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MONOMIALALGEBRAS

Marcel Dekker, Inc. New York BaselTM

MONOMIALALGEBRAS

Rafael H. VillarrealDepartamento de Matemticas, Centro de Investigacin y de Estudios Avanzadosdel Instituto Politcnico Nacional (IPN) Mexico City, Mexico

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PURE AND APPLIED MATHEMATICS

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ed.; K. Makowski, trans.) (1971)5. L. Narici et al., Functional Analysis and Valuation Theory (1971)6. S. S. Passman, Infinite Group Rings (1971)7. L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory.

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115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988)116. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1988)117. J. A. Huckaba, Commutative Rings with Zero Divisors (1988)118. W. D. Wallis, Combinatorial Designs (1988)119. W. Wies` aw, Topological Fields (1988)120. G. Karpilovsky, Field Theory (1988)121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded

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Liapunovs Matrix Functions (2002)247. B. G. Pachpatte, Inequalities for Finite Difference Equations (2002)248. A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural

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Preface

Let R = k[x] = k[xi,... ,xn] be a polynomial ring in the indeterminatesxi,... ,xn, over the field k. Let

t. ri _ -an ain _ 1ji x Xi xn , i i,. . . , q,

be a finite set of monomials of R. We are interested in studying severalalgebras and ideals associated to these monomials. Some of these are:

the monomial subring: k [ f i , . . . ,fq] C k[x], the Rees algebra: fc[x, fit,..., fqt] C fc[x, t], which is also a monomial

subring,

the face ring or Stanley-Reisner ring: k [ x ] / ( f i , . . . , fq), if the mono-mials are square-free, and

the toric ideal: the ideal of relations of a monomial subring.

In the following diagram we stress the most relevant relations betweenthe properties of those algebras that will occur in this text.

Rees algebra

Face ring Monomial subring

Toric ideal

If such monomials are square-free they are indexed by a hypergraphbuilt on the set of indeterminates, which provides a second combinatorialstructure in addition to the associated Stanley-Reisner simplicial complex.

in

iv Preface

This book was written with the aim of providing an introduction to themethods that can be used to study monomial algebras and their presentationideals, with emphasis on square-free monomials. We have striven to providemethods that are effective for computations.

A substantial part of this volume is dedicated to the case of monomialalgebras associated to graphs, that is, those defined by square-free quadraticmonomials denning a simple graph. We will systematically use graph theoryto study those algebras. Such a systematic treatment is a gap in the litera-ture that we intend to fill. Two outstanding references for graph theory are[28] and [139].

In the text special attention is paid to providing means to determinewhether a given monomial algebra or ideal is Cohen-Macaulay or normal.Those means include diverse characterizations and qualities of those twoproperties.

Throughout this work base rings are assumed to be Noetherian andmodules finitely generated.

An effort has been made to make the book self contained by includinga first chapter on commutative algebra that includes some detailed proofsand often points the reader to the appropriate references when proofs areomitted. However we make free use of the standard terminology and nota-tion of homological algebra (including Tor and Ext) as described in [250]and [311].

The first goal is to present basic properties of monomial algebras. Forthis purpose in Chapter 2 we study affine and graded algebras. The topicsinclude Noether normalizations and their applications, diverse attributes ofCohen-Macaulay graded algebras, Hilbert Nullstellensatz and affine vari-eties, some Grobner bases theory and minimal resolutions.

In Chapter 3 a thorough presentation of complete and normal idealsis given. Here the systematic use of Rees algebras and associated gradedalgebras makes clear their importance for the area.

Chapter 4 deals with Hilbert series of graded modules and algebras, atopic that is quite useful in Stanley's proof of the upper bound conjecturefor simplicial spheres. Here we introduce the h-vector and a-invariant ofgraded algebras and give several interpretations of the a-invariant when thealgebra is Cohen-Macaulay. Some optimal upper bounds for the numberof generators in least degree of Gorenstein and Cohen-Macaulay ideals arepresented, which naturally leads to the notion of an extremal algebra. Asan application the Koszul homology of Cohen-Macaulay ideals with pureresolutions is studied using Hilbert function techniques.

General monomial ideals and Stanley-Reisner rings are examined inChapter 5. The first version of this chapter was some notes originally pre-pared to teach a short course during the XXVII Congreso National de laSociedad Matemdtica Mexicana in October of 1994. In this course we pre-

Preface__________________________________________v

sented some applications of commutative algebra to combinatorics. We haveexpanded these notes to include a more complete treatment of shellable andCohen-Macaulay complexes. The presentation of the last two sections ofthis chapter, discussing Hilbert series of face rings and the upper boundconjectures, was inspired by [25, 44] and [271].

Since monomial algebras defined by square-free monomials of degree twohave an underlying graph theoretical structure it is natural that some inter-action will occur between monomial algebras, graph theory and polyhedraltheory. We have included three chapters that focus on monomial algebrasassociated to graphs. One of them is Chapter 6, where we present connec-tions between graphs and ideals and study the Cohen-Macaulay property ofthe face ring. Another is Chapter 8, where we present a combinatorial des-cription of the integral closure of the corresponding monomial subring andgive some applications to graph theory. In Chapter 9 we consider monomialsubrings and toric ideals of complete graphs with the aim of computing theirHilbert series, Noether normalizations and Grobner bases.

The central topic of Chapter 7 is the normality of monomial subringsand ideals; some features of toric ideals are presented here.

At the end Chapter 10 is devoted to studying monomial curves and theirtoric ideals, where the focus of our attention will be on monomial spacecurves and monomial curves in four variables. Affine toric varieties andtheir toric ideals are studied in Chapter 11.

Most material in this textbook has been written keeping in mind atypical graduate student with a basic knowledge of abstract algebra and anon-expert who wishes to learn the subject. We hope that this book canbe read by people from diverse subjects and fields, such as combinatorics,graph theory, and computer algebra. Various units are accessible to upperundergraduates.

In the last fifteen years a dramatic increase in the number of researcharticles and books in commutative algebra that stress its connections withcomputational issues in algebraic geometry and combinatorics has takenplace. Excellent references for computational and combinatorial aspectsthat complement some of the material included here are [74, 92, 298], [44,274, 278] and [39, 315].

A constant concern during the writing of this text was to give appro-priate credits for the proofs and results that were adapted from printedmaterial or communicated to us. We apologize for any inadvertent omis-sion and would appreciate any comments and suggestions in this regard.

During the fall of 1999 a course on monomial algebras associated tographs was given at the University of Messina covering Chapter 6 to Chapter9 with the support of the Istituto Nazionale Di Alta Matematica FrancescoSeveri. It is a pleasure to thank Vittoria Bonanzinga, Marilena Crupi, Gae-tana Restuccia, Rossana Utano, Maurizio Imbesi, Giancarlo Rinaldi, Fabio

vi Preface

Ciolli and Giovanni Molica for the opportunity to improve those chaptersand for their hospitality.

We thank Wolmer V. Vasconcelos for his comments and encouragementto write this book. A number of colleagues and students provided help-ful annotations to some early drafts. We are specially grateful to AdrianAlcantar, Joe Brennan, Alberto Corso, Jose Martmez-Bernal, Susan Morey,Carlos Renterfa, Enrique Reyes and Aron Simis. We are also grateful toLaura Valencia for her competent secretarial assistance.

The Consejo National de Ciencia y Tecnologia (CONACyT) and theSistema National de Investigadores (SNI) deserve special acknowledgementfor their generous support. It should be mentioned that the development ofthis book was included in the project Estudios sobre Algebras Monomiales,which was supported by the CONACyT grant 27931E.

In the homepage "http://www.math.cinvestav.mx/profesores/vila" wewill maintain an updated list of corrections.

RAFAEL H. VILLARREAL

Contents

Preface iii

1 Commutative Algebra 11.1 Module theory . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Graded modules . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Cohen-Macaulay modules . . . . . . . . . . . . . . . . . . . . 161.4 Normal rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.5 Koszul homology . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Afflne and Graded Algebras 312.1 Noether normalizations . . . . . . . . . . . . . . . . . . . . . 312.2 Cohen-Macaulay graded algebras . . . . . . . . . . . . . . . . 352 .3 Hilbert Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . 452 .4 Grobner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 .5 Minimal resolutions . . . . . . . . . . . . . . . . . . . . . . . 58

3 Rees Algebras and Normality 653 .1 Symmetric algebras . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Rees algebras and syzygetic ideals . . . . . . . . . . . . . . . 663.3 Complete and normal ideals . . . . . . . . . . . . . . . . . . . 693.4 A criterion of Jiirgen Herzog . . . . . . . . . . . . . . . . . . 823.5 Jacobian c r i t e r i o n . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Hilbert Series 974.1 Hilbert-Serre's Theorem . . . . . . . . . . . . . . . . . . . . . 974.2 a-invariants and h-vectors . . . . . . . . . . . . . . . . . . . . 1064.3 Extremal algebras . . . . . . . . . . . . . . . . . . . . . . . . 1104.4 Initial degrees of Gorenstein ideals . . . . . . . . . . . . . . . 1 1 84.5 A symbolic study of Koszul homology . . . . . . . . . . . . . 125

VII

viii_______________________________________Contents

5 Monomial Ideals and Stanley-Reisner Rings 1295.1 Primary decomposition . . . . . . . . . . . . . . . . . . . . . . 1295.2 Simplicial complexes and homology . . . . . . . . . . . . . . . 1385.3 Face rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.4 Hilbert series of face rings . . . . . . . . . . . . . . . . . . . . 1505.5 Upper bound conjectures . . . . . . . . . . . . . . . . . . . . 154

6 Edge Ideals 1616.1 Graphs and ideals . . . . . . . . . . . . . . . . . . . . . . . . 1616.2 Cohen-Macaulay graphs . . . . . . . . . . . . . . . . . . . . . 1726 .3 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.4 Bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . 1826.5 Links of some edge ideals . . . . . . . . . . . . . . . . . . . . 1856.6 First syzygy module of an edge ideal. . . . . . . . . . . . . . . 1886.7 Edge rings with linear resolutions . . . . . . . . . . . . . . . . 192

7 Monomial Subrings 2017.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.2 Integral closure of subrings . . . . . . . . . . . . . . . . . . . 2097.3 Integral closure of monomial ideals . . . . . . . . . . . . . . . 2337.4 Normality of some Rees algebras . . . . . . . . . . . . . . . . 2397.5 Ideals of mixed products . . . . . . . . . . . . . . . . . . . . . 2487.6 Degree bounds for some integral closures . . . . . . . . . . . . 2597.7 Degree bounds in the square-free case . . . . . . . . . . . . . 2617.8 Some lexicographical Grobner bases . . . . . . . . . . . . . . 276

8 Monomial Subrings of Graphs 2818.1 The subring associated to a graph . . . . . . . . . . . . . . . 2828.2 Rees algebras of edge ideals . . . . . . . . . . . . . . . . . . . 2868.3 Incidence matrix of a graph . . . . . . . . . . . . . . . . . . . 2948.4 Circuits of a graph and Grobner bases . . . . . . . . . . . . . 2968.5 Edge subrings of bipartite planar graphs . . . . . . . . . . . . 3038.6 Normality of bipartite graphs . . . . . . . . . . . . . . . . . . 3118.7 The integral closure of an edge subring . . . . . . . . . . . . . 3158.8 The equations of the edge cone . . . . . . . . . . . . . . . . . 325

9 Semigroup Rings of Complete Graphs 3359.1 Monomial subrings of bipartite graphs . . . . . . . . . . . . . 3359.2 Monomial subrings of complete graphs . . . . . . . . . . . . . 3509.3 Noether normalizations of edge subrings . . . . . . . . . . . . 359

Contents________________________________________ix

10 Monomial Curves 36710.1 Defining equations of monomial curves . . . . . . . . . . . . . 36710.2 Symmetric semigroups . . . . . . . . . . . . . . . . . . . . . . 37710.3 Ideals generated by critical binomials . . . . . . . . . . . . . . 38510.4 An algorithm for critical binomials . . . . . . . . . . . . . . . 399

11 Affine Toric Varieties and Toric Ideals 40311.1 Systems of binomials in toric ideals . . . . . . . . . . . . . . . 40311.2 Affine toric varieties . . . . . . . . . . . . . . . . . . . . . . . 40911.3 Curves in positive characteristic . . . . . . . . . . . . . . . . . 414

A Graph Diagrams 421A.I Cohen-Macaulay graphs . . . . . . . . . . . . . . . . . . . . . 4 2 1A.2 Unmixed graphs . . . . . . . . . . . . . . . . . . . . . . . . . 424

Bibliography 425

Notation 447

Index 449

Chapter 1

Commutative Algebra

In this chapter some basic notions and results from commutative algebrawill be introduced. All rings considered in this book are commutative andNoetherian and modules are finitely generated. Our main references are theworks of W. Bruns and J. Herzog [44], D. Eisenbud [92], H. Matsumura [218]and W. Vasconcelos [298]. Some of the results presented below are juststated without giving proofs, if need be, the reader may locate the missingproofs in one of those references.

1.1 Module theoryNoetherian modules and localizationsA Noetherian ring R is a commutative ring with unit with the propertythat every ideal of / is finitely generated, that is, given an ideal I of R thereexists a finite number of generators fi,..., fq such that

/= a,i G R, Vi

Let R be a commutative ring with unit and let M be an .R-module. Recallthat M is called Noetherian if every submodule TV of M is finitely generated,that is, N = Rfi + + R f g , for some / i , . . . , fv in N.Proposition 1.1.1 If M is a finitely generated R-module over a Noetherianring R, then M is a Noetherian module.

Corollary 1.1.2 If R is a Noetherian ring and I is an ideal of R, thenR/I and Rn are Noetherian R-modules. In particular any submodule of Rnis finitely generated.

2________________________________________Chapter 1

Theorem 1.1.3 (Hilbert basis theorem) A polynomial ring A[x] overa Noetherian ring A is Noetherian.

One of the important examples of a Noetherian ring is a polynomial ringover a field k. Often we will denote a polynomial ring in several variables byA;[x] and a polynomial ring in one variable by k[x\. Unless otherwise statedthe letters k and K will always denote fields.

Theorem 1.1.4 An R-module M is Noetherian if and only if M satisfiesthe ascending chain condition for submodules, that is, M is Noetherian ifand only if for every ascending chain of submodules of M

NoC^C-'-CNnC Nn+l C C M

there exists an integer k such that N{ = Nk for every i > k.In this book by a ring (resp. module) we shall always mean a Noetherian

ring (resp. finitely generated module).The spectrum of a ring R will be denoted by Spec(R), it is the set of

prime ideals of R. The minimal primes of R are the minimal elementsof Spec(.R) with respect to inclusion and the maximal ideals of R are themaximal elements of the set of proper ideals of R with respect to inclusion.

Let I? be a ring and X = Spec(R). Given an ideal / of R, the set of allprime ideals of R containing 7 will be denoted by V ( I ) . It is not hard toverify that the pair (X, Z) is a topological space, where Z is the family ofopen sets of X, and where U is in Z iff U = X \ V(/), for some ideal /.This topology is called the Zariski topology of the prime spectrum of R.

A local ring (R, m, k) is a Noetherian ring R with exactly one maximalideal m, the field k R/m is called the residue field of R.

A homomorphism of rings is a map 1.

Commutative Algebra

Proof. Let n = char(.R) > 2. If n = p

4 ________________________________________Chapter 1

Krull dimension and heightBy a chain of prime ideals of a ring R we mean a finite strictly increasingsequence of prime ideals

p o C p i C C p n ,

the integer n is called the length of the chain. The Krull dimension of R,denoted by d\m(R), is the supremum of the lengths of all chains of primeideals in R. Let p be a prime ideal of R, the height of p, denoted by ht (p)is the supremum of the lengths of all chains of prime ideals

po C pi C C pn = p

which end at p. Note dim(/?p) = ht (p). If / is an ideal of R, then ht (I),the height of /, is denned as

ht(J) = min{ht(p)| / C p and p e Spec(fi)}.In general dim(.R/7) +ht (/) < dim(^). The difference dim(E) - dim(R/I)is called the codimension of /.

Let M be an .R-module. The annihilator of M is given by

annfl(M) = {x g R\ xM = 0},if m G M the annihilator of m is ann (TO) = ann (Rm). It is convenient togeneralize the notion of annihilator to ideals and submodules. Let NI andTV'2 be two submodules of M, their ideal quotient or colon is defined as

(Nl:RNi) = {xeR\xN2 C A^i}.Let us recall that the dimension of an .R-module M is

dim(M) = dim(fl/ann(M))and the codimension of M is

codim(M) = dim(E) - dim(M).Theorem 1.1.9 // A[x] is a polynomial ring over a Noetherian ring A,then dimA[x} = dim(A) + 1.

Primary decomposition of modulesLet / be an ideal of a ring R. The radical of / is

rad (I) = {x 6 R\ xn / for some n > 0},the radical is also denoted by \fl. In particular y^Oj, denoted by Tl/?, isthe set of nilpotent elements of R and is called the nilradical of R. A ringis reduced if its nilradical is zero.

Commutative Algebra

Proposition 1.1.10 // / is a proper ideal of a ring R, then rad (/) is theintersection of all prime ideals containing I.Definition 1.1.11 Let M be a module over a ring R, the set of associatedprimes of M, denoted by Ass#(M), is the set of all prime ideals p of R withthe property that there is a monomorphism (f> of /^-modules:

> M.

Note that p = ann ((!)).If M = R/I it is usual to say that an associated prime ideal of R/I is

an associated prime ideal of / and to set Ass(7) = A.ss(R/I).Proposition 1.1.12 Let R be a ring and S a multiplicatively closed subsetof R. If M is an R-module and p is a prime ideal of R with Sflp = 0, thenp is an associated prime of M if and only if S~l$ is an associated prime ofS~1M.

Proof. If p is in Ass(M), then R/f M. Hence

and 5~1p is an associated prime of S~1M .For the converse assume Sr~1p = ann(m/l). Since R is Noetherian p is

generated by a finite set a\, . . . , an, hence for each i there is Si G S suchthat Sjttim = 0. Set s = si sn. We claim p = ann(sm). Clearly onehas p C ann(sm). To show the other containment take x ann(sm), thenxsm = 0 and x/1 ann(m/l) = 5-1p. Hence x 6 p. DDefinition 1.1.13 Let M be an .R-module, the support of M, denoted bySupp(M), is the set of all prime ideals p of R such that Mp ^ 0.

A sequence 0 -) M' -> M -4 M" -> 0 of jR-modules is called a shortexact sequence if / is a monomorphism, g is an epimorphism and im(/) =ker(g). A sequence of R modules and homomorphisms:

is said to be ea;acf at Mj if im(/j_i) = ker(/j), if the sequence is exact ateach Mi it is called an exact sequence.

Lemma 1.1.14 // 0 -> M' -> M - M" - 0 is a short exact sequence ofmodules over a ring R, then

Supp(M) = Supp(M') U Supp(M").

6 ________________________________________ Chapter 1

Proof. Let p be a prime ideal of R. It suffices to observe that from theexact sequence

0 -> Mp -> Mv -> Mp' -* 0,we get Mp 7^ 0 if and only if M'v ^ 0 or M^' ^ 0. D

Theorem 1.1.15 // M zs an R-module, then there is a filtration of sub-modules

(0) = M0 c Mi C C Mn = Mand prime ideals pi, . . . , pn of -R such that Mj/Mj_i ~ /?/pj for all i.

In general the primes pi, . . . ,pn that occur in a filtration of the typedescribed in the previous result are not associated primes of the module M;see [87] for a careful discussion of nitrations and some of their applicationsto combinatorics.

Lemma 1.1.16 // 0 - M' - M - M" -> 0 is a short exact sequence ofmodules over a ring R, then

Ass(M) C Ass(M') U Ass(M").Corollary 1.1.17 If M is an R-module, then Ass/j(M) is a finite set.Proof. Let pi , . . . ,pn be prime ideals as in Theorem 1.1.15. By a repeateduse of Lemma 1.1.16 one has Ass^j(M) C {pi, . . . , pra} C Supp(M). D

Let M be an .R-module. An element x e R is a zero divisor of M ifthere is 0 ^ m M such that xm = 0. The set of zero divisors of M isdenoted by Z(M). If x is not a zero divisor on M we say that x is a regularelement of M .

Lemma 1.1.18 If M is an R-module, then

Z(M] =pAssR(M)

Proof. See [218, Theorem 6.1]. D

Definition 1.1.19 Let M be an .R-module. A submodule JV of M is saidto be a primary submodule if Ass/j(M/./V) = {p}.

An ideal q of a ring R is a primary ideal if Ass/j(.R/q) = {p}, thus q isa primary ideal iff xy G q and x q implies yn q for some n > 1.

Definition 1.1.20 Let M be an .R-module. A submodule TV of M is said tobe irreducible if N cannot be written as an intersection of two submodulesof M that properly contain N.

Commutative Algebra

Proposition 1.1.21 Let M be an R-module. If Q ^ M is an irreduciblesubmodule of M , then Q is a primary submodule.Proof. Assume there are pi and p2 distinct associated prime ideals of M/Qand pick TO pi \ p2 (or vice versa). There is Xi in M \ Q such thatpi = arm (Si), where Xj = a;, + Q. We claim that

(Rxi + Q) n (Rx2 + Q) = Q.If z is in the intersection, then z = AII + qi = \ix-i + q%, for some AJ jRand

8 ________________________________________ Chapter 1

Definition 1.1.26 Let R be a ring and let S be the set of nonzero divisorsof R. The ring S~1R is called the total ring of fractions of R. If R is adomain, S~1R is the field of fractions of R.Proposition 1.1.27 Let R be a ring and let K be the total ring of fractionsof R. If R is reduced, then K is a direct product of fields.Proof. Let pi, . . . ,p r be the minimal primes of R and S = R\ U[=1pj.Since R is reduced one has (0) = pi f~l D pr and K S~1R. Define

(f>: K > S-lR/S~lpi. x x S^R/S'^rby (x) = (x + 5~1pi, . . . ,x + 5~1pr). As S~1fi, . . . , S~1pr are maximalideals and its intersection is zero, it follows from the Chinese remaindertheorem that is an isomorphism (see Exercise 1.1.38). DLemma 1.1.28 Let M be an R-module and L an ideal of R. If LM = M,then there is x = l(modL) such that xM = (0).Proof. Let M = Rai + + Ran, a e M. As LM = M, there are btjin L such that cnj = J^jLi ^ijai- Set a = (a\, . . . ,an) and H = (6^) I,where / is the identity matrix. Since Hof = 0 and .ffadj(-ff) = det(H)I,one concludes det(H)cti = 0 for all i. Hence xM = (0) and x = l(modL),where x = det(H). D

Lemma 1.1.29 (Nakayama) Let R be a ring and N a submodule of anR-module M. If I is an ideal of R contained in the intersection of all themaximal ideals of R such that M = IM + N , then M = N.Proof. Note I(M/N) = M/N. By Lemma 1.1.28 there exists an elementx = l(mod/) such that x(M/N) (0). To finish the proof note that xis a unit; otherwise x belongs to some maximal ideal m and this yields acontradiction because / C m . D

Let M be an .R-module. The minimum number of generators of M willbe denoted by v(M). A consequence of Nakayama's lemma is an expressionfor v(M], when the ring R is local (cf. Corollary 2.5.2).Corollary 1.1.30 If M is a module over a local ring (R,m), then

v(M] = dimfc(M/mM), where k = R/m.Proof. Let a\ , . . . , aq be a minimal generating set for M and a.i = aj+mM.After a permutation of the aj one may assume that ai,. . . ,ar is a basis forM/mM as a k- vector space, for some r < q. Set TV = Ra\ + + Rar.Note the equality M = N + mM, then by Nakayama's Lemma N M.Therefore r = q, as required. D

Commutative Algebra

Modules of finite lengthAn .R-module M has finite length if there is a composition series

(0) = M0CMlC---CMn = M,where Mj/Mj_i is a nonzero simple module (that is, Mj/Mj_i has no propersubmodules other than (0)) for all i. Note that M;/M;_i must be cyclicand thus isomorphic to R/m, for some maximal ideal m. The number n isindependent of the composition series and is called the length of M, it isusually denoted by IR(M) or simply t(M).Proposition 1.1.31 If M is an R-module, then

Ass(M) C Supp(M) = T/(ann(M)),and any minimal element o/Supp(M) is in Ass(M).Proof. If p is an associated prime of M, then there is a monomorphismE/pM-M and thus 0 7^ (R/p)pMv. Hence p is in the support of M, thisshows the first containment.

Let p e Supp(M) and let x 6 ann (M). If x $ p, then xm = 0 for allm M and Mp = (0), which is absurd. Therefore p is in V(ann(M)).Conversely let p be in V(ann(M)) and let mi,...,mr be a finite set ofgenerators of M. If Mp = (0), then for each i there is Sj ^ p so thatSirrii = 0, therefore si sr is in ann (M) C p, which is impossible. HenceMp ^ 0 and p is in the support of M.

To prove the last part take a minimal prime p in the support of M. AsMp ji (0) there is an associated prime pi-Rp of Mp, where pi is a prime idealof R contained in p. Since MP1 ~ (MP)P1 ^ (0), we get that pi is in thesupport of M and p = pi. Therefore using Proposition 1.1.12 one concludesp e Ass(M). DProposition 1.1.32 // 0 > M' > M > M" > 0 is an exact sequence ofmodules over a ring R, then

dim(M) = max{dim(M'), dim(M")}.Proof. Set d = dim(Af), d1 = dim(M') and d" = dim(M"). First note thatd = dim(J?/p) for some prime p containing ann (M), by Proposition 1.1.31we obtain Mp ^ (0). Therefore using Lemma 1.1.14 one has M^ ^ (0)or Mp" ^ (0), thus either p contains ann (M') or p contains ann(M").This proves d < max{d',d"}. On the other hand ann(M) is contained inann (M') n ann (M") and consequently max{d', d"} < d. D

Let M be an .R-module, the minimal primes of M are defined to be theminimal elements of Supp(M) with respect to inclusion. A minimal prime

10________________________________________Chapter 1

of M is called an isolated associated prime of M. An associated prime ofM which is not isolated is called an embedded prime.

If I? is a ring and / is an ideal, note that the minimal primes of /are precisely the minimal primes of Assp.(R/I). In particular the minimalprimes of R are precisely the minimal primes of Asspt(R).Proposition 1.1.33 If M is an R-module, then M has finite length if andonly if every prime ideal in Supp(M) is a maximal ideal.Definition 1.1.34 A ring R is Artinian if IR(R) < oo.Proposition 1.1.35 Let R be a ring. Then R is Artinian if and only if Ris Noetherian and every prime ideal of R is maximal.Definition 1.1.36 Let R be a ring with total ring of fractions Q. An R-module M is said to have rank r if MR Q is a free Q-module of rank equalto r.

Lemma 1.1.37 If M is an R-module of positive rank r, thendim(M) = dim(fl).

Proof. Let p i , . . . ,p ra be the associated primes of R and S = R\ U"=1pj.By hypothesis S~1M ~ (S~lR)r. Since

Mp; ~ (5-1M)pi ~ [(S-^Hp, - [(S'-1fl)pJr ^ (RfiY t (0),we obtain that all the minimal primes of R are in the support of M andconsequently one has dim(M) = dim(.R). D

Exercises1.1.38 (Chinese remainder theorem) Let / i , . . . , Ir be ideals of a ringR. If It + Ij = R for i ^ j, prove:

(a) / i n - - - n / r = / i - - - / r i(b) The rings R/Ii f~l n Ir and R/Ii x x R/IT are isomorphic.

1.1.39 Let / i , . . . , /n,p be ideals of a ring R. If p is prime and nf=1/j C p,prove that /, C p for some i. In particular if n=1/, = p, then p = /j forsome i.

1.1.40 Let /, PI, . . . , Pr be ideals of a ring R. I f / i s contained in U[=1Pj andPi is a prime ideal for i > 3, prove that I C Pi for some i (cf. Lemma 2.2.18).1.1.41 Let R be a ring and / an ideal. If P is a prime ideal such thatI C P, prove that ht(/) < ht(/p) and show an example where the strictinequality holds.

Commutative Algebra

1.1.42 Let R be a ring and let / be an ideal of R. If all the minimal primesof / have the same height and P is a prime ideal such that I C P, provethat ht(7) =ht(/p).

1.1.43 Let / be an ideal of a ring R. Then a prime ideal p of R is anassociated prime of R/I if and only if p = (/: x) for some x R.

1.1.44 Let M be an .R-module and / an ideal of R contained in ann^(M).Note that M inherits a structure of 7?//-module. Prove that p 6 Ass^(M)if and only if p// e Assfl///(M).

1.1.45 Let R' k[x2, ,xn] and R = R'[XI] be polynomial rings over afield k. If /' is an ideal of R' and p A S S R ( R / ( ! ' , X I ) ) , then

(a) p = xiR + tp'R, where p' is a prime ideal of R' , and(b) p' is an associated prime of R' /I' .

1.1.46 Let A be a ring and q a proper ideal of A. Prove that q is a primaryideal if and only if Z(A/q) C WA/%, where 9\4/q is the nilradical of A/q.

1.1.47 Let A be a ring and q a proper ideal of A. If every element in A/qis either nilpotent or invertible, prove that q is a primary ideal.

1.1.48 Let A be a ring and q an ideal of A such that rad(q) = m is amaximal ideal. Show that q is a primary ideal.

Hint Note Spec(A/q) = {m/q} and 91^/q = m/q.

1.1.49 If B = A[x] is a polynomial ring over a ring A and q a primary idealof A, then qB is a primary ideal of B.

Hint Note (A/q)[x] ~ B/qB. Use induction together with the next exercise.

1.1.50 Let A[x] be a polynomial ring in one variable over a ring A. Provethat a polynomial f ( x ) G A[x] is a zero divisor iff there exists a ^ 0 in Asuch that af(x) = 0.

1.1.51 Let M be an _R-module and S C T two multiplicatively closed sub-sets of R. Prove T~1M ~ T-l(S~1M) as T~1R modules.Hint Consider (f>(m/t) = (m/l)/t, which is induced by the canonical mapfrom M to S~1M.

1.1.52 Let M be an E-module and S a multiplicatively closed subset of R.VIf p is a prime ideal such that S n p = 0, prove Mp ~ (S l M ) f

12 _______________________________________ Chapter 1

1.1.53 Let R be a ring and / an ideal. If x 6 R \ /, then there is an exactsequence of ^-modules:

0 > R/(I:x) A R/I -A R/(I,x) > 0,

where (/'(r) = xr is multiplication by x and (r) = f.

1.1.54 If 0 - M' -4 M -4 M" -> 0 is an exact sequence of -R-modules,prove that the following sequence is also exact

0 > S-'M' S-^4 S^M S-^4 S^M" > 0.

1.1.55 Let TVi and 7V2 be submodules of an ^-module M. If 5 is a multi-plicatively closed subset of R, then

(b) S-HMn^a) = S1-1(Ar1)nS-1(A^2), and(c) S~1(Nl/N2) ~ S-1(Ni)/S-1(N2), as S-^

1.1.56 Let NI and N2 be submodules of an E-module M. If S is a multi-plicatively closed subset of R, then

(a) 5-1(ann(A^2)) = ann^"1^)), and(b) 5-1(JV1;]V2) = (S-^NJ-.S-^NJ).

Hint Use that NI is a finitely generated ^-module.

1.1.57 If / and J are ideals of a ring R and 5 is a multiplicatively closedsubset of R, then

and

1.2 Graded modulesLet (H, +) be an abelian semigroup. An H-graded ring is a ring R togetherwith a decomposition

R = H^ Ra (as a Z-module),

such that RaRb C Ra+b for all a,b H. A graded ring is by definition aZ-graded ring.

Commutative Algebra ______________________________ 13>

If R is an ff -graded ring and M is an .R-module with a decomposition

M = 0 Ma,aeH

such that RaMb C Ma+b for all a, b e H , we say that M is an H-gradedmodule,

An element / e M is said to be homogeneous of degree a if / 6 Ma, inthis case we set deg(/) = a. The elements in Ra are also called forms ofdegree a.

A map (p-.M-^-N between ff-graded modules is graded if

14 _______________________________________ Chapter 1

Proposition 1.2.4 Let M be an N-graded R-module and Q a p-primarysubmodule of M . //p is graded and Q* is the submodule of M generated bythe homogeneous elements in Q, then Q* is again a tp-primary submodule.

Proof. See [217, Chapter 4]. D

Theorem 1.2.5 Let M be an N-graded R-module. If N is a graded sub-module of M , then N has an irredundant primary decomposition

N = jVi n n Nrsuch that:

(a) Ni is a graded submodule and AsSft(M/Ni) = {pi} for all i,(b) N ^ NI n n JVj_i n Ni+i n n Nr for all i, and

Proof. Use Lemma 1.2.3, Proposition 1.2.4 and Theorem 1.1.23. QFinding primary decompositions of graded ideals in polynomial rings

over fields is a difficult task, the punch line is that there is not yet a bestgeneral strategy for computing primary decompositions [77]; for some of themain algorithms that can been used see [95, 127, 255, 313].

For a specially nice treatment of the principles of primary decompositionconsult the book of Wolmer V. Vasconcelos [298, Chapter 3].

Lemma 1.2.6 (Graded Nakayama lemma) Let M be an N-graded R-module and let m = R+. If N is a graded submodule of M and I C m is agraded ideal of R such that M = N + IM , then N M.

Proof. Since M/N = I(M/N), one may assume N = (Q). If x

Commutative Algebra ______________________________ 1J5

Theorem 1.2.8 (Hilbert) Let R = ~0^ be a homogeneous ring andlet M be a finitely generated N-graded R-module with d = dim(M). // R0is Artinian, then there is a unique polynomial PM(^) Q[t] of degree d 1such that PM(I) = H(M,i) for i 3> 0. The polynomial PM() is called theHilbert polynomial of M .Proof. See [44, Theorem 4.1.3]. D

Definition 1.2.9 If PM(t) = Qd-i^""1 + + a0 is the Hilbert polynomialof M, the integer (d l)!ad_i is called the multiplicity of M and is denotedby e(M).Lemma 1.2.10 Let f ( t ) (E Q[t] be a polynomial of degree d I such thatf ( n ) e TL for all n e TL, then there are unique integers OQ, . . . , ad-i such that

d-i , .a i f i ( t ) i where /(*) =

Proof. The polynomials /(), i N, are a basis for Q[i] as a Q- vectorspace. Hence

d-l

i=0

for some a^ 6 Q. Using the Pascal triangle we get

i=0 L v ' \ ' J i=0

thus by induction on the degree it follows that a i ^ T L for alH. n

Exercises1.2.11 Let / be a graded ideal of a polynomial ring R. Prove that theradical of / is also graded.

1.2.12 Let d, m e N. Prove the equality

1.2.13 Let /: Z -> Z be a numerical function, / is said to be afunction of degree d if there is a polynomial P(t) Q[t] such that P(i) =f ( i ) for i ^> 0. Prove that / is a polynomial function of degree d if andonly if the numerical function g: 1 -> Z given by g>(i) = /(i) - f ( i - 1) is apolynomial function of degree d 1.

!16_______________________________________Chapter 1

1.2.14 Let R = k[xi,... ,xn] be a polynomial ring over a field k and letdi,..., dn be a sequence of integers. Prove that R is a graded ring with theZ-grading:

where xa = x^ x^ and |a| = a\d\ + + andn for a = (QJ) 6 Nn.1.2.15 Let R = igz-Rz be a graded ring. Prove that RQ is a subring ofR with 1 RQ. A graded ring R with Ri = 0 for i < 0 is called a gradedRo-algebra.

1.2.16 Let R = k[xi, x^] be a polynomial ring over a field k with the gradinginduced by deg(xi) = (1)*. Determine the subring /?o-

1.3 Cohen-Macaulay modulesHere we introduce some special types of rings and modules and present thefollowing fundamental result of dimension theory.

Theorem 1.3.1 (Dimension theorem) Let (R,m) be a local ring and letM be an R-module. Set

Commutative Algebra ______________________________ 17

Proposition 1.3.5 Let M be an R-module and I an ideal of R such thatIM ^ M . If 0_ = 6*1 , . . . , 6r is an M-regular sequence in I, then 9_ can beextended to a maximal M-regular sequence in I .

Proof. By induction assume there is an M-regular sequence Q\ , . . . , Oi in/ for some i > r. Set_M = M/(0i, . . . ,(9 i)M. If 7 Z(M] pick Oi+l in 7which is regular on M. Since

(0i) C (0i,02) C C (0i, . . . ,0i) C (0i, . . ,0i+i) C Ris an increasing sequence of ideals in a Noetherian ring 7?, this inductiveconstruction must stop at a maximal M-regular sequence in 7. D

Lemma 1.3.6 Let M be a module over a local ring (R,m). If GI, . . . ,0r isan M-regular sequence in m, then r < dim(M).Proof. By induction on dim(M). If dim(M) = 0, then m is an associatedprime of M and every element of m is a zero divisor of M. Using thesecond part of the proof of Lemma 1.3.10 one has dim(M/0iM) < dim(M).Since 9%, . . . ,9r is a regular sequence on M/OiM by induction one derivesr < dim(M). D

Proposition 1.3.7 Let M be an R-module and let I be an ideal of R.(a) HomR(R/I, M) = (0) iff there is x 6 7 which is regular on M.(b) ExtrR(R/I,M) ~ HomR(R/I,M/0M), where 6 = 0 1 , . . . , 0 r is any

M-regular sequence in I.

Proof, (a) =>) Assume 7 C Z(M). Using Lemma 1.1.18 one has 7 C pfor some p 6 Ass^(M). Hence there is a monomorphism ip:R/p -> M. Toderive a contradiction note that the composition

is a nonzero map, where y> is the canonical map from R/I to 7?/p. Theconverse is left as an exercise.

(6) Consider the exact sequence

0 > M A- M > M = M/0iM > 0.

According to [250, Theorem 7.3] there is a long exact sequence with naturalconnecting homomorphisms

, M) ExtrR(R/I, M)

18 _______________________________________ Chapter 1

Since 0\ is in I, using [250, Theorem 7.16] it follows that in the last exactsequence the maps given by multiplication by 6\ are zero. Hence

) ~ ExtrR(R/I,M),and the proof follows by induction on r. D

Let M ^ (0) be a module over a local ring (R, m). The depth of M,denoted by depth (M), is the length of any maximal regular sequence on Mwhich is contained in m. From Proposition 1.3.7 one derives

depth(M) = inf{r| Ex.trR(R/m, M) ^ (0)}.In general by Lemma 1.3.6 we have depth(M) < dim(M).Definition 1.3.8 An ^-module M is called Cohen-Macaulay (C-M forshort) if depth(M) = dim(M), or if M = (0).Lemma 1.3.9 (Depth lemma) IfQ^N^M^-LtQisa short exactsequence of modules over a local ring R, then

(a) //depth(M) < depth(L), then depth(TV) = depth(M).(b) //depth(M) = depth (L), then depth(TV) > depth(M).(c) //depth(M) > depth(L), then depth(TV) = depth(L) + 1.

Proof. See [298, Corollary A.6.3]. D

Lemma 1.3.10 If M is a module over a local ring R and z is a regularelement of M , then(a) depth(M/zM) = depth(M) - 1; and(b) dim(MfzM) = dim(M) - 1.

Proof. As depth M > depth M/zM applying the depth lemma to the exactsequence

0 > M -^ M > M/zM > 0yields depth(M) = depth(M/^M) + l.

To prove the second equality first observe the validity of the inequalitydim(M) > dim(M/zM), otherwise there is a saturated chain of prime ideals

ann (M) C ann (M/zM) C po C C pd,where d is the dimension of M and po is minimal over ann (M). Accordingto Proposition 1.1.31 the ideal po consists of zero divisors, a contradictionsince z ann (M/zM) C po- On the other hand the reverse inequalitydim(M) < dim(M/zM) + 1 follows from Theorem 1.3.1. D

Commutative Algebra

Proposition 1.3.11 If M ^ (0) is a Cohen-Macaulay R-module over alocal ring (R,m) and p 6 Ass(M), then

dim(J?/p) = depth(M).

Definition 1.3.12 Let (jR,m) be a local ring of dimension d. A system ofparameters (s.o.p) of R is a set Oi, . . . ,0d generating an m-primary ideal.

Theorem 1.3.13 Let (R,m,k) be a local ring and let

S(R) = min{f(7) dirndl/ml) / is an ideal of R with rad (/) = m}.Thendim(R) = 6(R).Proof. This result is a particular instance of Theorem 1.3.1. D

Theorem 1.3.14 (Krull principal ideal theorem) Let I be an ideal ofa ring R generated by a sequence hi, . . . ,hr. Then(a) ht (p) < r for any minimal prime p of I.(b) If hi,...,hr is a regular sequence, then ht (p) = r for any minimal

prime p of I .Proof, (a) Since ^/IRp = p-Rp from Theorem 1.3.13 one has the inequalityht (p) = dim(/?p) < r.

(b) Set J = (hi,. . . ,hi-i) and K = (hi, . . . ,ht). Assume ht (P) i - Ifor any minimal prime P over J. Since hi is regular on R/J and K/ J is aprincipal ideal one has ht (K/ J) = 1, thus there is a prime ideal po minimalover J such that J C po C p and consequently ht (p) > i. Using (a) onegets ht (p) = i. DCorollary 1.3.15 If (R,m,k) is a local ring, then dim(R) < dinifc(m/m2)and R has finite Krull dimension.Proof. Let xi,. . . ,xq be a set of elements in m whose images in m/m2form a basis of this vector space. Then m = (xi,...,xq) + m2. Henceby Nakayama's lemma we get m = (x\, . . . ,xq). Hence dim R < q byTheorem 1.3.13. DRemark If R is a Noetherian ring, then dim Rf < oo for all p Spec(jR).However, there are examples of Noetherian rings of infinite Krull dimension.

Definition 1.3.16 A local ring (R,m, k) is called regular it

dim(-R) = dim f c(m/m2).

A ring R is regular if Rp is a regular local ring for every p G Spec(.R).

20_______________________________________Chapter 1

Cohen-Macaulay rings A local ring (R, m) is called Cohen-Macaulay ifR is Cohen-Macaulay as an 7?-module. If R is non local and Rp is a C-Mlocal ring for all p Spec(.R), then we say that R is a Cohen-Macaulay ring.An ideal / of R is Cohen-Macaulay if R/I is a Cohen-Macaulay .R-module.

If R is a Cohen-Macaulay ring and 5 is a multiplicatively closed subsetof R, then S~1(R) is a Cohen-Macaulay ring (see [44, Theorem 2.1.3]).

Proposition 1.3.17 Let M be a module of dimension d over a local ring(R, m) and let 9_ = Q\,..., 9^ be a system of parameters of M. Then M isCohen-Macaulay if and only if 9_ is an M-regular sequence.Proof. =>) Let p be an associated prime of M. By Proposition 1.3.11one has dim(.R/p) = d. We claim that Q\ is not in p. If 9i p, then byNakayama's lemma one has (M/diM)v 7^ (0). Hence p is in the supportof M/OiM, a contradiction because by Corollary 1.3.3 one has the equalitydim(M/0iM) = d - 1. Therefore Oi p and Q\ is regular on M. Thus theproof follows by induction, because according to Lemma 1.3.10 one has thatM/OiM is a C-M module of dimension d 1 and the images of 0 % , . . . ,64in R/OI form a system of parameters of M/(9i)M.

Commutative Algebra______________________________21

Proof. Set d = dim(R) and r = hi (I). By Lemma 1.3.10 R/I is Cohen-Macaulay and dim(7?/7) = d r. Let p be an associated prime of R/I, thenusing Proposition 1.3.11 yields

dim(7?) - ht (p) > dim(7?/p) = depths/I) =d-r.As a consequence ht (p) < r and thus ht (p) = r. Hence 7 is unmixed. DTheorem 1.3.23 Let R be a Cohen-Macaulay ring and I a proper ideal ofR of height r. If I is generated by r elements f i , , f r , then I is unmixed.Proof. It is enough to prove that / has no embedded primes, becauseby Krull's theorem all minimal primes of 7 have height r. Let p and q betwo associated primes of R/I and assume p C q, then IRq C p7?q C q72q.Since 77?q has height r and is generated by r elements, one obtains (usingLemma 1.3.18) that / i / l , . . . , /r/l is part of a system of parameters of Rq.Therefore by Proposition 1.3.17 the sequence / i / l , . . . ,/r/l is a regularsequence and thus Proposition 1.3.22 proves that 77?q is unmixed. Noticingthat p/iq and q7?q are both associated primes of IRq, one derives p7?q = q7?qand hence p = q. D

Proposition 1.3.24 /// is an ideal of height r in a Cohen-Macaulay ringR, then there is a regular sequence / i , . . . , fr in I.Proof. By induction assume that /i , . . . , / s is a regular sequence in 7such that s < r. Note that ( / i , . . . , /) is unmixed by Theorem 1.3.23. If7 C Z(R/(fi,..., /s)), then ht (7) = s, which is a contradiction. Thereforethere is an element fs+i in 7 which is regular modulo ( / i , . . . , /s). D

Let A be a (Noetherian) ring one says that A is a catenary ring if forevery pair p C q of prime ideals ht (q/p) is equal to the length of anymaximal chain of prime ideals between p and q. If A is a domain, then Ais catenary if and only if ht (q/p) = ht (q) ht (p) for every pair of primeideals p C q.

Theorem 1.3.25 If R is a Cohen-Macaulay ring, then R is catenary.Proof. See [44, Theorem 2.1.12]. Q

Gorenstein rings Let M ^ (0) be a module over a local ring (R, m) andlet k = R/m be the residue field of 7?. The socle of M is denned as

Soc(M) = (0:M m) = {z e M\ mz = (0)},and the type of M is defined as

type(M) = dimfc Soc(M/xM),

22 _______________________________________ Chapter 1

where x is a maximal M-sequence in m. Observe that the type of M is welldefined because by Proposition 1.3.7 one has:

ExtrR(k,M) ~ EomR(k, M/xM) ~ Soc(M/zM),where r = depth(M). The ring R is said to be Gorenstein if R is a Cohen-Macaulay ring of type 1, and an ideal / C R is called Gorenstem if .R// isa Gorenstein ring.

For a thorough study of Gorenstein rings see [14, 44]; a vivid presentationof the ubiquity of Gorenstein rings is given in [184].

Exercises1.3.26 Let

Commutative Algebra______________________________23

1.3.32 Let M be a module over a local ring (R,m). If _ = Oi,... ,0r isan M-regular sequence in m, prove that can be extended to a system ofparameters of M.

1.3.33 Let M be an .R-module and let / be an ideal of R. If there is x /which is regular on M, show that Hom/j(.R//, M) = (0).1.3.34 Let M be an .R-module and let / be an ideal of R. If J C /, then:

HomR(R/I,M/JM) ~ (JM:MI}/JM.

1.4 Normal ringsLet A and B be two rings. One says that B is an A-algebra if there is ahomomorphism of rings

(f>:A > B.Note that B has an /1-module structure (compatible with its ring structure)given by

a b = tp(a)bfor all a A and b B.

Thus if A C B is a ring extension, that is, A is a subring of B, then Bhas in a natural way an A-algebra structure induced by the inclusion map.If A = K is a field and B is a /^-algebra, then (f> is injective, in this caseone may always assume that K C B is a ring extension.

Given F = { f i , . . . , fq} a finite subset of B, we denote the subring ojB generated by F and C of A-algebras is a map 0 which is botha homomorphism of rings and a homomorphism of ^-modules. Note that4>(a 1) = a 1 for all a A.

Definition 1.4.1 Let A be a subring of B. An element b 6 B is integralover A, if there is a monic polynomial 0 ^ /(a;) G .A[a;] such that /(&) = 0.The set A of all elements b e B which are integral over A is called theintegral closure of A in B.

Let .A be a subring of B, one says that A C B is an integral extensionof rings or that B is integral over A, if 6 is integral over A for all b . B.A homomorphism of rings B is called integral if S is integral over

in this case one also says that B is integral over A.

24 _______________________________________ Chapter 1

Proposition 1.4.2 Let A be a subring of B. If B is a finitely generatedA-module, then B is integral over A.

Proof. Let /3 B. There are ai, . . . , an B such that

B = Aai + - - - + Aan.

One can write

where m^- 6 A. Set M = (m,./), N = M f l l and a (ai, . . . , an). Here/ denotes the identity matrix. Since Not* = 0, one can use the formulaN&dj(N) = det(N)I to conclude at det(TV) = 0 for all i. Hence det(iV) = 0.To complete the proof note that

f ( x ) = (-l)ndet(M-x/)is a monic polynomial in A[x] and /(/?) = (-l)ndet(JV) = 0. nCorollary 1.4.3 If A is a subring of B, then the integral closure A of Ain B is a subring of B .Proof. Let a,/3 A. If a and /? satisfy monic polynomials with coefficientsin A of degree m and n respectively, then A[a, /3] is a finitely generated A-module with basis

{a1 ft 0 < i < m - 1 and 0 < j < n - I } .Hence by Proposition 1.4.2 A[a,/?] is integral over A. In particular a /?and a/3 are integral over A. D

Corollary 1.4.4 If B is an A-algebra of finite type, then B is integral overA if and only if B is finite over A.Proof. =>) This implication follows using the arguments given in the proofof Corollary 1.4.3.

4=) It follows from Proposition 1.4.2. DLemma 1.4.5 Let A C B be a ring extension. If B is a domain and b isan element of B which is integral over A, then A[b] is a field if and only ifA is a field.Proof. =>) Let a A \ {0} and c = a~l its inverse in ^4 [6]. Since A[b] isintegral over A, there is an equation

cn + an-icn~l +---+a1c + a0 - 0 (at A),

Commutative Algebra ______________________________ 25>

multiplying by a it follows rapidly that c 6 A.4=) Let A[x] be a polynomial ring and let

V>: 4 [a] A [6]

be the epimorphism given by if>(f(x)) = /(&). Note that ker(^) is a nonzeroprime ideal. As A[x] is a principal ideal domain, ker(^) is a maximal idealand consequently A[b] is a field. D

Proposition 1.4.6 Let A C B be an integral extension of rings. If B is adomain, then A is a field if and only if B is a field.

Proof. It follows from Lemma 1.4.5. D

Corollary 1.4.7 Let A C B be an integral extension of rings. If P is aprime ideal of B and p = PdA, then P is a maximal ideal of B if and onlyif p is a maximal ideal of A.

Proof. The result is a direct consequence of Proposition 1.4.6, becauseA/p C B/P is an integral extension of rings. D

Corollary 1.4.8 Let A C B be an integral extension of rings. If P C Qare two prime ideals of B such that P n A = Q n A, then P = Q.

Proof. Set p = P n A. As Bp is integral over Av and

using Corollary 1.4.7 we get that PB9 is maximal and hence P = Q. D

Lemma 1.4.9 If A C B is an integral extension of rings andp is a maximalideal of A, then pB ^ B and p = P n A for any maximal ideal P of Bcontaining pB.

Proof. If pB = B, one can write 1 = a\b\ + + aqbq with GJ pand bi B. Set S = A[bi , . . . , b q ] , the subring of B generated by b\ , . . . , bq.Since 5 is a finitely generated A-module and pS = 5, then by Lemma 1.1.28there is x = l(modp) such that xS = (0). As 1 G B, one derives x = 0 and1 G p, which is impossible. Hence p? ^ B. Note that this part of proofholds if p is a proper ideal of A.

If pB C P is a maximal ideal of B, then p c P n A and thus p = A n Pby the maximality of p. D

Proposition 1.4.10 If A C B is an integral extension of rings and p is aprime ideal of A, then there is a prime ideal P of B such that p = P fl A.

26_______________________________________Chapter 1

Proof. Since Af C Bp is an integral extension, then by Lemma 1.4.9pf?p ^ Bf. Note that P n A = p for any prime ideal P of B such that P?pis prime and contains p?p (see Exercise 1.4.23). nTheorem 1.4.11 (Going-up) Lei A C B be an integral ring extension.If p C q are iwo prime ideals of A and p = P n A for some P in Spec(B),then there is Q 6 Spec(B) such that q = Q n A and P C Q.Proof. Since A/p C B/P is an integral extension, by Proposition 1.4.10one has q/p = (Q/P) n (A/p) for some prime ideal Q of B containing P. Itfollows readily that q = A n Q. OProposition 1.4.12 Let A C B be an integral extension of rings. If B isintegral over A, then dim (A) = dim (5).Proof. The formula follows using Theorem 1.4.11, Corollary 1.4.8 andProposition 1.4.10. d

Let A be an integral domain and let KA be_its field of fractions. Theintegral closure or normalization of A, denoted A, is the set of all / e KAsatisfying an equation of the form

/" + an-ifn~l + + ai/ + a0 = 0 (a,i e A and n > I).By Corollary 1.4.3 the integral closure A of A is a subring of KA- If A = Awe say that A is integrally closed or normal. If A is not a domain we saythat A is normal if Ap is a normal domain for every prime ideal p of A.

A useful fact is that a normal ring is a direct product of finitely manynormal domains (see [92]).Proposition 1.4.13 Let A C B be an integral extension of rings. If B isa domain and A is a normal domain, then

(a) (going-down) if p C q are two prime ideals of A and q = Q D A forsome Q in Spec(S), then there is P G Spec(B) such that p = P n Aand P C Q.

(b) ht (7) = ht (/ n A) for any ideal I of B.Proof. See [217, Theorem 5] and [217, Theorem 20]. DProposition 1.4.14 Let A be a domain and let x be an indeterminate overA. Then A is normal if and only if A[x] is normal.Theorem 1.4.15 (Serre's criterion) A ring A is normal if and only if

(S2) depth(^p) > inf{2,ht(p)} for all p e Spec(A), and(Ri) Av is regular for all p G Spec(A) with ht (p) < 1.

Theorem 1.4.16 (Auslander-Buchsbaum) // (A,m) is a regular localring, then A is a unique factorization domain.

Commutative Algebra______________________________27

Flat and faithfully flat algebras Let B-primary ideal.

Proof. See [217, Sections 3.H, 5.D and 9.C]. D

Theorem 1.4.18 Let (j>: A > B be a faithfully flat homomorphism of rings.The following holds:(a) The map ^*:Spec(5) -> Spec(A), *(P) = $~l(P] is surjective.(b) IB n A = I and ht (/) = ht (IB) for any ideal I of A.

Proof. [217, Sections 4.C and 13.B]. D

Exercises1.4.19 Let K be a field and B an isomorphism of J^-algebras, note(p(r) = r for all r 6 K. If A = i->$Ai is a graded /C-algebra, then B is alsoa graded /sT-algebra graded by B = i

1.4.20 Let

28 Chapter 1

1.4.21 If A is a unique factorization domain, show that A is normal.

1.4.22 Show an example of an integral extension of rings A C B, such thatA is a field and B is not an integral domain (cf. Lemma 1.4.5).1.4.23 Let A C B be a ring extension and p a prime ideal of A. If S = A\pand Bp = S~~1B, prove that the map P i> PB$ gives a bijection betweenthe set of prime ideals of P of B such that P n A = p and the set of primeideals of Bp containing p5p.

1.4.24 Let B = A[x] be a polynomial ring over a ring A and let / be anideal of A. Prove (A/J)[x] ~ B/IB, where the left hand side is a polynomialring with coefficients in A/1.

1.4.25 If B = A[x] is a polynomial ring over a ring A and q is a p-primaryideal of A, then qB is a p5-primary ideal of B.Hint Use Theorem 1.4.17.

1.4.26 Let B = A[x] be a polynomial ring over a ring A and let J be an idealof A. If / = n[=1qj is a primary decomposition of 7, then /B = r\Ti=lc\iB isa primary decomposition of IB.

1.5 Koszul homologyLet a be an element of the ring R and Kit(a) be the complex defined as

R for = 0,1,* "* 0 otherwise,

with di:Ki(a) > -K"o(a) being multiplication by a.Let / be an ideal of R generated by the sequence z = {x\ ,... ,xn}. The

ordinary Koszul complex associated to z is defined as

#*(z; R) = A"*(zi) /T*(zn).For an R- module M we shall write AT*(z; M) for K*(x; R)M. The Koszulcomplex K*(x; R) is then the exterior algebra complex associated to E = Rnand the map

9:E >R,defined as

+ZnXn.

That is, 6 defines a differential d = dO on the exterior algebra /\(E) of Egiven in degree r by

r

d(ei / \ - - - f \ e r ) = '(-l)i~l9(ei)ei A A &i A A er.

Commutative Algebra ______________________________ 29

A consequence of the definition of the differential of K+ (x_; R) is that if wand w1 are homogeneous elements of f \ ( E ) , of degrees p and q respectively,then

d(w A w') = (-l)pw A d(w') + d(w) A w1.This implies that the cycles Z(fiT*)'form a subalgebra of /\(E), and thatthe boundaries B(K+) form a two-sided ideal of Z(K^). As a consequencethe homology of the Koszul complex, H*(x), inherits a skew commutativeR- algebra structure.

One can also see that H*(x) is annihilated by I = (x). Indeed, if e Eand uj Zr(K+~), we have from the last formula .

The ordinary Koszul complex K*(x) = K*(x_;R) is simply the complexof free modules

#*(z) : o -> A" -R" - A71"1 #n - > - A1 -R" -> A Rn -> >where f\kRn is the fct/i exterior power of f?"; thus f\kRn is a free .R-moduleof rank () with basis

{ejj A A ejjl < ij < < i* < n}.

Proposition 1.5.1 // x_ = X i , . . . ,xn is a regular sequence in R, then theKoszul complex is acyclic, that is, the complex K*(x) is exact.Proof. See [290, Theorem 2.3]. n

Sliding depth Let (R, m) be a Cohen-Macaulay local ring and let / bean ideal of R generated by xi, . . . ,xn, denote by H*(x) the homology of theordinary Koszul complex built on the sequence x_ = {xi, . . . , xn}.

Definition 1.5.2 (i) (SD) I satisfies sliding depth if

depth Hi(x) > dim(R) -n + i, Vi > 0.

(ii) (SCM) I is strongly Cohen-Macaulay if Hi(x) are C-M, Vi > 0.(Depths are computed with respect to maximal ideals. It is usual to set

depth(O) equal to oo.)

Remark 1.5.3 (a) The (SD) condition localizes [156], (b) If / satisfies (SD)with respect to some generating set, then it will satisfy (SD) with respectto any other generating set of /. This follows from the isomorphisms:

..,xn,0}) ~ f l j ( z ) #i_i(z), andHi({xi,...,xn,y}) ~ Hi({xi,...,xn,Q}), where y (x).

30 Chapter 1

Linkage Let / and J be two ideals in a Cohen-Macaulay local ring R. Theideals J and J are said to be (algebraically) linked if there is an /^-sequencex = {xi,..., xn} in / n J such that

I = ( ( x ) : J ] and J=((x):I],

if in addition / and J are unmixed ideals of the same height n withoutcommon components and such that / fl J = (x), then / and J are said tobe geometrically linked.

When / and J are linked we shall write I ~ J. We say that J is in thelinkage class of / if there are ideals l\,..., Im such that

Ir, J.The ideal J is said to be in the even linkage class of / if m is odd.

Let _R be a Gorenstein ring and let / be a Cohen-Macaulay ideal of R.If J is linked to I, then Peskine and Szpiro [233] showed that J is Cohen-Macaulay. A dramatic result of Huneke [179] proves that (SCM) is preservedunder even linkage, his method can be adapted to prove the following.

Proposition 1.5.4 Let I and J be two ideals in a Gorenstein local ring Rof dimension d, and let x_ = {x\,..., xn} be a generating set for I. Assumethat J is evenly linked to I. If I satisfies the condition(SDk) depth Hi(x;R) > d-n + i, 0

Chapter 2

Affine and GradedAlgebras

A few topics connected with affine and graded algebras are studied in thischapter, e.g., Grobner bases, Hilbert Nullstellensatz, minimal resolutionsand Betti numbers. We present the affine and graded versions of the Noethernormalization lemma and some of their applications to affine and Cohen-Macaulay graded algebras.

As before all base rings considered here are Noetherian and modules arefinitely generated.

2.1 Noether normalizationsAffine algebras occur naturally in algebraic combinatorics and geometry.One of the key results to understand their behavior is the famous Noethernormalization lemma, first we present its affine version.

Definition 2.1.1 Let k be a field and let S be a /c-algebra. We say that 5is an affine k-algebra if S = k [ y i , . . . , yr] for some y\,..., yr 5.

Definition 2.1.2 Let a and /? be in N. The lexicographical order in N"is obtained by declaring

/3>a

if the last nonzero entry of /? a is positive.

Notation The set of positive integers will be denoted by N+. If a,/3 G En,here a f3 will denote the usual inner product of a and /?.

31

32 _______________________________________ Chapter 2

Lemma 2.1.3 Let ai > > am be a sequence of m distinct points in Nordered lexicographically. Then there is w = (w\ ,..., wn) 6 N such thatWi = 1 and w cti > w on for i > 2.Proof. Let a; = (am, . . . ,ain) and fa = (an, . . . ,*(_!)). We proceed byinduction on n > 2. There is k so that ccin = = a^n and a.kn > Q-in fori > k. One may assume k < m; for otherwise fa > > fan and one canuse induction.

Since fa > > fa, by induction there is w' = ( l , u>2 , . . . , wn-i) suchthat w' fa > w' fa for 2 < i < k. On the other hand for every i > A; onecan choose Si N+ so that

w1 fa + ain8i > w' fa + ainSi.To finish the proof set

u;n = max{(5j| k 0, and Cj A; [2/2, ,2/n] /or > 1.Proof. The polynomial / can be written as

where 0 ^ 6j A; for all i. One may assume that a.\ > > am are orderedlexicographically. By Lemma 2.1.3 there is w N" such that Xj = xWi +yisatisfies the required properties. Note r = w a\. d

Lemma 2.1.5 If R = k [ x \ , . . . ,xn] is an affine k-algebra of dimension n,then R is a polynomial ring.

Proof. One may assume R ~ B/I, where B is a polynomial ring in nvariables with coefficients in the field k and / is an ideal of B. Let

/ C po C C pnbe a chain of prime ideals of B of length n. If / ^ (0), then adding (0)to the chain yields a chain of length n + 1, which is impossible becausedim(.B) = n. Hence / = (0). n

Affine and Graded Algebras _________________________ 33

Theorem 2.1.6 Let R = k[xi, . . . ,xn] be a polynomial ring over a field kand let I ^ R be an ideal of R. Then there are Zi, . . . ,zn in R such that(a) k[x\ , . . . , xn] is integral over k\z\ , , . . ,zn], and(b) lnk[zi,...,zn] = zik[zi,...,zn] + + zgk[zi,...,zn].

Proof. The proof is by induction on n. If n = 1 and 7 = ( f ( x i ) ) ^ 0, thenone sets z\ = f ( x \ ) . Assume n > 2 and 7 ^ 0 . One may assume that 7contains a monic polynomial in x\. Otherwise take a nonzero polynomial gin 7 and apply Proposition 2.1.4 to get an isomorphism of /c-algebras

k[xi,...,xn] -A k[xi ,y2...,yn]induced by ip(x\) = x\ and ip(xi) = xf + yi for i > 2, such that tp(g) ismonic in x\. Let

+ + Cr^iXi + CT

be a polynomial in 7 with Cj e k[x2, , xn] and r > 0. Set7i = Ir\k[x2,...,xn}.

By induction there are z%, . . . ,zn such that:(i) k[x2, , xn] is integral over k[z%, . . . , xn], and(ii) 7i n k[z2, ...,zn] = (z-2, . . . , Zn).

Set z\ = /. It is not hard to see that 7? is integral over k [ z i , . . . ,zn] andk [ z i , . . . , z n ] n 7 = (z i , . . . , ^ a ) ,

as required D

Corollary 2.1.7 If R = k [ x i , . . . ,xn] is a polynomial ring over a field kand I j^ R is an ideal of R, then

dim(7?/7) =dim(7?)-ht(7).Proof. One may assume that there are Zi, . . . , zn in 7? and an integer gsuch that the conditions (a) and (b) of Theorem 2.1.6 are satisfied. Wewill show that g is equal to the height of 7. By Lemma 2.1.5 the Zi's arealgebraically independent. Hence Proposition 1.4.13 yields

ht(7) = ht(lnk[zi,...,zn]) =g.Note that there is an integral extension

k[zg+i,...,zn] ~k[zi,...,zn]/(zi,...,zg) A k[xi,...,xn]/I.Therefore n-g = dim(7?/7). D

34_______________________________________Chapter 2

Corollary 2.1.8 (Noether Normalization Lemma) If R = k\x\ is apolynomial ring over a field k and I ^ R is an ideal, then there is anintegral extension

where hi,... ,hd are in R and d = dim(_R/7).

Proof. By Theorem 2.1.6 there is an integral extension

k[zg+i,...,zn] ~k[z1,...,zn]/(zi,...,zg) A R/I.

To conclude the argument set hi = zg+i for i = 1 , . . . , d. D

Corollary 2.1.9 If R = k[xi,... ,xn] is a polynomial ring over a field k,then R is a catenary ring.

Proof. Note ht (q/p) = dim(E/p) - dim(/?/q) for any two prime idealsp C q in R. Hence by Corollary 2.1.7 we get ht (q/p) = ht (q) - ht (p). D

Definition 2.1.10 Let k C L be a field extension. A subset of L whichis algebraically independent and is maximal with respect to inclusions iscalled a transcendence basis of L over k.

Theorem 2.1.11 If k C L is afield extension, then any two transcendencebasis have the same cardinality.

Proof. See [191, Theorem 8.35]. n

Definition 2.1.12 Let k C L be a field extension. The transcendencedegree of L over k, denoted trdegf t(L), is the cardinality of any transcendencebasis of L over k.

Corollary 2.1.13 Let k be a field and let A be a finitely generated k-algebra. If A is a domain with field of fractions L, then

dim(A) = trdeg fc(L).

Exercises2.1.14 If A and B are affine algebras over a field k, then

dim(A fc B) - dim(A) + dim(B).

Hint Use the normalization lemma.

Afflne and Graded Algebras_________________________35

2.2 Cohen-Macaulay graded algebrasIn this section we will emphasize the relationship between graded Cohen-Macaulay rings and their homogeneous Noether normalizations. Then someuseful characterizations of those rings will be given.

Definition 2.2.1 Let k be a field. A standard algebra or homogeneousalgebra is a finitely generated N-graded /c-algebra

4=0

such that j/j 5i for all i and So = k. If we only require j/j homogeneousand deg(j/i) > 0 for all i, we say that 5 is a positively graded fc-algebra.

The irrelevant maximal ideal m of 5 is defined by

t=l

Definition 2.2.2 Let A; be a field and S = k [ y \ , . . . ,yr] a positively graded^-graded with j/j homogeneous of degree di. There is a graded epimorphism

ip: R = k[xi,..., xr] > S

given by

where R is a polynomial ring graded by deg(i) = di, the presentation of5 is the fc-algebra R/ker(tf>). The graded ideal ker(

36_______________________________________Chapter 2

As R is a catenary domain one obtains ht (p) = r + g and consequentlyht (J) < r + g. It is not hard to verify that this inequality is equivalent tothe inequality

dim(S) < ht (L) + dim(5/L).To conclude the proof recall that the inequality

dim(S) > ht (L) + dim(5/L)holds in general. D

Corollary 2.2.4 Let R be a positively graded polynomial ring over a fieldk and I a graded ideal of R. If R/I is Cohen-Macaulay, then I is unmixed.Proof. It follows from Proposition 1.3.11 and Proposition 2.2.3. D

Definition 2.2.5 Let k be a field and let S be a positively graded fc-algebra.A set of homogeneous elements 6_ = { O i , . . . , & d } is called a homogeneoussystem of parameters (h.s.o.p for short) if d = dim(S) and rad (0) = S+.

Corollary 2.2.6 Let S be a positively graded algebra over a field k andhi,..., hci a homogeneous system of parameters for S. Then

dim S/(hi,..., hi) = dim(S) i,

forl

Affine and Graded Algebras_________________________37

Proof. By Corollary 2.2.6 dimS/(0i , . . . , 0*) = d - i. If S is C-M, thenby Proposition 2.2.3 one has ht ( 6 1 , . . . ,$;) = i. Assume # i , . . . , 0 j _ i is aregular sequence. Next we show that Q{ is regular on A = S / (# i , . . . , #j_i) .Otherwise if 0$ is a zero divisor of A, then #j belongs to some minimalprime p of ($1 , . . . , 0j_i) . Since ht (p) < i 1 (see Theorem 1.3.14) weobtain ht ($1, . . . ,$,) < i 1, which is impossible. Conversely if 9_ is aregular sequence, then depth(S) = d and S is C-M. D

Proposition 2.2.8 Let S be a positively graded algebra over afield k. If Sis Cohen-Macaulay and 9_ = &i,..., 9q is a regular sequence of homogeneouselements in S+, then S/(0) is Cohen-Macaulay.

Proof. It suffices to prove the case q = 1, which is a direct consequence ofLemma 1.3.10. D

Lemma 2.2.9 Let V ^ {0} be a vector space over an infinite field K. ThenV is not a finite union of proper subspaces of V.

Proof. We proceed by contradiction. Assume that there are proper sub-spaces Vi,..., Vm of V such that

1=1

where m is the least positive integer with this property. Let

vi Vi \ (V2 U U Vm) and v2 6 V2 \ (Vi U V3 U U Vm).Pick m + 1 distinct nonzero scalars ko, , km in K. Consider the vectors

fa = V i

for i 0, . . . ,m. By the pigeon-hole principle there are distinct vectors /3r,/3S in Vj for some j. Since /3r - /3S Vj we get u2 Vj. Thus j = 2 by thechoice of v% . To finish the proof observe that /3r V% imply v\ 6 V^ , whichcontradicts the choice of vi . n

Proposition 2.2.10 Let R = k[x\, . . . ,xn] be a polynomial ring over afieldk with a positive grading and let I be a graded ideal of R. Then there arehomogeneous polynomials hi, . . . , h

38 _______________________________________ Chapter 2

Proof. Assume d > 0. Let pi, . . . , pr be the set of minimal primes of Iof height g = ht (/). We claim that there is a homogeneous polynomial hinot in U[=1pj. To show it we use induction on r. Since pi - p r _ i 0.By induction there is g e fid2 and 5 U^pj, d% > 0. Assume Ri C U=1pifor all i > 0, hence g p r. To complete the proof of the claim considerh = fd2 gdl to derive a contradiction.

Note dim(R/I) > dim R/ (I, hi), by the choice of hi. Hence a repeatedapplication of the claim rapidly yields a sequence hi , . . . , hs of homogeneouspolynomials in R+ with s < d and such that ht (/, hi, . . . , hs) = dim(R).Therefore by Theorem 1.3.13 one concludes d = s, as required.

If k is infinite and deg(rEi) = 1 for all i, then there is hi in RI and notin U[=1pi; for otherwise one has RI = U[=1(pj)i and since .Ri cannot be afinite union of prop