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91 J . P . May et al. , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e memor y of Rober t J . Piacenza , 199 6
90 Joh n Roe , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6
89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6
88 Crai g Huneke, Tigh t closur e an d it s applications , 199 6
87 Joh n Erik Fornaess , Dynamic s i n severa l comple x variables , 199 6
86 Sori n Popa, Classificatio n o f subfactor s an d thei r endomorphisms , 199 5
85 Michl o Jimbo and Tetsuji Miwa , Algebrai c analysi s o f solvable lattic e models , 199 4
84 Hug h L. Montgomery, Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d
harmonic analysis , 199 4
83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e
problems, 199 4
82 Susa n Montgomery, Hop f algebra s an d thei r action s o n rings , 199 3
81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3
80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1
79 Michae l Frazier , Bjor n Jawerth, an d Guid o Weiss , Littlewood-Pale y theor y an d th e stud y
of functio n spaces , 199 1
78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1
77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0
76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0
75 F . Thoma s Rarrel l and L. Edwin Jones, Classica l aspherica l manifolds , 199 0
74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l
equations, 199 0
73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9
72 Pete r Orlik , Introductio n t o arrangements , 198 9
71 Harr y Dym , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d
interpolation, 198 9
70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9
69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d
superalgebras, 198 7
68 J . Willia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,
Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7
67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics ,
1987
66 G . Andrews , ^-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,
combinatorics, physic s an d compute r algebra , 198 6
65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o
differential equations , 198 6
64 Donal d S . Passman , Grou p rings , crossed product s an d Galoi s theory , 198 6
63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6
62 Bel a Boliobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6
61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6
60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6
59 Roge r How e an d Alle n Moy, Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5
58 H . Blain e Lawson , Jr., Th e theor y o f gauge fields i n fou r dimensions , 198 5
57 Jerr y L . Kazdan, Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5
56 Har i Bercovici, Ciprian Foia§, and Carl Pearcy, Dua l algebra s wit h application s t o invarian t subspaces an d dilatio n theory , 198 5
{Continued in the back of this publication)
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Conference Boar d o f the Mathematica l Science s
CBMS Regional Conference Serie s in Mathematic s
Number 2 4
Topics in the Homologica l Theory of Modules
Over Commutative Ring s
Melvin Hochste r
Published fo r th e Conference Boar d o f the Mathematica l Science s
by th e ^^Sfc. America n Mathematica l Societ y ^^ji3o.
inl^^im Providence , Rhod e Islan d ^ J ^ ^ t * C ^ ^ | f wit h suppor t fro m th e VQwC "
^ g s ^ Nationa l Scienc e Foundatio n °<%itZ x°
http://dx.doi.org/10.1090/cbms/024
Expository Lecture s from th e CBMS Regional Conferenc e
held a t th e University of Nebraska June 24-28, 1974
2000 Mathematics Subject Classification, Primary 13-XX ; Secondary 20-XX .
Library o f Congres s Cataloging- in-Publ icat io n D a t a
Hochster, Melvin , 1943 -Topics i n th e homologica l theor y o f module s ove r commutativ e rings . (Regional conferenc e serie s i n mathematics ; no . 24 ) "Expository lecture s fro m th e CBM S regiona l conferenc e hel d a t th e Universit y o f Nebraska ,
June 24-28 , 1974. " Bibliography: p . 1. Commutativ e rings . 2 . Module s (Algebra ) 3 . Homolog y theory . I . Conferenc e Boar d o f
the Mathematica l Sciences . II . Title . III . Series . QA1.R33 no . 2 4 [QA251.3 ] 510'.8 s (512'.4 ] 75-132 5 ISBN 0-8218-1674- 8
CIP
Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e source i s given .
Republication, systemati c copying , o r multiple reproductio n of any materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissioiiCaiiis.org .
© Copyrigh t 197 5 by th e America n Mathematica l Society . Printed i n th e Unite d State s o f America .
The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government .
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Visit th e AM S hom e pag e a t URL : http://wwv.ams.org /
11 1 0 9 8 7 6 5 0 6 0 5 0 4 0 3 0 2 0 1
Contents
Acknowledgements vi i
0. Introductio n . . 1
1. Preliminarie s 3
2. Som e Ope n Question s 8
3. Consequence s o f th e Exis tenc e o f Cohen-Macaula y Module s 1 5
4. Modification s an d th e Exis tenc e o f Bi g Cohen-Macaula y Module s i n Charac -
terist ic p > 0 2 0
5. Henselia n Rings , M . Artin ' s Approximatio n Theorem , an d Bi g Cohen-Macaula y
Modules Ove r F ie ld s o f Characteris t i c 0 2 7
6. Depth-Sensitivit y Theorem s an d Exactnes s Criteri a 3 4
7. Module s o f Project iv e Dimensio n 2 , Generi c Module s o f F in i t e Projectvi e Di -
mension, an d th e Buchsbaum-Eisenbu d Structur e Theorem s 44
8. Linea r Algebrai c Group s 5 2
9. Application s o f Homologica l Method s an d Some More Open Question s 6 1
10. Supplemen t 7 0
References 7 2
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Acknowledgements
I wan t t o than k th e man y peopl e a t Nebrask a wh o di d s o muc h i n arrangin g th e con -
ference an d makin g i t s o pleasan t t o vis i t Lincoln .
Part icular thank s g o t o Ma x Larsen , wh o di d a n excellen t jo b o f organizing .
I a ls o wan t t o expres s m y appreciatio n t o Roge r Wiegan d wh o rea d an d edite d th e
manuscript an d arrange d fo r i t t o b e reproduce d o n extremel y shor t not ice . Th e simpl e
form o f th e argumen t o n pp . 32—3 3 i s du e t o him .
A specia l wor d o f thank s goe s t o Caro l Por tsch e wh o di d a remarkabl e jo b o f ac -
curate typin g i n a ver y shor t t ime . o
I wan t t o than k th e student s an d facult y o f Arhu s University , particularl y Sve n Tof t
Jensen , Niel s Nygaard , an d Torgn y Svanes , wh o l is tene d patientl y t o man y version s
(both correc t an d incorrect ) o f argument s give n i n thes e notes .
Final ly, I wan t t o than k th e Rock y Mountai n Mathematic s Consortiu m an d th e Nation -
al Scienc e Foundatio n fo r sponsorin g an d supportin g th e conference .
vii
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Selected Title s i n Thi s Serie s (Continued from the front of this publication)
55 Willia m Arveson , Te n lecture s o n operato r algebras , 198 4
54 Willia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4
53 Wilhel m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3
52 Tsit-Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3
51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3
50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3
49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2
48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2
47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1
46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1
45 Ronal d L . Graham, Rudiment s o f Ramse y theory , 198 1
44 Philli p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,
1980
43 Willia m Jaco, Lecture s o n three-manifol d topology , 198 0
42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0
41 D . J. Newman , Approximatio n wit h rationa l functions , 197 9
40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9
39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8
38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8
37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8
36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8
35 R . Bowen , O n Axio m A diffeomorphisms , 197 8
34 L . Auslander, Lectur e note s o n nil-thet a functions , 197 7
33 G . Glauberman , Factorization s i n loca l subgroup s o f finite groups , 197 7
32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7
31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7
30 A . Pelczynski , Banac h space s o f analyti c function s an d absolutel y summin g operators ,
1977
29 A . Weinstein , Lecture s o n symplecti c manifolds , 197 7
28 T . A . Chapman , Lecture s o n Hilber t cub e manifolds , 197 6
27 H . Blain e Lawson , Jr., Th e quantitativ e theor y o f foliations , 197 7
26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d orders , 197 6
25 K . W . Gruenberg , Relatio n module s o f finite groups , 197 6
24 M . Hochster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings , 197 5
23 M . E . Rudin , Lecture s o n se t theoreti c topology , 197 5
22 O . T. O'Meara , Lecture s o n linea r groups , 197 4
21 W . Stoll , Holomorphi c function s o f finite orde r i n severa l comple x variables , 197 4
20 H . Bass , Introductio n t o som e method s o f algebrai c /{'-theory , 197 4
19 B . Sz.-Nagy , Unitar y dilation s o f Hilber t spac e operator s an d relate d topics , 197 4
18 A . Friedman , Differentia l games , 197 4
17 L * Nirenberg, Lecture s o n linea r partia l differentia l equations , 197 3
16 J . L . Taylor, Measur e algebras , 197 3
15 R . G . Douglas , Banac h algebr a technique s i n th e theor y o f Toeplit z operators , 197 3
14 S . Helgason , Analysi s o n Li e group s an d homogeneou s spaces , 197 2
13 M . Rabin , Automat a o n infinit e object s an d Church' s problem , 197 2
(See th e A MS catalo g fo r earlie r titles )
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