n∈N0Rn is positively Z-graded, and

take S :=⋃

n∈NRn, the set of homogeneous elements of R of positive de-grees, together with the zero element of R; then AS is the set of all gradedideals of R which are contained in R+.

Show that T := Proj(R) = *Spec(R) \ Var(R+) is essential with respectto S, and that the S-topology on T is the Zariski topology.

The next exercise suggests how the notion of S-topology on T can be re-garded as a generalization of the Zariski topology on an affine variety.

20.1.5 Exercise. Let V be an affine variety over the algebraically closed fieldK. Take R := O(V ), and S := R; by 20.1.3(i), we know that T := Spec(R)

is essential with respect to S. Let maxT denote the set of all maximal idealsof R; note that maxT is also essential with respect to S (since R is a Hilbertring), and that the S-topology on maxT is the topology induced from theZariski topology on T .

Use the Nullstellensatz to show that there is a homeomorphism j : V≈−→

maxT given by j(p) = IV (p) for all p ∈ V .Show that the assignment U �→ U ∩maxT defines a bijection between U (S)

T

and U (S)maxT .

We next introduce the notion of ‘S-local family of fractions’: this general-izes the concept of regular function on a quasi-affine variety.

20.1.6 Notation and Terminology. (The hypotheses of 20.1.2 apply.) Notethat, if T = ∅, then

∏p∈T (S \ p) = ∅: a member of this set is called a family

of denominators in S.

(i) Let p ∈ T . Since S\p = ∅, we can form the commutative ring (S\p)−1R

and the (S \ p)−1R-module (S \ p)−1M .(ii) Let ∅ = U ∈ U (S)

T . The elements of∏

p∈U (S \ p)−1M are called fam-ilies of fractions over U with numerators in M and denominators in S.Such a family γ = (γp)p∈U ∈

∏p∈U (S \ p)−1M is called local, or an

S-local family of fractions over U , if and only if, for each q ∈ U , there


exist W ∈ U (S)T,q and (s,m) ∈ S ×M such that, for each p ∈ U ∩W , it

is the case that s ∈ S \ p and γp = m/s in (S \ p)−1M .

The set of all S-local families of fractions overU is denoted by M(U).We set M(∅) = 0.

Note that∏

p∈U (S \ p)−1M has a natural structure as a module over

the commutative ring∏

p∈U (S \ p)−1R. It is easy to check that R(U)

is a subring of∏

p∈U (S \ p)−1R and that M(U) is an R(U)-submoduleof∏

p∈U (S \ p)−1M .

Note that R(∅) is a trivial ring; of course, M(∅) = 0 is an R(∅)-module.

(iii) Now let m ∈ M and let U be as in part (ii). Choose a family of de-nominators (sp)p∈U ∈

∏p∈U (S \ p). For each p ∈ U , the fraction

spm/sp ∈ (S \ p)−1M is independent of the choice of denominatorsp ∈ S \ p. We can therefore define a family of fractions

m :=

(spm

sp

)p∈U

∈∏p∈U

(S \ p)−1M,

which does not depend on the choice of family of denominators (sp)p∈U .Moreover, for q ∈ U , we have, for each p ∈ U \ Var((sq)), the relationspm/sp = spsqm/spsq = sqm/sq in (S \p)−1M . As U \Var((sq)) ∈U (S)T,q , we thus see that the family of fractions m is local. We can therefore

define a map εUM : M −→ M(U) by εUM (m) = m = (spm/sp)p∈U forall m ∈M . Of course, we define ε∅M :M −→ M(∅) = 0 to be the zerohomomorphism.

Note that εUR : R −→ R(U) is a ring homomorphism, and so turnsR(U) into anR-algebra; furthermore, εUM is anR-homomorphism. Sim-ilar comments apply to ε∅R and ε∅M .

(iv) Now let U, V ∈ U (S)T with V ⊆ U . Suppose that V = ∅, and let γ =

(γp)p∈U ∈ M(U). Then it is clear that the restriction γ� V := (γp)p∈Vbelongs to M(V ). We therefore have a restriction map ρUV (= ρUV,M ) :

M(U) −→ M(V ) for which ρUV (γ) = γ� V for all γ ∈ M(U). Ofcourse, we define ρU∅ : M(U) −→ M(∅) = 0 to be the zero map.

It is easy to see that ρUV,R : R(U) −→ R(V ) is a homomorphism ofR-algebras and that ρUV,M : M(U) −→ M(V ) is an R(U)-homomor-


phism. Note also that the diagram

M

�

εUM εVM

M(U)ρUV

M(V )

��

��

��

commutes. Statements similar to those in this paragraph hold if V = ∅.

The next exercise shows that the notion of local family of fractions, as in-troduced in 20.1.6, can be viewed as a generalization of the notion of regularfunction on a quasi-affine variety.

20.1.7 Exercise. Let V be an affine variety over the algebraically closed fieldK, and take R := O(V ), S := R and T := Spec(R), as in Exercise 20.1.5;let j : V ≈−→ maxT be the homeomorphism introduced in that exercise.

Let b be a non-zero ideal of O(V ), and let U be the non-empty open subset{p ∈ V : f(p) = 0 for some f ∈ b} of V . We have j(U) = maxT \Var(b).

(i) Recall that, for p ∈ U , we have OV,p = O(V )IV (p) = O(V )j(p) =

(S \ j(p))−1R, and that each f ∈ O(U) can be viewed as an elementof the ring OV,p. Thus we can define a ring homomorphism max ιU :

O(U) −→ R(j(U)) by the assignment f �→ (f)m∈j(U). Show thatmax ιU is an isomorphism.

(ii) Now let U be the open subset T \Var(b) of T = Spec(R). Given p ∈ U ,there exists p ∈ U such that p ⊆ j(p), so that (S\j(p))−1R = Rj(p) canbe viewed as a subring of Rp = (S \ p)−1R. Thus we can define a ringhomomorphism ιU : O(U) −→ R(U) by the assignment f �→ (f)p∈U .Show that ιU is an isomorphism.

In 2.3.2, we described, in terms of ideal transforms, the ring of regular func-tions on a non-empty open subset of an affine algebraic variety over an al-gebraically closed field. The last exercise therefore shows that, in one specialcase at least, a ring of local families of fractions can be described in terms ofideal transforms. The next major aim for this chapter is a description of generalmodules of local families of fractions in terms of ideal transforms, and we nowembark on the preparations for this result.

20.1.8 Lemma. (The standard hypotheses of 20.1.2 apply.) Set T \Var(a) =:

U ; assume that U = ∅. Then


(i) for each γ = (γp)p∈U ∈ M(U), there exist r ∈ N, s1, . . . , sr ∈ S ∩ a

and m1, . . . ,mr ∈ M such that U =⋃r

i=1

(T \ Var((si))

)and γp =

mi/si for all p ∈ T \Var((si)) (for i = 1, . . . , r);(ii) Γa(M(U)) = 0; and

(iii) Ker(εUM ) = Γa(M), where εUM : M −→ M(U) is the homomorphismdefined in 20.1.6(iii).

Proof. (i) As U is quasi-compact (see 20.1.1(iii)), there exist n ∈ N,W1, . . . ,

Wn ∈ U (S)T , t1, . . . , tn ∈ S and l1, . . . , ln ∈ M such that U =

⋃nj=1Wj and,

for each j = 1, . . . , n and each p ∈Wj , we have tj ∈ p and γp = lj/tj .Let j ∈ {1, . . . , n}. Since Wj ∈ U (S)

T , there exists bj ∈ AS such thatWj = T \Var(bj). Now Wj ⊆ U = T \Var(a), and so

Wj =Wj ∩ U = (T \Var(bj)) ∩ (T \Var(a)) = T \Var(abj).

We therefore can, and do, assume that bj ⊆ a. Suppose that bj is generated bywj1, . . . , wjnj ∈ S ∩ a. Set Wjk := T \ Var((wjk)) for all k = 1, . . . , nj ;then Wj =

⋃nj

k=1Wjk, and, for k ∈ {1, . . . , nj} and all p ∈ Wjk, we havetjwjk ∈ S \ p (so that Wjk = T \ Var((wjk)) = T \ Var((tjwjk))) and wecan write

γp =ljtj

=wjkljwjktj

in (S \ p)−1M.

We have therefore only to relabel the pairs

(wjktj , wjklj) ∈ (S ∩ a)×M (k = 1, . . . , nj , j = 1, . . . , n)

in order to complete the proof of part (i).(ii) Let γ = (γp)p∈U ∈ Γa(M(U)). There exists n ∈ N such that anγ = 0.

Consider a p ∈ U = T \ Var(an). Since an ⊆ p and an can be generated byelements of S, there exists sp ∈ (S∩an)\p. Since spγ = 0, we have spγp = 0

and we see that, in (S \ p)−1M ,

γp =1

sp(spγp) = 0.

Hence γ = 0.(iii) As εUM :M −→ M(U) is an R-homomorphism, we have

εUM (Γa(M)) ⊆ Γa(M(U)) = 0

(we have used part (ii) here), and so Ker(εUM ) ⊇ Γa(M).Now let m ∈ Ker(εUM ). Choose a family of denominators

(sp)p∈U ∈∏p∈U

(S \ p).


Then 0 = εUM (m) = (spm/sp)p∈U , and so, for each p ∈ U , there existstp ∈ S \ p such that tpspm = 0.

Let b =∑

p∈U Rtpsp; then b is an ideal belonging to AS and bm = 0. As,for each p ∈ U , we have tpsp ∈ p, it follows that U ∩ Var(b) = ∅, so thatT ∩ Var(b) ⊆ Var(a) and Var((S)) ∪ (T ∩ Var(b)) ⊆ Var(a). Therefore,since T is essential with respect to S,

√b =

⋂p∈Var((S))∪(T∩Var(b))

p ⊇⋂

p∈Var(a)

p =√a ⊇ a.

Thus there exists h ∈ N such that ah ⊆ b; therefore ahm ⊆ bm = 0 andm ∈ Γa(M).

20.1.9 Lemma. (The standard hypotheses of 20.1.2 apply.) Set T \Var(a) =:

U , assume that U = ∅, and let W ∈ U (S)T be such that U ⊆ W . Consider

the restriction homomorphism ρWU : M(W ) −→ M(U) of 20.1.6(iv). ThenKer(ρWU ) = Γa(M(W )).

Proof. By 20.1.8(ii), we have ρWU (Γa(M(W ))) ⊆ Γa(M(U)) = 0, so thatKer(ρWU ) ⊇ Γa(M(W )).

There exists c ∈ AS such that W = T \ Var(c). Let γ = (γp)p∈W ∈Ker(ρWU ). By Lemma 20.1.8(i), there exist r ∈ N, s1, . . . , sr ∈ S ∩ c

and m1, . . . ,mr ∈ M such that W =⋃r

i=1

(T \ Var((si))

)and, for each

i = 1, . . . , r and each p ∈ T \ Var((si)), we have γp = mi/si. Sinceγ ∈ Ker(ρWU ), we have γp = 0 for all p ∈ U = T \Var(a).

Let i ∈ {1, . . . , r}. Set

U ′ := (T \Var(a)) ∩(T \Var((si))

)= T \Var(sia).

Then mi/si = 0 in (S \ p)−1M for all p ∈ U ′. This means that εU′

M (mi) = 0,so that there exists hi ∈ N such that (sia)himi = 0 by 20.1.8(iii). Let h :=

max{hi : i = 1, . . . , r}.Now let p ∈ W . There exists i ∈ {1, . . . , r} with p ∈ T \ Var((si)), and

then, for all d ∈ ah, we have, in (S \ p)−1M ,

dγp =dmi

si=dshimi

sh+1i

= 0.

Hence ahγ = 0, and γ ∈ Γa(M(W )).

20.1.10 Lemma. (The standard hypotheses of 20.1.2 are in force.) Set U :=

T \ Var(a). The homomorphism εUM : M −→ M(U) of 20.1.6(iii) has a-torsion cokernel.


Proof. We can assume that U = ∅.Let γ = (γp)p∈U ∈ M(U). By 20.1.8(i), there exist r ∈ N, s1, . . . , sr

∈ S ∩ a and m1, . . . ,mr ∈ M such that⋃r

i=1

(T \ Var((si))

)= U and, for

each i = 1, . . . , r and each p ∈ T \Var((si)), we have γp = mi/si.Let i ∈ {1, . . . , r}, and let Ui := T \ Var((si)). Then, for each p ∈ Ui, we

have siγp = simi/si in (S \ p)−1M . But this means that

ρUUi(siγ) = siρUUi(γ) = εUi

M (mi) = ρUUi(εUM (mi)),

so that siγ−εUM (mi) ∈ Ker(ρUUi) = Γ(si)(M(U)) by Lemma 20.1.9. Hence

there exists ni ∈ N such that snii (siγ − εUM (mi)) = 0. Define

n := max{ni + 1 : i = 1, . . . , r}.

Then, for all i = 1, . . . , r, we have

sni γ = sn−1−nii sni

i siγ = sn−1−nii sni

i εUM (mi) = εUM (sn−1

i mi) ∈ εUM (M).

As

T \Var(

r∑i=1

Rsni

)= T \Var

(r∑

i=1

Rsi

)=

r⋃i=1

Ui = U = T \Var(a),

it follows that T ∩Var (∑r

i=1Rsni ) ⊆ Var(a). Therefore, since T is essential

with respect to S,√∑ri=1Rs

ni =

⋂p∈Var((S))∪(T∩Var(

∑ri=1 Rsni ))

p ⊇⋂

p∈Var(a)

p =√a ⊇ a.

Thus there exists h ∈ N such that ah ⊆∑r

i=1Rsni , and

ahγ ⊆ (∑r

i=1Rsni ) γ =

∑ri=1Rs

ni γ ⊆ εUM (M).

20.1.11 Lemma. (The standard hypotheses of 20.1.2 apply.) Set T \Var(a)=: U , and let W ∈ U (S)

T be such that U ⊆W . Consider the restriction homo-morphism ρWU : M(W ) −→ M(U) of 20.1.6(iv). TheR-module Coker ρWU

is a-torsion.

Proof. Since εUM = ρWU ◦ εWM by 20.1.6(iv), we see that Im εUM ⊆ Im ρWU

and Coker ρWU is a homomorphic image of Coker εUM . The claim thereforefollows from Lemma 20.1.10.

20.1.12 Remark and Notation. With the hypotheses of 20.1.2, let h :M →N be a homomorphism of R-modules, and let ∅ = U ∈ U (S)

T .


(i) Now h induces, for each p ∈ U , an (S \ p)−1R-homomorphism

(S \ p)−1h : (S \ p)−1M −→ (S \ p)−1N.

There is therefore induced a∏

p∈U (S \ p)−1R-homomorphism∏p∈U

(S \ p)−1h :∏p∈U

(S \ p)−1M −→∏p∈U

(S \ p)−1N.

It is easy to see that the image under this map of an S-local family offractions over U with numerators in M is an S-local family of fractionsover U with numerators in N . Thus h induces an R(U)-homomorphismh(U) : M(U) −→ N(U) for which

h(U) ((γp)p∈U ) = ((S \ p)−1h(γp))p∈U for all (γp)p∈U ∈ M(U).

Of course, we define h(∅) : M(∅) −→ N(∅) to be the zero homomor-phism.

(ii) It is clear that IdM (U) = IdM(U)

, and that, if g : N −→ L is another

homomorphism of R-modules, then g ◦ h(U) = g(U) ◦ h(U). Thus• (U) is a covariant functor from C(R) to C(R(U)). It is straightforwardto check that this functor is R-linear (note that an R(U)-module can beregarded as an R-module by means of εUR).

Clearly, similar comments can be made when U is replaced by ∅.(iii) Now consider a second open set V ∈ U (S)

T such that V ⊆ U . It is clearthat (even if V = ∅) the diagram

M(U) N(U)�

ρUV,M ρUV,N

M(V ) N(V )��

h(U)

h(V )

commutes, and so ρUV : • (U) −→ • (V ) is a natural transformation offunctors (from C(R) to C(R(U))). Also, ρ∅∅ is a natural transformation.

(iv) Finally, the diagram

M N�

εUM εUN

M(U) N(U)��

h

h(U)


also commutes, and so εU : Id −→ • (U) is also a natural transformationof functors (from C(R) to itself). In addition, ε∅ : Id −→ • (∅) is anatural transformation.

20.1.13 �Exercise. Show that, in the situation of 20.1.12, the functor • (U) :

C(R) −→ C(R(U)) is left exact.

We are now ready to establish the central result of this chapter; this result isa version of the Deligne Isomorphism.

20.1.14 The Deligne Isomorphism Theorem. (The standard hypotheses of20.1.2 apply.) Set U := T \Var(a); assume that U = ∅.

(i) There is a unique R-isomorphism νa,M : M(U)∼=−→ Da(M) such that

the diagram

MεUM

M(U)�

ηa,Mνa,M∼=

Da(M)

��

commutes. (Occasionally, νa,M will be written as νUa,M when it is impor-tant to stress the dependence on the open set U .) Moreover, if h :M −→N is a homomorphism of R-modules, then the diagram

M(U) N(U)�

νa,M∼= νa,N∼=

Da(M) Da(N)��

h(U)

Da(h)

commutes, and so νa : • (U) −→ Da is a natural equivalence of func-tors (from C(R) to itself).

(ii) The map νa,R : R(U)∼=−→ Da(R) is an isomorphism of R-algebras.

Proof. By 20.1.8(iii) and 20.1.10, both the kernel and cokernel of εUM area-torsion. It is therefore immediate from 2.2.15(ii) that there is a unique


R-homomorphism νa,M : M(U) −→ Da(M) such that the diagram

MεUM

M(U)�

ηa,Mνa,M

Da(M)

��

commutes, and, in fact, it follows from the formula for νa,M provided by2.2.15(ii) that this map is monomorphic, since we know from 20.1.8(ii) thatΓa(M(U)) = 0. Note also that it follows from 2.2.17 that νa,R is a homomor-phism of R-algebras, and so part (ii) will follow from part (i).

To show that νa,M is surjective, let y ∈ Da(M). Then there exists n ∈ Nand h ∈ HomR(a

n,M) such that y is the natural image of h in Da(M). Asa is generated by elements of S, there exists, for each p ∈ U , an elementsp ∈ (a ∩ S) \ p. Note that

δ :=

(h(snp )

snp

)p∈U

∈∏p∈U

(S \ p)−1M.

Now, for q ∈ U , we have, for each p ∈ U \Var((sq)), that, in (S \ p)−1M ,

h(snp )

snp=snqh(s

np )

snqsnp

=h(snqs

np )

snqsnp

=snph(s

nq )

snpsnq

=h(snq )

snq,

so that δ is an S-local family of fractions over U , that is, δ ∈ M(U). Ourimmediate aim is to show that νa,M (δ) = y.

For each r ∈ an, we have

rδ =

(rh(snp )

snp

)p∈U

=

(h(rsnp )

snp

)p∈U

=

(snph(r)

snp

)p∈U

= εUM (h(r)).

Hence rνa,M (δ) = νa,M (rδ) = νa,M (εUM (h(r))) = ηa,M (h(r)), and thisis just the natural image in Da(M) of h′ ∈ HomR(a

n,M), where h′(r′) =

r′h(r) = rh(r′) for all r′ ∈ an. We thus see that rνa,M (δ) = ry for all r ∈ an.Hence νa,M (δ)− y ∈ Γa(Da(M)), which is zero by 2.2.10(iv). Thus νa,M issurjective, and so is an isomorphism.


To prove the second part, we wish to show that, in the diagram

M

εUM

M(U)

Da(M)

N

εUN

N(U)

Da(N) ,

�

�

��

νa,M νa,N

h

h(U)

Da(h)

�

ηa,M

ηa,N

�

��

��

the front square commutes. We know from the first part of this proof thatthe two side triangles commute; furthermore, the top square commutes by20.1.12(iv), while the sloping rectangle on the underside commutes becauseηa is a natural transformation. Therefore

νa,N ◦ h(U) ◦ εUM = νa,N ◦ εUN ◦ h = ηa,N ◦ h= Da(h) ◦ ηa,M = Da(h) ◦ νa,M ◦ εUM .

However, by 2.2.13(ii), 20.1.8(iii) and 20.1.10, there is a unique R-homomor-phism h′ : M(U)→ Da(N) such that the diagram

MεUM

M(U)�

h h′

Nηa,N

Da(N)��

commutes, and so νa,N ◦ h(U) = Da(h) ◦ νa,M , as required.

The observant reader might have noticed that, in the Deligne IsomorphismTheorem 20.1.14, we did not consider the case where U = ∅, that is, whereT ⊆ Var(a). The next exercise shows that, in some circumstances, there is a‘Deligne Isomorphism’ in the case where U = ∅.

20.1.15 Definition and �Exercise. (The standard hypotheses of 20.1.2 ap-ply.) We say that T is large with respect to S if and only if, whenever c ∈ AS

is such that T ⊆ Var(c), then Var(c) = Spec(R), that is, c is nilpotent. Sup-pose that this is the case and that T ⊆ Var(a). Show that

(i) Ker(ε∅M ) = Γa(M) (compare 20.1.8(iii));(ii) if W ∈ U (S)

T , then Ker(ρW∅) = Γa(M(W )) (compare 20.1.9); and


(iii) there is a natural equivalence of functors ν∅a : • (∅) −→ Da (compare20.1.14(i)).

In several subsequent results that depend on the Deligne Isomorphism The-orem 20.1.14, the reader will find the hypothesis (about U := T \Var(a)) that‘U = ∅ (or U = ∅ and T is large with respect to S)’. The facts that the resultsconcerned are still valid under the alternative hypothesis in parentheses are inmost cases easy consequences of Exercise 20.1.15.

20.1.16 Exercise. Consider the special case of the situation of 20.1.12 inwhich R = K[X,Y ] is the polynomial ring in two indeterminates over a fieldK, and take S = R and T = Spec(R), so that T is essential with respect toS (see 20.1.3(i)). Let U := T \ Var((X,Y )). Use the natural homomorphismR→ R/XR to show that the functor • (U) : C(R) −→ C(R(U)) is not exact.

The following exercise shows that Theorem 20.1.14 can indeed be viewedas a natural generalization of the isomorphism established in 2.3.2.

20.1.17 Exercise. Consider again the situation of 20.1.5 and 20.1.7, so that Vis an affine variety over the algebraically closed field K, R := O(V ), S := R,T := Spec(R), b is a non-zero ideal of O(V ), U is the open subset of Vdetermined by b, and U is the open subset T \Var(b) of T = Spec(R).

Let j : V≈−→ maxT be the homeomorphism of 20.1.5, and consider the

ring isomorphisms ιU : O(U) −→ R(U) and max ιU : O(U) −→ R(j(U))

of 20.1.7. Show that both νUb,R ◦ ιU : O(U) −→ Db(R) and νj(U)b,R ◦max ιU :

O(U) −→ Db(R) coincide with the isomorphism νV,b of 2.3.2.

20.1.18 �Exercise. Consider again the situation and notation of the DeligneIsomorphism Theorem 20.1.14.

(i) Show that, when Da(M) is regarded as an R(U)-module via νa,R, themap νa,M : M(U)

∼=−→ Da(M) is an R(U)-isomorphism.(ii) Let b ∈ AS be such that a ⊆ b, and set W := T \Var(b). Show that the

diagram

M(W ) M(U)�

νb,M∼= νa,M∼=

Db(M) Da(M) ,��

ρWU

αb,a,M

in which αb,a,M is the natural map of 2.2.23, commutes.


20.1.19 �Exercise. Consider once more the situation and notation of theDeligne Isomorphism Theorem 20.1.14.

(i) Show that, if a contains a poor M -sequence of length 2, then the homo-morphism εUM :M −→ M(U) of 20.1.6(iii) is an isomorphism.

(ii) Deduce that, if S contains a unit of R, then εTM : M −→ M(T ) is anisomorphism.

20.1.20 Exercise. Let V be an affine variety over the algebraically closedfield K, take R := O(V ), let (as in 20.1.2) S be a non-empty subset of Rwhich is closed under multiplication, and let T be a subset of Spec(R) whichis essential with respect to S. Let b be a non-zero ideal of AS , and let U =

T \Var(b); assume that U = ∅.Show that the functor • (U) : C(R) −→ C(R(U)) is exact if and only if

the open subset V \ V (b) of V determined by b (the notation is as in 6.4.1) isaffine.

20.2 The Graded Deligne Isomorphism

We now intend to ‘add graded frills’ to the Deligne Isomorphism Theorem20.1.14: we shall call the refined version the ‘Graded Deligne IsomorphismTheorem’. One can view this refinement process as analogous to the improve-ment, in the case of rings of regular functions on affine varieties, in Theorem2.3.2 afforded, in the Z-graded case, by Proposition 15.1.3. For this work, weintroduce the concept of ‘homogeneous S-local family of fractions’.

20.2.1 Hypotheses for the section. The standard hypotheses of 20.1.2 willbe in force throughout this section, and, in addition, we shall assume that Gis a finitely generated torsion-free Abelian group and that R =

⊕g∈GRg

is G-graded, that M =⊕

g∈GMg is a graded R-module, and that S consistsentirely of homogeneous elements (so that that all the ideals in AS are graded).

20.2.2 Remark and Notation. (The hypotheses of 20.2.1 apply.) Let ∅ =U ∈ U (S)

T .

(i) For each p ∈ U , the non-empty set S \ p consists entirely of homo-geneous elements, so that, in the light of 13.1.1, the ring (S \ p)−1R

carries a natural G-grading, and the (S \ p)−1R-module (S \ p)−1M

is also naturally graded; note also that the natural ring homomorphismR −→ (S \ p)−1R is homogeneous in the sense of 14.1.3. For g ∈ G,we use ((S \ p)−1R)g and ((S \ p)−1M)g to denote the components ofdegree g of (S \ p)−1R and (S \ p)−1M respectively.

20.2 The Graded Deligne Isomorphism 453

(ii) Let g ∈ G. The elements of∏

p∈U ((S \ p)−1M)g are called familiesof homogeneous fractions of degree g over U with numerators in M

and denominators in S. The set M(U) ∩∏

p∈U ((S \ p)−1M)g of such

families which are also local will be denoted by M(U)g; the members ofM(U)g are called homogeneous S-local families of fractions of degreeg over U . Of course, we set M(∅)g = 0.

(iii) Let g, h ∈ G. It is easy to check that R(U)0 is an R0-subalgebra ofR(U), that M(U)g is an R(U)0-submodule of M(U), and that σγ ∈M(U)g+h for all σ ∈ R(U)g and all γ ∈ M(U)h. An important aim forus is to establish that the R(U)g (g ∈ G) provide a G-grading on R(U),and that the M(U)g (g ∈ G) provide a grading on the R(U)-moduleM(U). Lemma 20.2.3 below provides a key for the establishment of thisaim.

(iv) Let γ = (γp)p∈U ∈∏

p∈U (S \ p)−1M , and let g ∈ G. Then, foreach p ∈ U , the element γp ∈ (S \ p)−1M has g-th component, de-noted by (γp)g, in ((S \ p)−1M)g . We shall refer to the family γg :=

((γp)g)p∈U ∈∏

p∈U ((S \ p)−1M)g as the g-th homogeneous part ofthe family γ.

20.2.3 Lemma. (The hypotheses of 20.2.1 apply.) Let ∅ = U ∈ U (S)T and let

γ = (γp)p∈U ∈ M(U). Then

(i) γg := ((γp)g)p∈U ∈ M(U)g for each g ∈ G;(ii) the set {g ∈ G : γg = 0} is finite;

(iii) γ =∑

g∈G γg; and(iv) γ = 0 if and only if γg = 0 for all g ∈ G.

Proof. Because U is quasi-compact (see 20.1.1(iii)), there exist r ∈ N, W1,

. . . ,Wr ∈ U (S)T , t1, . . . , tr ∈ S and l1, . . . , lr ∈ M such that U =

⋃rj=1Wi

and, for each j = 1, . . . , r and each p ∈ Wj , we have tj ∈ p and γp = lj/tj .For each j = 1, . . . , r, let hj ∈ G be such that tj ∈ Rhj (recall that S consistsof homogeneous elements of R), and let (lj)g denote the g-th homogeneouscomponent of lj (for all g ∈ G).

(i) It is obvious that, for each j = 1, . . . , r,

(γp)g =

(ljtj

)g

=(lj)g+hj

tjfor all p ∈Wj .

Hence the family γg = ((γp)g)p∈U is local, and so lies in M(U)g .(ii) For each j = 1, . . . , r, let Hj denote {g ∈ G : (lj)g+hj = 0}, a finite

set, and observe that (γp)g = 0 for all g ∈ G \ Hj and all p ∈ Wj . Henceγg = 0 for all g ∈ G \ (H1 ∪ · · · ∪Hr).


(iii) As γp =∑

g∈G(γp)g for each p ∈ U , this is immediate from part (ii).(iv) Just note that γ = 0 if and only if γp = 0 for all p ∈ U , and that this is

the case if and only if (γp)g = 0 for all p ∈ U and all g ∈ G.

20.2.4 Remark and Definition. (The hypotheses of 20.2.1 apply.) Let ∅ =U ∈ U (S)

T . It is now immediate from 20.2.2(iii) and 20.2.3 that(R(U)g

)g∈G

provides a grading on the ring R(U), and that(M(U)g

)g∈G

provides M(U)

with the structure of a graded R(U)-module: we refer to these gradings as thenatural gradings, and any unexplained references to gradings on R(U) andM(U) should always be interpreted as references to these natural gradings.

Trivially, similar conclusions apply to R(∅) and M(∅).

20.2.5 Remarks. (The hypotheses of 20.2.1 apply.) Let U ∈ U (S)T .

(i) It is clear that the map εUM : M −→ M(U) of 20.1.6(iii) is homoge-neous.

(ii) Let V ∈ U (S)T with V ⊆ U . It is also clear that the restriction map

ρUV (= ρUV,M ) : M(U) −→ M(V ) of 20.1.6(iv) is homogeneous.(iii) Now let N be a second graded R-module and let h : M −→ N be

a homogeneous R-homomorphism. It is easy to check that the R(U)-homomorphism h(U) : M(U) −→ N(U) of 20.1.12(i) is homoge-neous.

20.2.6 �Exercise. (The hypotheses of 20.2.1 apply.) Let U ∈ U (S)T and g ∈

G. This exercise involves the g-th shift functor described in 13.1.1.

(i) Use the fact that (S \ p)−1(M(g)) = ((S \ p)−1M)(g) for all p ∈ U to

show that M(g)(U) = M(U)(g).(ii) Let V ∈ U (S)

T with V ⊆ U . Show that ρUV,M(g) = ρUV,M (g), that is,

that the restriction homomorphism ρUV,M(g) : M(g)(U) −→ M(g)(V )

is the g-th shift of the restriction homomorphism ρUV,M : M(U) −→M(V ).

(iii) Let N be a second graded R-module and let h : M −→ N be a homo-geneous R-homomorphism. Show that h(g)(U) = h(U)(g).

Our promised Graded Deligne Isomorphism Theorem can now be obtainedvery quickly.

20.2.7 The Graded Deligne Isomorphism Theorem. (The hypotheses of20.2.1 apply.) Define U := T \Var(a); assume that U = ∅.

Then the Deligne Isomorphism νa,M (= νUa,M ) : M(U)∼=−→ Da(M) of

20.3 Links with sheaf theory 455

20.1.14 is a homogeneousR-homomorphism (with respect to the natural grad-ing of 20.2.4 on M(U) and the grading of 13.3.14 on Da(M)).

Proof. Since a is graded, M(U) is graded by 20.2.4, and εUM :M −→ M(U)

is homogeneous by 20.2.5(i), this result is now immediate from 13.5.4(ii) andthe Deligne Isomorphism Theorem 20.1.14.

20.2.8 Remarks. (The hypotheses of 20.2.1 apply.) Set T \Var(a) =: U .

(i) It is clear from 20.2.4 and 20.2.5(iii) that the functor • (U) : C(R) −→C(R(U)) has the *restriction property of 13.3.6, and (by 20.2.5(i)) thatthe natural transformation εU : Id −→ • (U) has the *restriction prop-erty of 13.3.7.

(ii) Let V ∈ U (S)T with V ⊆ U . By 20.2.5(ii), the natural transformation of

functors ρUV : • (U) −→ • (V ) (from C(R) to C(R(U))) of 20.1.12(iii)has the *restriction property.

(iii) Assume now that U = ∅ (or U = ∅ and T is large with respect toS). We see from 20.2.7 (or 20.1.15) that νa,M is a homogeneous R-isomorphism; by 20.1.18(i), it is also an R(U)-isomorphism; since thenatural grading on M(U) is a grading of this as an R(U)-module, it istherefore automatic that the grading of 13.3.14 on Da(M) is a gradingof Da(M) as an R(U)-module. Hence Da : C(R) −→ C(R(U)) has the*restriction property, and νa : • (U) −→ Da, when viewed as a naturalequivalence of functors from C(R) to C(R(U)) (or, for that matter, fromC(R) to itself), has the *restriction property.

20.3 Links with sheaf theory

Most serious readers of this chapter will by now have realised that, in viewof the results we have obtained about local families of fractions, we have es-sentially started to discuss sheaves, even if only implicitly. We now intend tomake the connection more explicit, and to formulate 20.1.14 and 20.2.7 insheaf-theoretic terms. Henceforth, we assume that the reader has some famil-iarity with basic knowledge about schemes and sheaves, although we providenumerous references to Hartshorne’s book [30].

20.3.1 Remarks and �Exercise: sheaf-theoretic interpretations. (The stan-dard hypotheses of 20.1.2 apply.)

(i) It is not difficult to check that the family(R(U))U∈U(S)

T

, together with


the restriction maps ρUV,R : R(U) −→ R(V ) (U, V ∈ U (S)T with V ⊆

U) defines a sheaf R of R-algebras on the topological space T , so that(T, R) is a ringed space (see [30, Chapter II, §2, p. 72]).

(ii) Similarly, the family(M(U))U∈U(S)

T

, together with the restriction maps

ρUV,M : M(U) −→ M(V ) (U, V ∈ U (S)T with V ⊆ U), defines a

sheaf M of R-modules on T (see [30, Chapter II, §5, p. 109]). We shallcall this sheaf M the S-sheaf induced by M . We shall sometimes usestandard notation from sheaf theory (see [30, Chapter II, §1, p. 61]) anddenote M(U) by Γ(U, M): the elements of Γ(U, M) are the sections ofthe sheaf M over U .

(iii) Now let h : M −→ N be a homomorphism of R-modules. It fol-lows from 20.1.12(i),(iii) that the homomorphisms h(U) : M(U) −→N(U) (U ∈ U (S)

T ) define a morphism h : M −→ N of sheaves of R-modules (see [30, Chapter II, §5, p. 109]), which we call the morphisminduced by h.

(iv) It is now clear that • is a functor from C(R) to the category S(R) ofsheaves of R-modules.

(v) Let p ∈ T . The stalk Rp := lim−→

U∈U(S)T,p

R(U) of the sheaf R at p (see [30,

Chapter II, §1, p. 62]) has a natural structure as an R-algebra. Also, thestalk Mp := lim

−→U∈U(S)

T,p

M(U) of M at p has a natural structure as an Rp-

module. For U ∈ U (S)T,p , we shall use ρU,p (= ρU,p,M ) : M(U) −→ Mp

to denote the natural map; thus ρU,p,R is a homomorphism ofR-algebrasand ρU,p,M is an R(U)-homomorphism.

The universal property of direct limits leads to a map ψpM : Mp −→

(S \ p)−1M for which ψpM (ρU,p,M (γ)) = γp for all U ∈ U (S)

T,p and γ =

(γq)q∈U ∈ M(U). Show that ψpR is an isomorphism of R-algebras, and

that, as M varies through C(R), the ψpM constitute a natural equivalence

ψp : • p −→ (S \ p)−1 of functors from C(R) to itself.Deduce that the functor • : C(R) −→ S(R) of part (iv) is exact (see

[30, Chapter II, Exercise 1.2(c), p. 66]).

20.3.2 �Exercise. (The standard hypotheses of 20.1.2 apply.) Let h : L −→N be a homomorphism of R-modules. Assume that T = ∅.

(i) Show that the induced sheaf M is zero if and only if the R-module M is(S)-torsion.

(ii) Show that the induced morphism h : L −→ N of sheaves of R-modules


is injective (respectively surjective) if and only if Kerh (respectivelyCokerh) is (S)-torsion.

20.3.3 Remarks. (The standard hypotheses of 20.1.2 are in force.) Let U :=

T \ Var(a); assume that U = ∅ (or U = ∅ and T is large with respect to S).Here, among other things, we reformulate the Deligne Isomorphism Theoremin the language of sheaves.

(i) The functor • (U) : C(R) −→ C(R(U)) of 20.1.12(ii) can be regardedas the composition of the functor • : C(R) −→ S(R) of 20.3.1(iv)and the section functor Γ(U, • ) : S(R) −→ C(R(U)). Therefore, theDeligne Isomorphism Theorem 20.1.14, with the refinement afforded byExercise 20.1.18(i), provides a natural equivalence νa : Γ(U, • )

∼=−→ Da

of functors from C(R) to C(R(U)). Of course, we can interpret both Da

and Γ(U, • ) as functors from C(R) to itself (strictly, we should thenwrite Γ(U, • )� R in the latter case, but we shall often omit the ‘� R’ inthe interests of notational simplicity), so that νa can also be interpretedas a natural equivalence of functors from C(R) to itself.

(ii) The above interpretation can be refined in the case of the Graded DeligneIsomorphism Theorem 20.2.7, for which we assume, in addition, that Ris G-graded, where G is a finitely generated torsion-free Abelian group,and that S consists entirely of homogeneous elements. Then, in view of20.2.4, the sheaf R becomes a sheaf of G-graded R-algebras, and thefunctor • : C(R) −→ S(R) of 20.3.1(iv) restricts to a functor from*C(R) to the category *S(R) of sheaves of graded R-modules. We shallagain denote this functor by • , as we do not expect this to cause confu-sion. It follows from 20.2.8 that Γ(U, • ), considered as a functor fromC(R) to either C(R) or C(R(U)), has the *restriction property, and that,also in the two cases, the natural equivalence νa : Γ(U, • )

∼=−→ Da ofpart (i) has the *restriction property.

(iii) Note that, with the notation of part (ii),(R(U)0)U∈U(S)

T

defines a sheaf

R0 =(R)0

of R0-algebras, so that(T, R0

)is also a ringed space. Sim-

ilarly, for a graded R-module M and g ∈ G,(M(U)g

)U∈U(S)

T

defines a

sheaf Mg =(M)g

of R0-modules.(iv) Again with the notation of parts (ii) and (iii), for each p ∈ U , the stalk

Mp inherits a structure as a G-graded R-module for which the naturalhomomorphism ρU,p,M : M(U) −→ Mp is homogeneous. It is straight-forward to check that there is a natural isomorphism of R0-modules(Mg

)p∼=(Mp

)g

for all g ∈ G. We shall use these isomorphisms asidentifications.


A special case of 20.3.3(i) yields the ‘classical’ form of Deligne’s Isomor-phism.

20.3.4 Example: Deligne’s Isomorphism for Affine Schemes. Take S = R

and T := Spec(R) (which is essential with respect to S by 20.1.3(i)). Thenthe induced sheaf R of 20.3.1(i) is just the structure sheaf OT of the affinescheme (T,OT ) defined by R (see [30, Chapter II, §2, p. 70]). Moreover, ifM is an R-module, then the induced sheaf M of 20.3.1(ii) is just the sheafof OT -modules associated to M (see [30, Chapter II, §5, p. 110]). It followsfrom [30, Chapter II, §5, p. 111, and Proposition 5.4, p. 113] that the inducedsheaves in the sense of 20.3.1(ii) are, up to isomorphism, precisely the quasi-coherent sheaves of OT -modules, and that the sheaves which are induced byfinitely generated R-modules are, up to isomorphism, precisely the coherentsheaves of OT -modules.

An arbitrary ideal b of R automatically belongs to AS in this case. Let U :=

T \Var(b) = Spec(R)\Var(b). In this special case, T is large with respect toS, and the isomorphism νb,M : Γ(U, M)

∼=−→ Db(M) of 20.1.14 (or 20.1.15)and 20.3.3(i) is just the classical isomorphism of Deligne (see [30, Chapter III,Exercise 3.7, p. 217]).

20.3.5 �Exercise. Let (T,OT ) be the affine scheme defined by R, and letqcohT denote the category of all quasi-coherent sheaves of OT -modules. LetM be an R-module. Use Exercise 20.1.19 to show that εTM :M −→ Γ(T, M)

is an isomorphism, and deduce that the functor • : C(R) −→ qcohT is anequivalence of categories. (See [30, Chapter II, Corollary 5.5, p. 113].)

20.3.6 �Exercise. (The standard hypotheses of 20.1.2 apply.) Let b be a sec-ond ideal of AS . Set U := T \Var(a) and Z := T ∩Var(b). Set

ΓZ(U, M) := {γ ∈ Γ(U, M) : ρU,p,M (γ) = 0 for all p ∈ U \ Z}.

(See [30, Chapter II, Exercise 1.14, p. 67, and Exercise 1.20, p. 68].) Note thatΓZ(U, • ) is a functor from C(R) to C(R), and also a functor from C(R) toC(R(U)).

Now assume that U = ∅ (or U = ∅ and T is large with respect to S).Use Lemma 20.1.10 to show that the restriction of the isomorphism νa,M :

Γ(U, M)∼=−→ Da(M) of 20.1.14 and 20.3.3(i) provides an isomorphism

νa,b,M : ΓZ(U, M)∼=−→ Γb(Da(M)),

and deduce that the functors ΓZ(U, • ) and Γb(Da( • )), whether considered asfunctors from C(R) to itself or from C(R) to C(R(U)), are naturally equivalent.


20.3.7 Example. Let (T,OT ) be the affine scheme defined by R. (We in-terpret S as R here and T as Spec(R), as in 20.3.4, so that R as well as a

belongs to AS .) Set Z := Var(a) and U := T \Var(a) = Spec(R) \ Z. If weapply 20.3.6 to the open set T and the closed set Z, we obtain an isomorphismνR,a,M : ΓZ(T, M)

∼=−→ Γa(DR(M)). Bear in mind that ηR,M : M −→DR(M) is an isomorphism. Also bear in mind 20.1.15. The diagram

0 ΓZ(T, M) Γ(T, M) Γ(U, M)� � �ρTU

0 Γa(M) M Da(M) H1a(M) 0,� � �ηa,M � �

�

μa,M ∼=

�

(εTM )−1 ∼=

�

νa,M ∼=

in which μa,M = Γa((ηR,M )−1) ◦ νR,a,M , the second map in the top row isthe inclusion map, and the lower row comes from 2.2.6(i)(c), has exact rowsand commutes.

20.3.8 Exercise. Let (T,OT ) be the affine scheme defined by R, let Z bea closed subset of T = Spec(R), and let U := Spec(R) \ Z. Let F be acoherent sheaf of OT -modules (see Example 20.3.4). Show that the restrictionmap ρTU : Γ(T,F) −→ Γ(U,F) associated with the sheaf F is injective ifand only if, for all p ∈ Z, the stalkFp ofF at p has positive depth. (Rememberthat the depth of a zero module over a local ring is interpreted as ∞.) Showfurther that ρTU is bijective if and only if depthFp > 1 for all p ∈ Z.

The following exercise and reminders are in preparation for an extension ofthe sheaf-theoretic version of Deligne’s Isomorphism, as presented in 20.3.3,to higher cohomology. This work will lead to what we shall call the DeligneCorrespondence.

20.3.9 �Exercise. (The standard hypotheses of 20.1.2 apply.)

(i) Let I be an injective R-module. Show that the S-sheaf induced by I isflasque, that is (see [30, Chapter II, Exercise 1.16, p. 67]), for every pairof open sets U, V ∈ U (S)

T with V ⊆ U , the restriction map ρUV,I :

Γ(U, I ) −→ Γ(V, I ) is surjective. (You might find 20.1.18(ii) helpful.)(ii) Now assume in addition that R =

⊕g∈GRg is G-graded, where G is a

finitely generated torsion-free Abelian group, and that S consists entirelyof homogeneous elements. Let J be a *injective gradedR-module. Showthat the S-sheaf induced by J is flasque.

(iii) Let R, S and J be as in part (ii), and let g ∈ G. Thus J is a sheaf ofgraded R-modules. Now (R)0 is a sheaf of R0-algebras and the g-th


homogeneous component(J)g

of this sheaf carries a natural structure

as a sheaf of(R)0-modules: show that

(J)g

is flasque.

20.3.10 Reminders. (The standard hypotheses of 20.1.2 apply.) Set U :=

T \Var(a). Recall [30, Chapter II, Exercise 1.8, p. 66] that the section functorΓ(U, • ) (from the category of sheaves of Abelian groups on T to the categoryC(Z) of Abelian groups) is left exact: for each i ∈ N0, the i-th right derivedfunctor of Γ(U, • ) is denoted by Hi(U, • ) and is referred to as the i-th sheafcohomology functor on U . See [30, Chapter III, §2, p. 207]. Thus Hi(U, • ) isagain a functor from the category of sheaves of Abelian groups on T to C(Z);however, if F is a sheaf of R-modules on T , then, for all i ∈ N0, the AbeliangroupHi(U,F) carries a natural structure as R(U)-module which is such that,for all λ ∈ R(U) and all σ ∈ Hi(U,F), we have λσ = Hi(U, λ IdF )(σ). Itthen follows that

(Hi(U, • )

)i∈N0

is a negative strongly connected sequence of

covariant functors from S(R) to C(R(U)).We noted in 20.3.1(v) that the functor • : C(R) −→ S(R) of 20.3.1(iv)

is exact. If we follow this with the above sheaf cohomology functors, we findthat(Hi(U, • )

)i∈N0

is a negative strongly connected sequence of covariant

functors from C(R) to C(R(U)). Of course, for anR-moduleM , we can regardthe R(U)-module Hi(U, M) as an R-module by restriction of scalars: thisdevice leads to a negative strongly connected sequence of covariant functors(Hi(U, • )� R

)i∈N0

from C(R) to itself (although we shall normally drop the‘� R’ from the notation).

We are now ready to present the promised Deligne Correspondence. Thiscorrespondence presents a fundamental connection between sheaf cohomologyand local cohomology.

20.3.11 The Deligne Correspondence Theorem. (The standard hypothesesof 20.1.2 apply.) Set U := T \ Var(a); assume that U = ∅ (or U = ∅ and Tis large with respect to S).

There is a unique isomorphism

Θ =(θi)i∈N0

:(Hi(U, • )

)i∈N0

∼=−→(RiDa

)i∈N0

of negative strongly connected sequences of covariant functors from C(R) toitself for which θ0 is the natural equivalence νa : Γ(U, • )

∼=−→ Da of 20.3.3(i).Consequently, by 2.2.6(iii), for each i ∈ N, the functorsHi(U, • ) andHi+1

a

from C(R) to itself are naturally equivalent.

Proof. Let I be an injective R-module. Of course, RiDa(I) = 0 for alli ∈ N. By 20.3.9(i), the induced S-sheaf I is flasque, so that Hi(U, I) = 0 for


all i ∈ N, by [30, Chapter III, Proposition 2.5, p. 208]. The claim is thereforeimmediate from 1.3.4(ii).

20.3.12 Remark. In the situation of 20.3.11, we can, on account of 20.1.18,consider Da as a functor from C(R) to C(R(U)), so that

(RiDa

)i∈N0

is actu-ally a negative strongly connected sequence of covariant functors from C(R) toC(R(U)); furthermore, νa : Γ(U, • )

∼=−→ Da is actually a natural equivalenceof functors from C(R) to C(R(U)) (see 20.3.3(i)).

One can therefore argue as in the above proof of 20.3.11 to see that there is aunique isomorphism

(δi)i∈N0

:(Hi(U, • )

)i∈N0

∼=−→(RiDa

)i∈N0

of negative

strongly connected sequences of covariant functors from C(R) to C(R(U)) forwhich δ0 is νa; moreover, the uniqueness aspect of Theorem 20.3.11 showsthat δi� R = θi for all i ∈ N0, so that, for each such i and each R-module N ,the maps δiN and θiN coincide and θiN is an R(U)-isomorphism.

20.3.13 A Graded Version of the Deligne Correspondence. Here, we pre-sent refinements of 20.3.11 which are available in the graded case, and so weassume, in addition to the hypotheses of 20.3.11, that R =

⊕g∈GRg is G-

graded, where G is a finitely generated torsion-free Abelian group, and that Sconsists entirely of homogeneous elements. In particular, this means that a isgraded. We consider the isomorphism

Θ =(θi)i∈N0

:(Hi(U, • )

)i∈N0

∼=−→(RiDa

)i∈N0

of negative strongly connected sequences of covariant functors from C(R)to C(R(U)) of 20.3.11 and 20.3.12: recall that θ0 is the natural equivalenceνa : Γ(U, • )

∼=−→ Da of 20.3.3(i), which, by 20.3.3(ii), has the *restrictionproperty.

(i) Whenever I is a *injective graded R-module, we have

Hi(U, I ) ∼= RiDa(I) ∼= Hi+1a (I) = 0 for all i ∈ N,

by 20.3.11, 2.2.6(iii) and 13.2.6. Since Γ(U, • ) has the *restriction prop-erty (see 20.2.8(i)), we can use Theorem 13.3.15 to deduce that there isexactly one choice of gradings on the R(U)-modules Hi(U, N) (i ∈ N,N a graded R-module) with respect to which

(Hi(U, • )

)i∈N0

has the*restriction property of 13.3.9. We shall refer to these gradings as thenatural gradings, and any unexplained gradings on these modules areto be interpreted as these natural ones. Notice that they provide grad-ings over R, and so must be the unique choice of gradings with respectto which

(Hi(U, • )

)i∈N0

, considered as a negative strongly connectedsequence of functors from C(R) to itself, has the *restriction property.


(ii) Similarly, Da : C(R) −→ C(R(U)) has the *restriction property (see20.2.8(iii)), and there is exactly one choice of gradings on the R(U)-modulesRiDa(N) (i ∈ N,N a gradedR-module) with respect to which(RiDa

)i∈N0

has the *restriction property. We again refer to these grad-ings as the natural gradings; the uniqueness aspect of Theorem 13.3.15means that these natural gradings, which work over both R(U) and R,must be the ones found in 13.5.7(ii).

Now θ0 has the *restriction property. We can define gradings on theRiDa(N) (i ∈ N, N a graded R-module) in such a way that all the θiN(i ∈ N, N a graded R-module) are homogeneous. Since Θ is an iso-morphism of connected sequences, it follows that, with respect to thesegradings,

(RiDa

)i∈N0

has the *restriction property, and so these grad-ings must be the natural gradings.

Thus, with respect to the natural gradings, for all i ∈ N0, the naturalequivalence θi : Hi(U, • )

∼=−→ RiDa of 20.3.11 has the *restrictionproperty.

(iii) Since, by 13.5.7(iii), for i ∈ N, the natural equivalence RiDa

∼=→ Hi+1a

of 2.2.6(ii) has the *restriction property, it follows that the natural equiv-alenceHi(U, • )

∼=−→ Hi+1a of the final paragraph of 20.3.11 also has the

*restriction property. This gives a satisfactory extension, to the gradedcase, of our fundamental connection between sheaf cohomology and lo-cal cohomology.

Next, we are going to study the cohomology of homogeneous components ofinduced S-sheaves in the graded situation. This theme has great importance inthe study of sheaf cohomology over projective varieties and, more generally,over projective schemes. Our aim is to produce a link between cohomologyof homogeneous components of induced S-sheaves and homogeneous com-ponents of graded ideal transforms and local cohomology modules. We beginwith some preparations.

20.3.14 Remarks and Notation. Here, the hypotheses of 20.2.1 apply. Letg ∈ G, and set U := T \Var(a); assume that U = ∅ (or U = ∅ and T is largewith respect to S).

(i) In 20.3.3(ii), we noted that R is a sheaf of G-graded R-algebras, and weintroduced the functor • from *C(R) to the category *S(R) of sheavesof graded R-modules; this functor is exact, by 20.3.1(v).

We shall use ( • )g : *S(R) −→ S((R)0) to denote the functor whichassociates to each sheaf F of graded R-modules its g-th component Fg,which is a sheaf of (R)0-modules.


It is a consequence of Exercise 20.2.6 that the functors ( • )g and(•(g))0 (from *C(R) to S((R)0)) are equal. (Here, ( • )(g) : *C(R) −→*C(R) denotes the g-th shift functor.)

(ii) It follows from the final paragraph of part (i) that, in C((R(U))0), we

have Γ(U, (M(g))0) = Γ(U, (M)g) = Γ(U, M)g (for all M ∈ *C(R)).In alternative notation, (M(g))0(U) = (M)g(U) = M(U)g for allM ∈*C(R). The natural equivalence νa : Γ(U, • )

∼=−→ Da of 20.3.3 has the*restriction property, by 20.3.3(ii). Hence, on taking g-th components,we see that

νa,g : Γ(U, • )g∼=−→ Da( • )g

is a natural equivalence of functors from *C(R) to C((R(U))0), and alsoof functors from *C(R) to C(R0).

(iii) Since the functor(•(g))0

from *C(R) to S((R)0) is exact, it follows that(Hi(U,(•(g))0

))i∈N0

can be considered as a negative strongly con-

nected sequence of covariant functors from *C(R) to either C((R(U))0)

or C(R0). We propose, in the Serre–Grothendieck CorrespondenceTheorem 20.3.15 below, to compare this with

(RiDa( • )g

)i∈N0

, whichalso can be considered as a negative strongly connected sequence of co-variant functors from *C(R) to either C((R(U))0) or C(R0).

20.3.15 The Serre–Grothendieck Correspondence Theorem. For this, weadopt the hypotheses of 20.2.1. Let g ∈ G, and set U := T \ Var(a); assumethat U = ∅ (or U = ∅ and T is large with respect to S).

There is a unique isomorphism

Ωg =(ωig

)i∈N0

:(Hi(U,(

• (g))0

))i∈N0

∼=−→(RiDa( • )g

)i∈N0

of negative strongly connected sequences of covariant functors from *C(R) toC((R(U))0) for which ω0

g is the natural equivalence

νa,g : Γ(U,(• (g))0

) ∼=−→ Da( • )g

of functors of 20.3.14(ii).

Proof. Let I be a *injective graded R-module. Let i ∈ N. Then

RiDa(I) ∼= Hi+1a (I) = 0,

by 2.2.6(ii) and 13.2.6. Also, by 20.3.14(i), we have(I(g))0=(I)g, and, by

20.3.9(iii), this is a flasque sheaf of (R)0-modules. HenceHi(U,(I(g))0

)= 0

by [30, Chapter III, Proposition 2.5, p. 208].


If we now endow(R(U))0

with the trivial grading, we can use 13.3.5(ii) tocomplete the proof.

20.3.16 Remarks. Here we consider further the situation of, and use thenotation of, the Serre–Grothendieck Correspondence 20.3.15.

(i) Observe that Ωg can be regarded as an isomorphism of negative stronglyconnected sequences of covariant functors from *C(R) to C(R0).

(ii) Let d ∈ G and rd ∈ Rd. For each graded R-module N , let μrd =

μrd,N : N −→ N(d) be the homogeneous R-homomorphism given bymultiplication by rd. Set u = g + d. It is straightforward to check that(Hi(U,( ˜μrd, • (g)

)0

))i∈N0

:(Hi(U,(

• (g))0

))i∈N0

−→(Hi(U,(

• (u))0

))i∈N0

is a homomorphism of connected sequences of functors from *C(R) toC(R0).

(iii) Let i ∈ N0. One can check that⊕

j∈GHi(U, (M(j))0

)has a structure

as a graded R-module such that rdzg = Hi(U,(μrd(g))0

)(zg) for (d ∈

G and) rd ∈ Rd and zg ∈ Hi(U,(M(g))0

). It is also straightforward to

use the uniqueness aspect of 13.3.5(i) to show that, with respect to thisgraded R-module structure, the R0-isomorphism⊕j∈G

ωij,M :⊕j∈G

Hi(U,(M(j))0

) ∼=−→⊕j∈G

RiDa(M)j = RiDa(M)

given by the Serre–Grothendieck Correspondence 20.3.15 (and part (i))is actually a homogeneous isomorphism of graded R-modules. For i =0, this is just the homogeneous R-isomorphism⊕

j∈Gω0j,M = νa,M :

⊕j∈G

Γ(U,(M(j))0

)= Γ(U, M) = M(U)

∼=−→⊕j∈G

Da(M)j = Da(M)

of 20.3.3(ii).(iv) Let i ∈ N. It follows from part (iii) and 13.5.7(iii) that there is a homoge-

neous isomorphism⊕

j∈GHi(U,(M(j))0

) ∼=−→ Hi+1a (M) of graded

R-modules.

20.3.17 �Exercise. Here the hypotheses of 20.1.2 and 20.2.1 apply. Considera p ∈ T .

20.4 Applications to projective schemes 465

(i) Show that the isomorphism of R-algebras ψpR : Rp −→ (S \ p)−1R of

20.3.1(v) is homogeneous, and that the natural equivalence

ψp : • p −→ (S \ p)−1

of functors from C(R) to itself (again of 20.3.1(v)) has the *restrictionproperty.

(ii) Let U := T \ Var(a) and assume that p ∈ U . Show that there is acommutative diagram with homogeneous maps

M(U)εUM

M�

ρU,p,M φMS

∼=Mp

ψpM

(S \ p)−1M��

in which φMS is the canonical map and the homogeneous homomor-phisms εUM and ρU,p,M are defined by 20.1.6(iii) and 20.3.1(v) respec-tively.

20.4 Applications to projective schemes

The Serre–Grothendieck Correspondence 20.3.15, as exploited in 20.3.16, canbe used in a very effective manner, in various situations, to derive results aboutthe cohomology of induced sheaves from purely algebraic results obtained ear-lier in the book. We have neither the space nor the intention to present a com-prehensive approach to sheaf cohomology in this book, and so we shall contentourselves, in this section, with some applications of the Serre–GrothendieckCorrespondence to the particular case of projective schemes induced by homo-geneous, positively Z-graded (Noetherian) rings. This situation is particularlyfertile in this context because the quasi-coherent sheaves are, up to isomor-phism, just the sheaves induced by graded modules.

We hope that the illustrations which we present in this section will convincethe reader of the value of the Serre–Grothendieck Correspondence, and willwhet her or his appetite for exploration of applications of the correspondencein other situations.

20.4.1 Hypotheses for the section. We shall assume throughout this sectionthat R =

⊕n∈N0

Rn is positively Z-graded and homogeneous; we shall take


S :=⋃

n∈NRn, the set of homogeneous elements of R of positive degrees to-gether with 0 (so that AS is the set of all graded ideals ofRwhich are containedin R+ (see 20.1.4)); we shall take T := Proj(R) = *Spec(R) \Var(R+) andwe shall assume that T = ∅; also, we shall assume that the R-module M isgraded. Note that our assumptions imply that T is large (see 20.1.15) withrespect to S.

Recall from 20.1.4 that the S-topology on Proj(R) is the Zariski topology.For p ∈ Proj(R), the natural ring homomorphism (S \ p)−1R −→ R(p) is anisomorphism.

20.4.2 Remarks. (The hypotheses of 20.4.1 apply.)

(i) The sheaf (R)0 of R0-algebras of 20.3.9(iii) is just the structure sheafOT of the projective scheme (T,OT ) defined by R: see [30, Chapter II,§2, p. 76]. Note that, for each open subset U of T , we have OT (U) =

(R(U))0, and, for p ∈ T , the stalk of OT at p is (R0)p = (Rp)0 (weare using 20.3.3(iv) here), and so 20.3.17(ii) yields an isomorphism ofR0-algebras (ψp

R)0 : OT,p

∼=−→ (R(p))0. The reader should notice thatour use of the notation R(p) is different from Hartshorne’s in [30, p. 18].From now on, we shall identifyOT,p with (R(p))0 by means of the aboveisomorphism.

(ii) The sheaf (M)0 of (R)0-modules of 20.3.14(i) is just the sheaf of OT -modules associated to M on Proj(R), as defined by Hartshorne in [30,Chapter II, §5, p. 116]. Although Hartshorne’s notation for this sheafis commonly used, we shall continue to use the notation (M)0, in anattempt to avoid confusing readers.

Note that, for each open subsetU of T and t ∈ Z, we have (M)t(U) =

M(U)t. Moreover, by 20.3.3(iv) and 20.3.17(i), for each p ∈ T andt ∈ Z, there is an isomorphism of R0-modules (ψp

M )t : ((M)t)p∼=−→

(M(p))t. We may consider ((M)t)p as an (R(p))0-module via the iso-morphism ((ψp

R)0)−1 of part (i). It then follows easily from 20.3.17(ii)

that the above isomorphism (ψpM )t is an isomorphism of modules over

(R(p))0 = OT,p. Observe also, that by 20.3.14(i) we have (M(t))0 =

(M)t for all t ∈ Z. Consequently, (ψpM(t))0 = (ψp

M )t for all t ∈ Z.

20.4.3 �Exercise. Let the situation be as in 20.4.2(ii), and let p ∈ T andt ∈ Z. Note that (M(p))t denotes the t-th component of M(p), the homoge-neous localization of M at p. Use the diagram of 20.3.17(ii), the fact that themaps in the diagrams in 20.1.14(i) are homogeneous, and the identifications(M(t))0=(M)t

and (ψpM(t))0 = (ψp

M )t of 20.4.2(ii), to show that there is a


commutative diagram(M(t))0(T )

(νTR+,M )t

DR+(M)t

(β(p))t

�

ρT,p,(M(t))0

∼=

∼=((M(t))0

)p

(ψpM )t (

M(p)

)t

��

in which β(p) : DR+(M) −→M(p) is defined as in 17.2.2.

20.4.4 Theorem: the classical form of the Serre–Grothendieck

Correspondence. (The hypotheses of 20.4.1 apply.) It follows from the Serre–Grothendieck Correspondence 20.3.15 and 20.3.16(iii) (with a = R+) thatthere are homogeneous R-isomorphisms⊕

j∈ZHi(T, (M(j))0)

∼=−→ RiDR+(M)∼=−→ Hi+1

R+(M) for all i ∈ N

and⊕

j∈Z Γ(T, (M(j))0) =⊕

j∈Z M(T )j = M(T )∼=−→ DR+(M). �

We recall now some facts about sheaves over projective schemes whichmean that the classical form of the Serre–Grothendieck Correspondence 20.4.4is a powerful tool for translation of algebraic results about local cohomologyinto geometric results about sheaf cohomology.

20.4.5 Reminders. (The hypotheses of 20.4.1 apply.) Let t ∈ Z.

(i) By [30, Chapter II, Proposition 5.11(c), p. 116], the sheaf (M)0 is aquasi-coherent sheaf of OT -modules; furthermore, if M is finitely gen-erated, then (M)0 is a coherent sheaf of OT -modules.

(ii) The twisted sheaf (M)0(t), as defined by Hartshorne [30, Chapter II, §5,p. 117] is naturally isomorphic (as a sheaf of OT -modules) to (M(t))0,the sheaf associated to the shifted module M(t): see [30, Chapter II,Proposition 5.12(b), p. 117]

(iii) LetF be a sheaf ofOT -modules. Recall from [30, Chapter II, §5, p. 118]that Γ∗(T,F) := Γ∗(F) :=

⊕j∈Z Γ(T,F(j)) carries a natural struc-

ture as a gradedR-module and is called the gradedR-module associatedto F . We shall use the notation Γ∗(T,F) instead of Hartshorne’s Γ∗(F).

(iv) Let F be a quasi-coherent sheaf of OT -modules. Then, by [30, ChapterII, Proposition 5.15, p. 119], there is a natural isomorphism(

˜Γ∗(T,F))0

∼=−→ F


of sheaves of OT -modules. This shows, in particular, that each quasi-coherent sheafF ofOT -modules is isomorphic to (N)0 for some gradedR-module N . Also, in this situation, the induced R0-isomorphism

Γ∗(T,F) :=⊕j∈Z

Γ(T,F(j)) ∼=⊕j∈Z

Γ(T, (N(j))0)

(which arises in view of (ii) above) is a homogeneous R-isomorphism.This is easily seen by comparing, by means of the natural isomorphismof part (ii), the action of a homogeneous element rd ∈ Rd (where d ∈N0) on Γ∗(T,F) as described in [30, Chapter II, §5, p. 118] with theaction of rd on

⊕j∈Z Γ(T, (N(j))0) as described in 20.3.16(iii).

20.4.6 �Exercise. (The hypotheses of 20.4.1 apply.) Let F be a coherentsheaf of OT -modules. Then by 20.4.5(iv) we can write F ∼= (M)0, where Mis a graded R-module. Our aim here is to show that we can choose M to befinitely generated. We do this in several steps.

(i) Show that (N)0 = 0 for each R+-torsion graded R-module N .(ii) Show that we can replaceM byM≥0 (see 16.1.1) and hence assume that

beg(M) ≥ 0. (You may find 20.3.14(i) helpful.)(iii) Write R = R0[f1, . . . , fr] with fi ∈ R1 \ {0}. Let i ∈ {1, . . . , r}.

Observe that, by [30, Chapter II, Proposition 2.5, pp. 76,77], the openset Ui := T \ Var(fiR) is (empty or) affine with O(Ui) = (Rfi)0, aNoetherian ring. Use 20.3.14(ii) to show that Γ(Ui,F) ∼= DfiR(M)0 ∼=(Mfi)0. Conclude by [30, Chapter II, Corollary 5.5 p. 113] that the(Rfi)0-module (Mfi)0 is finitely generated. This is true for each i =

1, . . . , r.(iv) Deduce that there is a finitely generated graded R-submodule P of M

such that M/P is R+-torsion.(v) Now use part (i) to show that F ∼= (P )0.

20.4.7 �Exercise. (The hypotheses of 20.4.1 apply.)

(i) Let b ⊆ R be a graded ideal. Show (by use of 20.3.14(i)) that the inclu-sion map b −→ R yields a monomorphism of sheaves of OT -modulesJ := (b)0 −→ (R)0 = OT . This means that, by definition, J is a sheafof ideals on T (see [30, p. 109]). Conclude that this sheaf is coherent.

(ii) Let J be a coherent sheaf of OT -modules which is a sheaf of ide-als on T , so that, by definition, there is a monomorphism of sheavesJ −→ OT . Use the exactness of the twisting functors on the categoryof sheaves of OT -modules, the left exactness of the section functor andthe Serre–Grothendieck Correspondence (see 20.3.16(iii)) to show that


d := Γ∗(T,J ) is homogeneously R-isomorphic to a graded ideal c ofthe graded R-algebra DR+(R). Let b ⊆ R be the inverse image of c

under the natural homomorphism of graded rings ηR : R −→ DR+(R).Show that there are isomorphisms of sheaves of OT -modules (b)0 ∼=(c)0 ∼= (d)0 ∼= J , so that J is induced by a graded ideal b of R.

We can now prove the fundamental Finiteness Theorem of Serre concerningthe cohomology of coherent sheaves over projective schemes.

20.4.8 Serre’s Finiteness Theorem. (See [77, §66, theoreme 1 and theoreme2(b)] and [30, Chapter III, Theorem 5.2, p. 228].) (The hypotheses of 20.4.1apply.) Let F be a coherent sheaf of OT -modules. Then

(i) Hi(T,F(j)) is a finitely generated R0-module, for all i ∈ N0 and allj ∈ Z; and

(ii) there exists r ∈ Z such that Hi(T,F(j)) = 0 for all i ∈ N and allj ≥ r.

Proof. By 20.4.6, there exists a finitely generated graded R-module N suchthat F ∼= (N)0; furthermore, by 20.4.5(iv), there is a homogeneous isomor-phism of graded R-modules

Γ∗(T,F) ∼=⊕j∈Z

Γ(T, (N(j))0).

Also, by 20.4.5(ii), for each i ∈ N0 and j ∈ Z, there is an isomorphism ofR0-modules Hi(T,F(j)) ∼= Hi(T, (N(j))0).

Let j ∈ Z and i ∈ N. We can now use the classical form of the Serre–Grothendieck Correspondence 20.4.4 to see that there are R0-isomorphismsΓ(T,F(j)) ∼= DR+(N)j and Hi(T,F(j)) ∼= Hi+1

R+(N)j . Since DR+(N)j

is a finitely generated R0-module by 16.1.6(ii), and Hi+1R+

(N)j is a finitelygeneratedR0-module by 16.1.5(i), part (i) is now proved. Part (ii) follows from16.1.5(ii).

20.4.9 Exercise. Consider the situation and use the notation of Serre’s Finite-ness Theorem 20.4.8. Show that, for all r ∈ Z, the graded R-module

Γ∗(T,F)≥r :=⊕

j∈Z, j≥r

Γ(T,F(j))

is finitely generated.

The philosophy of the above proof of Theorem 20.4.8 suggests that a similarapproach to Castelnuovo regularity of coherent sheaves of OT -modules mightbe profitable. This is indeed the case.


20.4.10 Definition. (The hypotheses of 20.4.1 apply.) Let F be a coherentsheaf of OT -modules, and let r ∈ Z.

We say that F is r-regular in the sense of Castelnuovo–Mumford if and onlyif Hi(T,F(s− i)) = 0 for all i ∈ N and all s ∈ Z with s ≥ r. In practice, thephrase ‘in the sense of Castelnuovo–Mumford’ is usually omitted.

We define the (Castelnuovo–Mumford) regularity reg(F) of F by

reg(F) = inf {r ∈ Z : F is r-regular} .

20.4.11 Remarks. In the situation, and with the notation, of 20.4.10, thereexists a finitely generated gradedR-moduleN such that F ∼= (N)0, by 20.4.6.Also, by 20.4.5(ii) and the classical form of the Serre–Grothendieck Corre-spondence 20.4.4, for each i ∈ N and each j ∈ Z, there exists anR0-isomorph-ism Hi(T,F(j)) ∼= Hi+1

R+(N)j ; in particular,

Hi(T,F(s− i)) ∼= Hi+1R+

(N)s+1−(i+1) for all i ∈ N and s ∈ Z.

This enables us to use results from Chapter 16 to make deductions about theregularity of coherent sheaves of OT -modules.

(i) By Theorem 16.2.5, if Hi(T,F(r − i)) = 0 for all i ∈ N, then F isr-regular.

(ii) In view of Definition 16.2.1(ii), we can say that F is r-regular if andonly if N is r-regular at and above level 2.

(iii) Hence reg(F) = reg2(N) (see Definition 16.2.9).

An important property that a sheaf of OT -modules might have is that ofbeing generated by global sections. We shall show that this property is closelyrelated to the concept of regularity.

20.4.12 Definition. Let the notation and hypotheses be as in 20.4.10. We saythat the coherent sheaf F is generated by its global sections precisely when,for each p ∈ T , the stalk Fp of F at p is generated ‘by germs of global sectionsof F’, that is, if and only if

Fp =∑

γ∈Γ(T,F)OT,pρT,p,F (γ) for all p ∈ T,

where ρT,p,F : Γ(T,F) −→ Fp is the natural map.

20.4.13 Serre’s Criterion for Generation by Global Sections. (See [77,§66, Theoreme 2].) The hypotheses of 20.4.1 apply. Let F be a coherent sheafof OT -modules and let t ∈ Z with t ≥ reg(F). Then the twisted sheaf F(t) isgenerated by its global sections.


Proof. By 20.4.6, we may write F =(M)0

for some finitely generated

gradedR-moduleM . By 20.4.5(ii), there is an isomorphism F(t) ∼=(M(t))0.

Then, by 17.2.2 and with the notation of that exercise, and in view of the iden-tification of 20.4.2(i), we have

(M(p))t =∑

m∈S(R(p))0(β(p))t(m) =∑

m∈SOT,p(β(p))t(m)

for some set S ⊆ DR+(M)t. Moreover, by 20.4.3 and 20.4.5(ii), we have acommutative diagram (

M(t))0(T )

(νTR+,M )t

DR+(M)t

(β(p))t

�

ρT,p,F(t)

∼=

∼=((M(t))0

)p

(ψpM )t (

M(p)

)t

.��

Γ(T, (F(t)))

(F(t))p

Set T :=((νTR+,M )t

)−1(S) ⊆ Γ(T, (F(t))). Then, since (ψp

M(t))0 = (ψpM )t

is an OT,p-isomorphism (by 20.4.2(ii)), it follows that

(F(t))p =∑

γ∈TOT,pρT,p,F(t)(γ),

and this proves the claim.

The above theorem extends a result of Serre [77, §66, Theoreme 2] whichstates that, if R is the homogeneous coordinate ring of a projective variety andF is a coherent sheaf of OT -modules, then F(t) is generated by its globalsections for all t� 0.

20.4.14 Definition and �Exercise. (The hypotheses of 20.4.1 apply.) Let Fbe a non-zero coherent sheaf of OT -modules. Recall that the support of F ,denoted by SuppF , is the set {p ∈ T : Fp = 0} (see [30, Chapter II, Exercise1.14, p. 67]); the dimension of F , denoted by dimF , is defined to be thedimension of SuppF (see 19.1.8).

By 20.4.6, there exists a finitely generated graded R-module N such thatF ∼= (N)0. Show that

(i) SuppF = SuppN ∩ Proj(R) (a closed subset of Proj(R)), and(ii) when R0 is Artinian, dimF = dimN − 1 (so that dimF is finite) (you

might find 19.7.1(i),(iv) helpful here).

20.4.15 Notation. (The hypotheses of 20.4.1 apply.) Assume that R0 isArtinian and let F be a coherent sheaf of OT -modules.


By Serre’s Finiteness Theorem 20.4.8, for each i ∈ N0 and n ∈ Z, the R0-module Hi(T,F(j)) is finitely generated, and so has finite length: we denotethis length by hi(T,F(j)).

We show next how the Serre–Grothendieck Correspondence enables us toproduce quickly, for a coherent sheafF ofOT -modules as in 20.4.15, a Hilbertpolynomial of F and, for each i ∈ N0, an i-th cohomological Hilbert polyno-mial of F which play similar roles to the corresponding polynomials in 17.1.8and 17.1.11.

20.4.16 Theorem and Definitions. (The hypotheses of 20.4.1 are in force.)Assume that R0 is Artinian and let F be a non-zero coherent sheaf of OT -modules. Set d := dimF .

Now hi(T,F) = 0 for all i � 0, and the Euler characteristic χ(F) of F isdefined by χ(F) :=

∑i∈N0

(−1)ihi(T,F) (see [30, Chapter III, Exercise 5.1,p. 230]).

There is a (necessarily uniquely determined) polynomial PF ∈ Q[X] ofdegree d = dimF such that PF (n) = χ(F(n)) for all n ∈ Z. This polynomialPF is called the Hilbert polynomial of F .

With the notation of 17.1.1(v), for each j = 0, . . . , d, we set ej(F) :=

ej(PF ) and refer to this as the j-th Hilbert coefficient of F . Thus

PF (X) =

d∑i=0

(−1)iei(F)(X + d− id− i

).

Moreover, if N is a finitely generated graded R-module with F =(N)0

(see 20.4.6), then we have dimN = d+1, PF = PN , χF = χN and ei(F) =ei(N) for all i ∈ {0, . . . , d}.

Proof. By 20.4.6, there exists a finitely generated graded R-module N suchthat F = (N)0; by 20.4.14, dimF = dimN − 1, so that dimN = d + 1.Let n ∈ Z and i ∈ N. We can now use 20.4.5(ii) and 20.3.15 to see thatthere are R0-isomorphisms Γ(T,F(n)) ∼= DR+(N)n and Hi(T,F(n)) ∼=RiDR+(N)n ∼= Hi+1

R+(N)n.

Hence hi(T,F(n)) = 0 for all i > d, and, with the notation of 17.1.4, wehave χ(F(n)) = χN (n). The result therefore follows from 17.1.7.

20.4.17 Definitions and Exercise. (The hypotheses of 20.4.1 are in force.)Assume that R0 is Artinian and let F be a non-zero coherent sheaf of OT -modules. Let i ∈ N0. The function hiF : Z −→ N0 defined by hiF (n) =

hi(T,F(n)) for all n ∈ Z is referred to as the i-th cohomological Hilbertfunction of F .


Show that there is a polynomial piF ∈ Q[X] of degree at most i such thathiF (n) = hi(T,F(n)) = piF (n) for all n � 0. The (uniquely determined)polynomial piF is called the i-th cohomological Hilbert polynomial of F .

Let d := dimF . Show that the leading term of pdF is

(−1)de0(F)d!

Xd,

where e0(F) is as defined in 20.4.16.

We come now to a basic bounding result, due to Mumford, for the regularityof coherent sheaves of ideals.

20.4.18 Theorem: Mumford’s Regularity Bound [54, Theorem, p. 101].Suppose that R = R0[X0, X1, . . . , Xd] is a polynomial ring in d+ 1 indeter-minates X0, X1, . . . , Xd over an Artinian local ring R0, where d ∈ N. RegardR =⊕

n∈N0Rn as N0-graded with degXi = 1 for all i = 0, . . . , d. As in

20.4.1, we set T = Proj(R). (Thus T is just PdR0

, projective d-space over R0:see [30, Chapter II, Example 2.5.1, p. 77].)

Then there is a function F : Zd+1 −→ Z such that, for each non-zerocoherent sheaf I of ideals on T , we have reg(I) ≤ F (e0(I), . . . , ed(I)).

Proof. By 20.4.7, the coherent sheaves of ideals on T are exactly the sheavesof the form I =

(b)0, where b is a graded ideal of R. If a = 0 is such a

graded ideal, corresponding to the coherent sheaf of ideals J , then dim a =

dim((a)0

)+1 = dimJ +1 = d+1 and ei(a) = ei(J ) for all i ∈ {0, . . . , d}

(see 20.4.16). Also, reg(J ) = reg((a)0

)= reg2(a) by 20.4.11(iii). The re-

sult therefore follows from 17.3.6.

So far in this chapter, the results which we have obtained about sheaf coho-mology have not been obviously related to local properties of the underlyingsheaves. In our next sequence of results and exercises, we aim for the so-calledSeveri–Enriques–Zariski–Serre Vanishing Theorem (see F. Severi [78], F. En-riques [13], O. Zariski [88], and J.-P. Serre [77, §76, Theoreme 4]), whichestablishes a fundamental link between the local structure of a coherent sheafof OT -modules (conveyed by information about the depths of its stalks, forexample) and global properties of the sheaf (described by the vanishing of cer-tain cohomology groups, for example). Our approach to this Vanishing Theo-rem again makes use of the Serre–Grothendieck Correspondence, this time inconjunction with the Graded Finiteness Theorem 14.3.10.

20.4.19 �Exercise. (The hypotheses of 20.4.1 apply.) Assume that the gradedR-module M is finitely generated, and set F := (M)0. Fix p = x ∈ T , and


let mT,x denote the maximal ideal (pR(p))0 of the local ring OT,x, which weidentify with (R(p))0 by means of the isomorphism of 20.4.2(i).

(i) Use 19.7.1(vii) to show that, if x is a closed point of T (so that p is amaximal member of Proj(R) with respect to inclusion), then p ∩ R0 isa maximal ideal of R0 and dimR/p = 1.

(ii) Since R is homogeneous, there exists t ∈ R1 \ p. Denote the ring offractions OT,x[X]X of the polynomial ring OT,x[X] by OT,x[X,X

−1].Show that there is a unique homogeneous isomorphism ofOT,x-algebrasφt : OT,x[X,X

−1]∼=−→ R(p) for which φt(X) = t.

(iii) Identify the stalk Fx with (M(p))0 by means of the isomorphism of20.4.3. Show that, whenM(p) is considered as anOT,x[X,X

−1]-moduleby means of the isomorphism φt of part (ii), there is a homogeneous iso-morphism of graded OT,x[X,X

−1]-modules

ψt,M : Fx ⊗OT,xOT,x[X,X

−1]∼=−→M(p)

for which

ψt,M (z ⊗ f) = φt(f)z for all z ∈ (M(p))0 and f ∈ OT,x[X,X−1].

(iv) Show that the isomorphism φt of part (ii) leads to a ring isomorphismOT,x[X]mT,xOT,x[X]

∼=−→ Rp. Conclude that dimOT,x = ht p, thatdepthOT,x = depthRp, and that OT,x is a domain, respectively nor-mal, Cohen–Macaulay, Gorenstein, regular, if and only if Rp has thesame property.

(v) Show that the isomorphism ψt,M of part (iii) gives rise to an isomor-phism Fx ⊗OT,x OT,x[X]mT,xOT,x[X]

∼=−→Mp of OT,x[X]mT,xOT,x[X]-modules. Conclude that depthOT,x

Fx = depthRpMp and that Fx is

free of rank r over OT,x if and only if Mp is free of rank r over Rp.(vi) Let q ∈ Spec(R) be such that q* ⊆ p. Show that Rq is a domain,

respectively normal, Cohen–Macaulay, Gorenstein, regular, if OT,x hasthe same property.

20.4.20 �Exercise. (The hypotheses of 20.4.1 are in force.) Let π : T =

Proj(R) −→ Spec(R0) be the natural map, defined by π(q) = q ∩ R0 for allq ∈ T . Fix p = x ∈ T . In the graded ring R/p, let Σ(p) denote the set of non-zero homogeneous elements of degree 0. Note that Σ(p)−1(R/p) is positivelygraded and homogeneous: set T〈x〉 := Proj(Σ(p)−1(R/p)).

(i) Show that there is a homeomorphism T〈x〉≈−→ π−1({π(x)}) ∩ {x},

where the ‘overline’ is used to indicate closure in T .


(ii) Show that dimT〈x〉 = ht(R+ + p)/p− 1.

20.4.21 Notation and Remark. (The hypotheses of 20.4.1 apply.) Let F bea coherent sheaf of OT -modules. We set

δ(F) := inf{depthOT,x

Fx + dimT〈x〉 : x ∈ T},

where T〈x〉, for x ∈ T , is as defined in 20.4.20.By 20.4.6, there exists a finitely generated graded R-module N such that

F ∼= (N)0; by 20.4.19(v), for x = p ∈ T , we have

depthOT,xFx = depthRp

Np.

Therefore, by 20.4.20, with the notation of 9.2.2,

δ(F) = inf {depthNp + ht(R+ + p)/p− 1 : p ∈ * Spec(R) \Var(R+)}

= inf{adjR+

depthNp − 1 : p ∈ *Spec(R) \Var(R+)}.

In the case whenR is a homomorphic image of a regular (commutative Noethe-rian) ring, we can deduce from the Graded Finiteness Theorem 14.3.10 thatδ(F) = fR+(N) − 1. This observation is the key to our proof of the Severi–Enriques–Zariski–Serre Vanishing Theorem 20.4.23 below.

20.4.22 Exercise. (The hypotheses of 20.4.1 apply.) Let π : T → Spec(R0)

be the natural map. Assume that R0 is Artinian, and let F be a coherent sheafof OT -modules.

(i) Show that π−1({π(x)}) ∩ {x} = {x} for all x ∈ T .(ii) Use 9.3.5 to show that

δ(F) = inf{depthOT,x

Fx : x is a closed point of T}.

20.4.23 The Severi–Enriques–Zariski–Serre Vanishing Theorem. (SeeSeveri [78], Enriques [13], Zariski [88], and Serre [77, §76, Theoreme 4]).(The hypotheses of 20.4.1 apply.) Assume that R0 is a homomorphic image ofa regular (commutative Noetherian) ring. Let F be a coherent sheaf of OT -modules, and let r ∈ N0. Then δ(F) > r if and only if Hi(T,F(j)) = 0 forall i ≤ r and all j � 0.

Proof. By 20.4.6, there is a finitely generated graded R-module N such thatF ∼= (N)0. By 20.4.4, there are homogeneous R-isomorphisms⊕

j∈ZHi(T, (N(j))0)

∼=−→ Hi+1R+

(N) for all i ∈ N

and⊕

j∈Z Γ(T, (N(j))0)∼=−→ DR+(N). Also, it follows from 2.2.6(i)(c) that

DR+(N) is finitely generated if and only if H1

R+(N) is.


In view of [7, Proposition 1.5.4], the hypothesis on R0 ensures that R is ahomomorphic image of a regular (commutative Noetherian) ring. It thereforefollows from 20.4.21 that δ(F) = fR+(N)−1, so that δ(F) > r if and only if

the gradedR-module⊕

j∈ZHi(T, (N(j))0) is finitely generated for all i ≤ r.

The desired conclusions follow from these observations and Serre’s FinitenessTheorem 20.4.8.

20.4.24 Exercise. (The hypotheses of 20.4.1 apply.) Assume that R0 is Ar-tinian, and let F be a coherent sheaf of OT -modules. Recall that AssF :=

{x ∈ T : depthOT,xFx = 0}. Show that the graded R-module Γ∗(T,F) =⊕

j∈Z Γ(T,F(j)) is finitely generated if and only if AssF contains no closedpoint of T .

20.5 Locally free sheaves

In this final section, we give some applications to locally free sheaves. Thesecorrespond to vector bundles in algebraic geometry, and form an importantclass of sheaves.

20.5.1 Hypotheses for the section. Throughout this section, the hypothesesof 20.4.1 will be in force.

20.5.2 Definition. A coherent sheaf F of OT -modules is called locally freeif the stalk Fx is a free OT,x-module (of finite rank, as F is coherent) for allx ∈ T . (See [30, Chapter II, §5, p. 109, and Exercise 5.7, p. 124].)

20.5.3 �Exercise. Suppose that T is connected and OT,x is an integral do-main for all x ∈ T . Show that T has a unique minimal member (with respectto inclusion of prime ideals) and conclude that T is irreducible.

20.5.4 Reminder. Recall from [30, Chapter II, Exercise 3.8, p. 91] that T issaid to be normal if and only ifOT,x is a normal integral domain for all x ∈ T .Similarly, T is said to be regular if and only if the local ring OT,x is regularfor all x ∈ T . (See [30, Chapter II, Remark 6.11.1A, p. 142].)

Observe that, if T is regular, then it is normal. Note also that, by 20.4.19(iv),Proj(K[X0, . . . , Xr]) is regular (where K is a field, r ∈ N, X0, . . . , Xr areindeterminates, and the polynomial ring is graded so that degXi = 1 for alli = 0, . . . , r and K is the component of degree 0).

20.5.5 Exercise. Assume that R0 is a field and that T = Proj(R) is con-nected. Set d := dimT .

Suppose that T is normal (see 20.5.4), and assume that d ≥ 2. Let F be a

20.5 Locally free sheaves 477

locally free coherent sheaf ofOT -modules. Show thatH1(T,F(j)) = 0 for allj � 0. (Compare this with Hartshorne’s presentation of the Enriques–Severi–Zariski Lemma in [30, Chapter III, Corollary 7.8, p. 244].)

20.5.6 Theorem: Serre’s Cohomological Criterion for Local Freeness

[77, §75, Theoreme 3]. Assume that R0 is a field, and that T = Proj(R) isconnected and regular (see 20.5.4). Let d := dimT . Let F be a coherent sheafof OT -modules. Then F is locally free if and only if Hi(T,F(n)) = 0 for alli < d and all n� 0.

Proof. By 20.4.23, it is enough for us to show thatF is locally free if and onlyif δ(F) ≥ d. (See 20.4.21 for the definition of δ(F).) First of all observe thatT has a unique minimal member q (by 20.5.3). SinceR0 is a field, Var(R+) =

{R+}. Since q ∈ T , it follows that q is the unique minimal prime of R. Letx = p ∈ T . Then we have q ⊆ p ⊆ p + R+ ∈ *Spec(R). So, on use of20.4.19(iv) and 20.4.20, we obtain

dimOT,x + dimT〈x〉 = ht p+ ht(R+ + p)/p− 1 = ht(R+ + p)− 1

= htR+ − 1 = dimR− 1 = d

because R is a finitely generated R0-algebra (and therefore catenary) and hasa unique minimal prime q. Thus dimOT,x + dimT〈x〉 = d for all x ∈ T .

Now assume that F is locally free. Then, for each x ∈ T ,

depthOT,xFx = depthOT,x = dimOT,x,

as OT,x is regular, and therefore Cohen–Macaulay. It follows that

depthOT,xFx + dimT〈x〉 = dimOT,x + dimT〈x〉 = d for all x ∈ T,

so that δ(F) = d.Conversely, suppose that δ(F) ≥ d. Then, for each x ∈ T , we have

depthOT,xFx + dimT〈x〉 ≥ d = dimOT,x + dimT〈x〉,

so that depthOT,xFx ≥ dimOT,x. As OT,x is regular, it follows from the

Auslander–Buchsbaum Theorem (see [50, Theorem 19.1], for example) thatthe OT,x-module Fx is projective, and therefore free (by [50, Theorem 2.5]).

Finally, we consider locally free sheaves over projective d-space PdK =

Proj(K[X0, . . . , Xd]) over K, where K is an algebraically closed field.

20.5.7 Reminder and �Exercise. (The hypotheses of 20.4.1 apply.)


(i) In the category S(OT ) of sheaves of OT -modules one may form directsums, and the functors ( • )0 (from *C(R) to S(OT )) and Hi(T, • ) (i ∈N0) (from S(OT ) to *C(R)) are additive. So, if r ∈ N, N1, . . . , Nr aregradedR-modules and F1, . . . ,Fr are sheaves ofOT -modules, we have(⊕r

j=1Nj

)0∼=⊕r

j=1

(Nj

)0

and

Hi(T,⊕r

j=1 Fj) ∼=⊕r

j=1Hi(T,Fj) for all i ∈ N0.

(ii) Assume now that d ∈ N, that K is a field and R = K[X0, . . . , Xd]

is a polynomial ring, so that T = Proj(R) becomes projective d-spacePdK over K. A sheaf F of OT -modules is said to split (completely) (the

word ‘completely’ is often omitted) if and only if F ∼=⊕r

j=1OT (aj)

for some r ∈ N and some integers a1, . . . , ar. Show that, if this is thecase, then F is coherent and locally free, and if also the ai are numberedso that a1 ≥ a2 ≥ · · · ≥ ar, then a := (a1, . . . , ar) ∈ Zr is uniquelydetermined by F . In this situation, a = (a1, . . . , ar) is called the split-ting type of F . (You might find it helpful to consider the cohomologicalHilbert functions n �→ h0(T,F(n)) and n �→ h0(T,OT (aj)(n)), inconjunction with the Serre–Grothendieck Correspondence 20.4.4).

20.5.8 Theorem: Horrocks’ Splitting Criterion [40]. Let d ∈ N, let K bea field, let T = Pd

K and let F be a non-zero coherent sheaf of OT -modules.Then the following statements are equivalent:

(i) F splits;(ii) H0(T,F(n)) = 0 for all n � 0 and Hi(T,F(n)) = 0 for all i ∈

{1, . . . , d− 1} and all n ∈ Z.

Proof. Here, T = Proj(R) where R is the polynomial ring K[X0, . . . , Xd]

with degXi = 1 for all i = 0, . . . , d.(i)⇒ (ii) Assume that F splits, so that F ∼=

⊕rj=1OT (aj) for some r ∈ N

and some integers a1, . . . , ar. On use of 20.5.7(i), we obtain

Hi(T,F(n)) ∼=⊕r

j=1Hi(T,OT (aj + n)) ∼=

⊕rj=1H

i(T, R0(aj + n)

)∼=⊕r

j=1Hi(T, ˜R(aj + n)0

)for all i ∈ N0 and all n ∈ Z.

By the Serre–Grothendieck Correspondence 20.4.4,

H0(T, ˜R(aj + n)0

) ∼= DR+(R)aj+n,

and DR+(R)aj+n∼= Raj+n as H0

R+(R) = H1

R+(R) = 0. Therefore

H0(T, ˜R(aj + n)0

)= 0 for all n < −aj ,

and so H0(T,F(n)) = 0 for all n� 0. Also, by Theorem 20.4.4 once again,

20.5 Locally free sheaves 479

for all i > 0 and all n ∈ Z, we have Hi(T, ˜R(aj + n)0

) ∼= Hi+1R+

(R)aj+n,and this vanishes for all i ∈ {1, . . . , d−1} as gradeR+ = d+1. It follows thatHi(T,F(n)) = 0 for all i ∈ {1, . . . , d − 1} and all n ∈ Z. Hence statement(ii) is true.

(ii) ⇒ (i) Assume that statement (ii) is true. By 20.4.6, we can write F =(M)0

for a finitely generated graded R-module M . Then it follows from theSerre–Grothendieck Correspondence 20.4.4 that Hi

R+(M) = 0 for all i ∈

{2, . . . , d} and DR+(M)n = 0 for all n� 0. Consider the exact sequence

0 −→ ΓR+(M) −→MηM−→ DR+(M) −→ H1

R+(M) −→ 0

of 13.5.4(i), in which all the homomorphisms are homogeneous. As the K-vector space H1

R+(M)n has finite dimension for all n ∈ Z and vanishes for all

large n (by 16.1.5), it follows that the graded R-module DR+(M) is finitelygenerated. Moreover, if we apply the exact functor ( • )0 to the above exactsequence and observe that both ΓR+

(M) and H1R+

(M) are R+-torsion (so

that( ˜ΓR+(M)

)0=( ˜H1

R+(M))0= 0 (by 20.3.2(i))), we get an isomorphism

of sheaves F =(M)0∼=( ˜DR+(M)

)0. Thus DR+(M) is finitely generated

and F ∼= ( ˜DR+(M))0. Now by 2.2.10(iv),(v), we have HiR+

(DR+(M)) = 0

for i = 0, 1 and HiR+

(DR+(M)) ∼= HiR+

(M) for all i > 1, As HiR+

(M) = 0

for all i ∈ {2, . . . , d}, we get that HiR+

(DR+(M)) = 0 for all i ∈ {0, . . . , d}.Therefore gradeDR+

(M)R+ ≥ d + 1. As 0 = F ∼=( ˜DR+(M)

)0, we have

DR+(M) = 0, so that gradeDR+(M)R+ = d + 1. So, by Hilbert’s Syzygy

Theorem (see [7, Corollary 2.2.15]), the finitely generated graded R-moduleDR+

(M) is free, and so DR+(M) ∼=

⊕rj=1R(aj) (in *C(R)) for some r ∈ N

and some a1, . . . , ar ∈ Z. Therefore, in view of 20.4.5(ii),

F ∼=(

˜⊕rj=1R(aj)

)0

∼=⊕r

j=1

(R(aj))0∼=⊕r

j=1

(R)0(aj)

=⊕r

j=1OT (aj).

Therefore F splits.

20.5.9 Corollary: Grothendieck’s Splitting Theorem [23, Theoreme 2.1].Let K be a field and let T = P1

K be the projective line over K. Then eachnon-zero coherent locally free sheaf F of OT -modules splits.

Proof. Here T = Proj(R) where R is the polynomial ring K[X0, X1] withdegX0 = degX1 = 1.

As T is regular (see 20.5.4) of dimension 1 and connected, Serre’s Criterionfor Local Freeness 20.5.6 implies that H0(T,F(n)) = 0 for all n � 0. Wecan therefore apply 20.5.8 to complete the proof.

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Index

a-adjusted depth, see depthb-minimum (λb

a(M)), see depthAbelian category, 255acyclic module, 65Γa-, 65–69, 71, 76, 260ΓB-, 66, 260

affinealgebraic cone, see conealgebraic set, 40n-space (An(K)), 39

complex (An), 39open set, 419, 452scheme, 458, 459surface, 162variety, 39, 93, 122–124, 126, 127, 410,

423, 441, 443, 451, 452a-invariant, 326analytic spread (spr(b)), 394–398, 402analytically independent, 396

in b, 392, 393, 395, 396analytically irreducible

closed set, at p, 420local ring, see local ring

analytically reducibleclosed set, at p, 420

Annihilator Theoremof Faltings, see Faltings’

Graded, see Faltings’arithmetic

depth, 339, 341properties, of a projective variety, 338

arithmetic rankof a closed subset, with respect to a larger

one (araW (Z)), 425local, at p (araW,p(Z)), 420

of an ideal (ara(a)), 56, 78, 398, 412–415arithmetically

Cohen–Macaulay, 339, 341, 342Gorenstein, 339, 341

Artinian

local ring, 108, 200, 201, 203, 241, 357,369, 380, 387, 473

module, 135–139, 141, 142, 144, 146, 158,159, 200–206, 208–210, 220

ring, 355, 367–371, 373, 377–379, 471,472, 475

associated graded ring (G(b)), 394, 437a-torsion

functor (Γa), see functormodule, see modulesubmodule (Γa(M)), 3

a-torsion-free module, see modulea-transform, see transform

functor (Da), see functorattached prime ideal, 140attached primes (Att), 140–142, 144, 145,

158, 159, 206, 210, 220–222

Baer Criterion, 18, 260, 261basic ideal, 396, 397Bass number (μi(p,M)), 213, 216, 236, 299,

301, 306–308, 316, 325B-closure, see closurebeginning of a graded module (beg(M)), see

graded moduleBertini’s Connectivity Theorem, 426Bertini–Grothendieck Connectivity Theorem,

425, 432b-finiteness dimension relative to a (fb

a (M)),see dimension

bounding system, 378minimal, 378

B-torsion module, see moduleB-torsion-free module, see moduleB-transform, see transform

functor (DB), see functorBuchsbaum

module, 187, 188ring, 187, 188

canonical module (ωR), 224, 226–231, 233,

486 Index

234, 236–244, 248–250, 314–317,320–322, 328

*canonical module, 313–330, 343Cartesian curve, see curveCastelnuovo(–Mumford) regularity

of a coherent sheaf of OT -modules, 470of a graded module (reg(M)), 351–363,

373, 383–387, 400–402, 404at and above level 1 (reg1(M)), 359,

360, 383–387at and above level 2 (reg2(M)), 360,

373–384at and above level l (regl(M)), 357–358,

369, 400–402, 404syzygetic characterization of, 362

of a projective variety (reg(V )), 360, 373category

Abelian, see Abelian categoryC(R), C(R′), xxi*C(R), *CG(R), *C(R′), *CG(R′), xxi,

252of all quasi-coherent sheaves of

OT -modules (qcohT ), 458of all sheaves of ˜R-modules (S( ˜R)), 456of all sheaves of graded ˜R-modules

(*S( ˜R)), 457catenary local ring, see local ringcatenary ring, 169, 181, 186, 242, 243, 414,

415Cech complex, see complexcharacteristic function, see functionChevalley’s Theorem, 49, 156closureB-, 247S-, 248

coarsening, 275coarser grading, 274codimension

of a closed subvariety, 122pure, 122

Cohen’s Structure Theorem, 110, 153, 157,221, 225, 232, 234, 372

Cohen–Macaulaylocal ring, 130, 132–134, 231, 235–241,

244, 245, 308, 474*local graded ring, 315–323, 326, 329, 330module, 115, 180, 185, 192, 228, 235

generalized, see generalizedCohen–Macaulay module

quasi-, see quasi-Cohen–Macaulaymodule

ring, 181, 185, 189, 191, 192, 237, 238,284, 308, 315, 404

cohomological dimension of an ideal(cohd(a)), 56, 71, 111, 112, 151,412–414

cohomological Hilbertfunction, see Hilbertpolynomial, see Hilbert

comparison exact sequence, 149, 292complementn-, 83

complete local domain, 110, 153, 155, 157,412

complete local ring, 49, 157, 204, 206, 216,222, 228, 294, 372, 413, 414

completion, 109, 158, 159, 161, 200, 203, 204,208, 209, 222, 225, 227, 232, 238, 244,250, 294, 435

complexCech, 82–91, 94, 96, 98, 101, 130, 273–275affine n-space (An), see affineKoszul, 94–99, 278projective r-space (Pr), see projective

cone, 332affine algebraic, 332–337, 407, 422–425,

427non-degenerate, 332punctured, 334

connected sequencenegative, 10, 261negative strongly, 10, 261positive, 12

connected sequenceshomomorphism of, 11, 263isomorphism of, 11, 263

connected topological space, 406Connectedness Bound for Complete Local

Rings, 414Connectedness Criterion for the Special Fibre,

435connectedness dimension (c(T )), 408–416,

429–431formal, at p (cp(W )), see formal

connectedness dimensionlocal, at p (cp(T )), see local connectedness

dimensionConnectedness Theorem of Grothendieck, see

Grothendieck’sConnectivity Theorem

of Bertini, see Bertini’s ConnectivityTheorem

of Bertini–Grothendieck, seeBertini–Grothendieck ConnectivityTheorem

of Fulton–Hansen and Faltings, 431coordinate ring, 40

homogeneous, see homogeneouscurve, 338, 341

Cartesian, 437cuspidal, 437fully branched, at p, see fully branched

curve

Index 487

Macaulay’s, 343rational normal, 341

cuspidal curve, see curve

Deligne Correspondence Theorem, 460Graded Version of, 461

Deligne Isomorphism Theorem, 448, 451, 457for Affine Schemes, 458Graded, 454, 457

depth, 114, 115, 154, 155, 169, 228, 232, 308,459, 474

a-adjusted, 169b-minimum (λb

a(M)), 169diagonal, 427

isomorphism, 427dimensionb-finiteness relative to a (fb

a (M)), 167finiteness relative to a (fa(M)), 166, 373injective, 215, 235of a coherent sheaf of OT -modules, 471of a module, 107, 114, 115, 143, 144, 230,

305of a Noetherian topological space, 408of a variety, 338projective, 177–179, 181, 362

disconnected topological space, 406discrete valuation ring, 350dual, see Matlisduality, see Matlis or local duality

end of a graded module (end(M)), see gradedmodule

endomorphism ring of canonical module,238–244, 248–250

equidimensional local ring, see local ringessential extension, 193, 194, 202, 212, 257essential subset (with respect to S), 440*essential extension, 257–259Euler characteristic (χ(F)), 472extended Rees ring (R[aT, T−1]), 389, 394extension of an ideal, under f , 37, 67

Faltings’Annihilator Theorem, 183, 191

Graded, 309Connectivity Theorem, see Connectivity

Theoremfamily of denominators, 441family of fractions, 441S-local, 441

family of homogeneous fractions, 453S-local, 453

fibre cone, 394filter-regular sequence, see R+-filter-regular

sequencefiniteness dimension relative to a (fa(M)),

see dimensionFiniteness Theorem

of Grothendieck, see Grothendieck’s

Graded, see Grothendieck’sFirst Uniqueness Theorem (for secondary

representation), see secondaryFlat Base Change Theorem, 75

Graded, 290flat ring homomorphism, 68, 69, 75, 77, 78,

109, 180, 221, 232, 242, 290, 352formal connectedness dimension, at p

(cp(W )), 419–421formal fibres, 188, 191, 192, 244F -rational local ring, see local ring*free graded module, see moduleFrobenius

action, 102, 103, 105, 131, 133, 134homomorphism, 102, 103power (of an ideal), 102

full ring of fractions, 64fully branched curve, at p, 162Fulton–Hansen Connectivity Theorem, see

Connectivity Theoremfunction

characteristic, 364, 368–371cohomological Hilbert, see Hilbertof polynomial type, 366of reverse polynomial type, 367

functora-torsion (Γa), 2a-transform (Da), 21, 92, 117–120, 124,

161, 268, 279commutes with direct limits, 64

B-transform (DB), 22, 268, 280faithful, 226φ-coarsening, 275generalized local cohomology (Hi

B), 22local cohomology, 3

commutes with direct limits, 63R-linear, 1, 6section, 457, 460, 468sheaf cohomology, see sheafshift, see shiftVeronesean, see Veronesean

generalized Cohen–Macaulay module,185–188, 192

generalized ideal transform, 22, 245is independent of the base ring, 37right derived functors of, 22universal property of, 29, 32–34, 278

generalized local cohomology functor (HiB),

see functorgenerating degree, 386Gorenstein

local ring, 5, 154–158, 212–221, 225, 227,231–237, 243–245, 308, 474

*local graded ring, 309–311, 313, 321–323,326, 330, 372

ring, 154, 216, 238, 308, 315Goto–Watanabe Theorems, 302–308, 328

488 Index

grade, 113, 180, 182–184, 239, 339M -, 113

graded local cohomology, 251–330and the shift functor, 291

graded local duality, see local dualitygraded module, 253–257

associated to a sheaf, 467beginning of (beg(M)), 287*canonical, see *canonical moduleend of (end(M)), 287*indecomposable, see *indecomposable

graded module*injective, see *injective

graded ring, 253associated, see associated graded ring*local, see *local graded ringpositively Z-graded, 282, 326, 332, 347,

388, 405homogeneous, 347

positively Zn-graded, 282*simple, see *simple graded ring

Grothendieck’sConnectedness Theorem, 415Finiteness Theorem, 183, 192

Graded, 309, 475Splitting Theorem, 479Vanishing Theorem, 107

group ring, 303

height (ht)of a proper ideal, 20, 42, 76, 79, 111, 118,

120, 134, 185, 238, 248–250, 305–309,328, 335, 397, 412

of the improper ideal, 20, 169, 248Hilbert

coefficients, 371, 378, 381–387, 472function

cohomological, 371, 373, 472polynomial, 370, 384, 385, 472

cohomological, 371, 373, 473homogeneous

coordinate ring, 337–339, 345, 360localization, 287positively Z-graded ring, see graded ringregular function, 333ring homomorphism, 287S-local family of fractions, see family of

homogeneous fractionsHomogeneous Prime Avoidance Lemma, 347homomorphism of connected sequences, see

connected sequencesHorrocks’ Splitting Criterion, 478hypersurface, 425

ideal transform, 21, 40, 93, 443, 448, 462conditions for exactness of, 117–121, 124geometrical significance of, 41, 44, 45is independent of the base ring, 37

right derived functors of, 22universal property of, 29, 32–34, 278

indecomposable module, 195–199, 213*indecomposable graded module, 295–302,

324–326Independence Theorem, 70

Graded, 289injective

cogenerator, 141, 236dimension, see dimensionenvelope (E( • )), 193–210, 212, 213, 216,

258, 300hull, 193module, 5, 12, 15, 18, 19, 21, 68–70, 75, 90,

100, 141, 148, 193–199, 202, 207resolution, 3, 18, 19, 21, 71, 75, 150

minimal, 212–217, 299, 301*injective

envelope (*E( • )), 257–259, 295–302graded module, 256–261, 289, 291, 292,

295–303, 324–326hull, 257resolution, 300, 325

minimal, 298–302integral closure, of a (a), 390integrally dependent, on a, 389–390Intersection Inequality for the Connectedness

Dimensionsof Affine Algebraic Cones, 429of Projective Algebraic Sets, 430

inverse family of ideals, 8, 13, 20, 48inverse polynomials, see module ofirreducible

component, 407ideal, 196topological space, 406

irrelevant ideal (R+), 332isomorphism of connected sequences, see

connected sequences

Koszul complex, see complexKrull dimension of a module, see dimension,

of a module

large with respect to S, 450Lichtenbaum–Hartshorne Vanishing Theorem,

156–159, 216, 294, 412Graded, 294, 433, 435

local arithmetic rank, see arithmetic ranklocal cohomology

functor, see functorgraded, see graded local cohomologymodule, see module

local connectedness dimension, at p (cp(T )),417–420

local dimension, at p (dimp T ), 417local duality, 216–221, 225, 234

graded, 310–312, 318–320, 323, 326

Index 489

Local Duality Theorem, 218Graded, 310, 372

local ringanalytically irreducible, 161, 162, 250Artinian, see Artiniancatenary, 242, 243Cohen–Macaulay, see Cohen–Macaulaycomplete, see complete local ringequidimensional, 181, 250F -rational, 134Gorenstein, see Gorensteinof a quasi-affine variety, at q (OW,q), 122regular, see regular local ringuniversally catenary, see universally

catenarylocal subdimension, at p (sdimp T ), 417–421local vanishing ideal, at p (IV,p(C)), 419–421Local-global Principle for Finiteness

Dimensions, 189, 191locally free coherent sheaf, see sheaf*local graded ring, 286

Cohen–Macaulay, see Cohen–MacaulayGorenstein, see Gorenstein

locusnon-S2, 248, 249, 344S2-, 244

Macaulay’s curve, see curveMatlis

dual, 200duality, 199–210, 225, 228, 229

graded, 312, 372Duality Theorem, 204

Partial, 209maximal M -sequence, see sequence*maximal graded ideal, 286Mayer–Vietoris sequence, 51–56, 64, 67, 79,

121, 412graded, 288, 435

metric topology, 46M -grade, see grademinimal graded free resolution, 362module

acyclic, see acyclicArtinian, see Artiniana-torsion, 3, 17–19, 37, 85, 92a-torsion-free, 3, 17, 67B-torsion, 21, 27, 29, 32, 33B-torsion-free, 21Buchsbaum, see Buchsbaumcanonical, see canonical module*canonical, see *canonical moduleCohen–Macaulay, see Cohen–Macaulayfaithful, 110, 201, 232, 249, 327, 328*free, 256, 257, 266, 303, 306, 307, 317,

318, 320, 344, 362, 381generalized Cohen–Macaulay, see

generalized Cohen–Macaulay module

graded, see graded moduleindecomposable, see indecomposable

moduleinjective, see injectivelocal cohomology, 3of deficiency, 228of inverse polynomials, 276, 277, 323, 325p-secondary, 139quasi-Cohen–Macaulay, see

quasi-Cohen–Macaulay modulerepresentable, see representable modulesecondary, see secondarysum-irreducible, see sum-irreducible

moduleMonomial Conjecture, 131morphism

of sheaves, 456induced, 456

of varieties, 122, 123M -regular sequence, see sequenceM -sequence, see sequence

maximal, see sequencepoor, see poor M -sequenceweak, see weak M -sequence

multiplicatively closed subset, xximultiplicity, 371, 386Mumford’s Regularity Bound, 473

natural gradings, 454, 461, 462n-complement, see complementnegative

connected sequence, see connectedsequence

strongly connected sequence, see connectedsequence

Noetherian topological space, 407–411, 417,418, 433, 440

dimension of, see dimensionNon-vanishing Theorem, 109, 144non-S2-locus, see locusnormal projective scheme, see projectivenumerical invariant, 378

finite, 378numerical polynomial, 366

open set, 179, 244affine, see affinequasi-affine, see quasi-affine

parameter, 186Partial Matlis Duality Theorem, see Matlisp-component, 417pole, 46

isolated, 46polynomial ring, 40, 58, 71, 72, 91, 159, 181,

192, 254, 275, 282, 284, 323, 324, 326,329, 330, 336, 339, 342, 343, 347, 350,353, 354, 357, 360, 362, 378–380, 386,387, 426, 451, 473, 476, 478, 479

490 Index

Veronesean subring of, see Veroneseanpoor M -sequence, 112, 127–130positive connected sequence, see connected

sequencepostulation number, 371, 385–387prime characteristic, 50, 102, 103, 127,

131–134product of varieties, 427projective

algebraic set, 336dimension, see dimensionr-space (Pr(K)), 336–343, 345

complex (Pr), 337scheme, 466–479

normal, 476regular, 476, 477

spectrum (Proj(R)), 375, 433–437, 441,466–479

variety, 336–343, 345, 360, 426, 431*projective graded module, 256projectivization, 337punctured

cone, see conespectrum, 409

quasi-affineopen set, 418variety, 39, 40, 42, 44–46, 122, 127, 162,

407analytically irreducible, at p, 162analytically reducible, at p, 162

quasi-Cohen–Macaulay module, 185quasi-compact topological space, 190, 406,

440, 444, 453quasi-projective variety, 337, 418, 419

radical ideal, 180, 419rational normal curve, see curverational normal surface scroll, 345Ratliff’s Theorem, 188, 191, 417reduction, 2, 388–398

minimal, 389–391, 395, 398, 404number, 389, 397, 402, 404

*reduction, 397–399Rees ring (R(a)), 436

extended, see extended Rees ringregular local ring, 5, 72, 73, 79, 110, 111,

115–117, 153, 154, 157, 159, 181, 185,222, 225, 232, 234, 306, 474

regular projective scheme, see projectiveregular ring, 76, 176, 180–182, 186, 188, 303,

304, 309, 335, 475regularity (Castelnuovo–Mumford), see

Castelnuovo(–Mumford) regularityRegularity Bound of Mumford, see

Mumford’s Regularity Boundrepresentable module, 140, 141restriction map, 442

*restriction property, 264–274, 277–280,287–291, 455, 457, 461, 462, 465

R+-filter-regular sequence, 399–402ring homomorphism

flat, see flat ring homomorphismhomogeneous, see homogeneous

ringed space, 456R-linear functor, see functorr-regular (in the sense of

Castelnuovo–Mumford), 351, 354, 356,357, 470

at and above level l, 351, 353, 354, 356, 358Rung’s display, 432, 434saturated ideal, 387saturation of an ideal, 387scheme

affine, see affineprojective, see projective

S-closure, see closureSecond Uniqueness Theorem (for secondary

representation), see secondarysecondary

module, 139representation, 139–142, 144, 158, 159

First Uniqueness Theorem for, 140minimal, 139, 221Second Uniqueness Theorem for, 141

section functor, see functorsections of a sheaf, see sheafsemi-local ring, 192, 250sequenceM -, 112maximal M -, 113M -regular, 112

Serre’sAffineness Criterion, 124Cohomological Criterion for Local

Freeness, 477, 479condition S2, 224, 233, 234, 242–245, 249,

250condition Si, 232Criterion for Generation by Global

Sections, 470Finiteness Theorem, 469, 476

Serre–Grothendieck Correspondence, 463,464, 468–472, 478

classical form of, 467Severi–Enriques–Zariski–Serre Vanishing

Theorem, 475sheaf

associated to M (˜M , (˜M)0), 458, 466coherent

generated by global sections, 470locally free, 476–479

cohomology functor, 460induced, 456

flasque, 459

Index 491

of ideals, 468of R-algebras, 456of ˜R-modules, 456sections of, 456splits (completely), 478twisted, 467

sheaves, morphism of, see morphismshift

functor, 253, 291, 321, 322, 324, 326, 454of a complex, 88

S2-ification, 245, 248–250in graded situations, 327–329, 343, 344

*simple graded ring, 298, 303, 304, 306simplicial complex, 255, 326S-local family of fractions, 441–455S2-locus, see locussocle, 202special fibre, 433–436splitting type, 478stalk, 456, 457, 459, 466, 470, 473, 474, 476Stanley–Reisner ring, 255, 326S-topology, 440, 441, 466structure sheaf of an affine scheme, 458subdimension (sdimT ), 410–417, 429, 430

local, at p (sdimp T ), see localsubdimension

subsystem of parameters, 130, 133sum-irreducible module, 141support of a coherent sheaf (SuppF ), 471surface, 338

affine, see affinesymbolic power (p(t)), 49, 50, 155, 156symmetric semigroup, 329, 330system of graded ideals, 267, 271, 277, 280system of ideals, 20, 21, 23, 32, 48–50, 66,

155, 156, 245–248subsystem of, 245, 246, 248

system of parameters, 130–133, 186–188, 228,229, 397

standard, 186–188syzygetic characterization of regularity, see

Castelnuovo(–Mumford) regularitytight closure (a∗), 131–134tightly closed ideal, 131–134topology

metric, see metric topologyS-, see S-topologyZariski, see Zariski topology

on Proj(R), see Zariski topologytotal order compatible with addition, 253, 255

transforma-, 21B-, 22, 29

trivial extension, 115, 237, 238, 315twisted

cubic, 341, 343quartic, 343sheaf, see sheaf

unique factorization domain (UFD), 118, 245,303, 304

universally catenarylocal ring, 188, 191, 192, 416, 417ring, 188, 192

vanishing ideal, 58, 332, 338, 360, 361, 373,419

local, at p (IV,p(C)), see local vanishingideal

Vanishing TheoremLichtenbaum–Hartshorne, see

Lichtenbaum–Hartshorne VanishingTheorem

of Grothendieck, see Grothendieck’sSeveri–Enriques–Zariski–Serre, see

Severi–Enriques–Zariski–SerreVanishing Theorem

variety, 337affine, see affineof an ideal (Var(a)), xxiprojective, see projective

arithmetically Cohen–Macaulay, seearithmetically

arithmetically Gorenstein, seearithmetically

quasi-affine, see quasi-affinequasi-projective, see quasi-projective

varietyVeronesean, 339, 341, 342, 360

functor, 281, 373, 402map, 340submodule (M(r,s)), 281, 341subring (R(r)), 281, 394

of a polynomial ring, 283, 284, 330, 339,342, 343, 356, 361

transformation, 340, 341weak M -sequence, 187Zariski topology, 39, 244, 336, 441

on Proj(R), 433Zariski’s Main Theorem on the Connectivity

of Fibres of Blowing-up, 437

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