Selected Title s i n Thi s Serie s

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 .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

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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[H ] , Grade'sensitive modules and perfect modules, P roc . Londo n Math .

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[H^] , Constraints on systems of parameters, Proc . Oklahom a Rin g Theor y

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[H ] } The equicharacteristic case of some homological conjectures on local

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[H«] , Grassmannians and their Schubert subvarieties are arithmetically

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[ H j , Rings of invariants of tori, Cohen-Macaulay rings generated by mono-

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24(1970) , 585 -595 . M R 4 3 #3275 .

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[zFFG] R . Fossum , H.-B . Foxb y an d P . Griffith , Minimal infective resolutions, Univ . o f

Illinois, Urbana-Champaig n (preprint) . A modified versio n wit h I . Reite n i s t o appea r i n Publ .

Math., I.H.E.S .

[zM] F . S . Macaulay , Some properties of enumeration in the theory of modular sys'

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[zPS] C . Peskin e an d L . Szpiro , C . R . Acad . Sci . Pari s Ser . A- B 27 8 (1974), 1421-1424 .

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(preprint).

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 )