Commutative Algebra: Durham 1981 1107087252, 9781107087255

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Mathematical

Professor I.M.

Institute,

James,

24-29 St Giles,Oxford

I. 4. 5.

General cohomology theory and K-theory, Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT

P.HILTON

8. 9. 10. II.

Integration and harmonic analysis on compact groups, Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.)

12.

Symposium on complex analysis, & W.K.HAYMAN (eds.)

13.

Combinatorics: Proceedings of the British Combinatorial Conference 1973, T.P.McDonough & V.C.MAVRON (eds.)

15. 16.

An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN

Canterbury,

1973,

R.E.EDWARDS

J.CLUNIE

17. Differential germs and catastrophes, Th.BROCKER & L.LANDER 18. A geometric approach to homology theory, S.BUONCRISTIANO, C.P. & B.J.SANDERSON 20. Sheaf theory, B.R.TENNISON

BOURKE

21. Automatic continuity of linear operators, A.M.SINCLAIR 23. Parallelisms of complete designs, P.J.CAMERON 24.

The topology of Stiefel manifolds,

I.M.JAMES

25. Lie groups and compact groups, J.F.PRICE 26. Transformation groups: Proceedings of the conference in the University 27. 28.

of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion. Hardy spaces and bounded mean oscillations,

K.E.PETERSEN Pontryagin duality and the structure of locally compact Abelian groups, S.A.MORRIS 30. Interaction models, N.L.BIGGS 31. Continuous crossed products and type III von Neumann algebras, A.VAN DAELE 32. Unifom algebras and Jensen measures, T.W.GAMELIN 33. Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE 34. Representation theory of Lie groups, M.F. ATIYAH et al. 29.

35. 36. 37.

Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry,

38. 39. 40. 41.

Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD,

42.

N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON

43. Graphs, codes and designs, 44. 45.

P.J.CAMERON &

G.W.BRUMFIEL

J.H.VAN LINT

Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications,

F.R.DRAKE

& S.S.WAINER (eds.) 46. p-adic analysis: a short course on recent work, N.KOBLITZ 47. Coding the Universe, A.BELLER, R.JENSEN & P.WELCH 48. Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds.)

49. Finite geometries and designs, P.CAMERON, J.W.P.HIRSCHFELD & D.R.HUGHES (eds.) 50. Commutator calculus and groups of homotopy classes, H.J.BAUES 51. Synthetic differential geometry, A.KOCK 52. Combinatorics, H.N.V.TEMPERLEY (ed.) 53. Singvilarity theory, V.I.ARNOLD 54. Markov processes and related problems of analysis, E.B.DYNKIN 55. Ordered permutation groups, A.M.W.GLASS 56. Journees arithmetiques 1980, J.V.ARMITAGE (ed.) 57. Techniques of geometric topology, R.A.FENN 58. Singularities of smooth functions and maps, J.MARTINET 59. Applicable differential geometry, F.A.E.PIRANI & M.CRAMPIN 60. Integrable systems, S.P.NOVIKOV et al. 61. The core model, A.DODD 62. Economics for mathematicians, J.W.S.CASSELS 63. Continuous semigroups in Banach algebras, A.M. SINCLAIR 64. Basic concepts of enriched category theory, G.M.KELLY 65. Several complex variables and complex manifolds I, M.J.FIELD 66. Several complex variables and complex manifolds II, M.J.FIELD 67. Classification problems in ergodic theory, W.PARRY & S.TUNCEL 68. Complex algebraic surfaces, A.BEAUVILLE 69. Representation theory, I.M.GELFAND et. al. 70. Stochastic differential equations on manifolds, K.D.ELWORTHY 71. Groups - St Andrews 1981, C.M.CAMPBELL & E.F.ROBERTSON (eds.) 72. Commutative algebra: Durham 1981, R.Y.SHARP (ed.) 73. Riemann surfaces: a view toward several complex variables, A.T.HUCKLEBERRY 74. Symmetric designs: an algebraic approach, E.S.LANDER

London Mathematical Society Lecture Note Series.

nla

Commutative Algebra; Durham 1981 Edited by R.Y. SHARP Reader in Pure Mathematics University of Sheffield

CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney

72

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 IRP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia © Cambridge University Press 1982 First published 1982 Printed in Great Britain at the University Press, Library of Congress catalogue card number:

82-12781

British Library Cataloguing in Publication Data Commutative algebra - (London Mathematical Society Lecture note series, ISSN 0076-0552; 72) 1. Mathematics - Congresses I. Sharp, R.Y. II. Series 510

QA3

ISBN O 521 27125 8

Qf\

oii-STiin myiYERSiin, eiii4bd LIBRART

Cambridge

CONTENTS

t

•

Preface

vii

Addresses of contributors

ix

List of participants

PART I:

THE LOCAL HOMOLOGICAL CONJECTURES, BIG

COHEN-MACAULAY MODULES, AND RELATED TOPICS The syzygy problem: a new proof and historical perspective E.G. EVANS and PHILLIP GRIFFITH The theory of homological dimensions of complexes

2

12

HANS-BJ0RN FOXBY Complexes of injective modules HANS-BJ0RN FOXBY

18

The local homological conjectures

32

MELVIN HOCHSTER The rank of a module G. HORROCKS

55

Modules of generalized fractions and balanced big Cohen-Macaulay modules R.Y.

SHARP and H.

Sur la theorie des complexes parfaits L.

61

ZAKERI 83

SZPIRO

PART II:

DETERMINANTAL IDEALS, FINITE FREE

RESOLUTIONS, AND RELATED TOPICS Some exact complexes and filtrations related to certain special Young diagrams KAAN AKIN and DAVID A.

92 BUCHSBAUM

The canonical module of a determinantal ring

109

WINFRIED BRUNS The MacRae invariant

121

HANS-BJ0RN FOXBY Finite free resolutions and some basic concepts of commutative algebra D.G.

NORTHCOTT

129

vi PART III:

MULTIPLICITY THEORY, HILBERT AND POINCAR^ SERIES,

ASSOCIATED GRADED RINGS, AND RELATED TOPICS Blowing-up of Buchsbaum rings

140

SHIRO GOTO Necessary conditions for an analytical algebra to be strict J.

163

HERZOG

Multiplicities,

Hilbert functions and degree functions

170

D. REES Finiteness conditions in commutative algebra and solution of a problem of Vasconcelos JAN-ERIK ROOS On the use of graded Lie algebras in the theory of local rings JAN-ERIK ROOS Reductions,

179

204

local cohomology and Hilbert functions of local

rings JUDITH D. SALLY

231

FURTHER PROBLEMS

243

PREFACE

A Symposium on Commutative Algebra was held at the University of Durham during the period 15-25 July,

1981, under the auspices of

the London Mathematical Society and with financial support from the Science and Engineering Research Council.

There were 71 partici¬

pants . The academic programme was built round a series of invited one hour lectures;

in addition, many participants volunteered lectures

at sessions of short talks. ations of space,

It was decided, on account of limit¬

to restrict this volume to articles by the invited

speakers related to lectures given at the Symposium, although all participants were welcome to contribute to the section of 'Further problems'

at the end of the book.

The articles have been grouped

together in sections in the hope that the result will reflect the flavour of the main themes of the Symposixmi.

The first group of

papers is concerned with the local homological conjectures, big Cohen-Macaulay modules,

and related topics and applications;

the

second group consists of articles related to determinantal ideals and finite free resolutions;

and the third group is concerned with

various topics in local algebra,

including multiplicity theory,

Hilbert and Poincare series, and associated graded rings. each section, authors'

Within

the papers are arranged in alphabetical order of

names.

Participants at the Symposium were invited to submit open problems in commutative algebra for inclusion in a Problem Section in these proceedings,

and the response to that invitation is

contained in the final section of the book, problems'.

entitled 'Further

I am grateful to the contributors for their efforts.

I am also very grateful to Mrs.

Elsie Benson and Mrs.

Janet

via williams

for their beautiful typing of the camera-ready copy for

this book. It is a pleasure co-organizer,

D.G.

to record my gratitude,

Northcott,

and that of my

to the numerous members of staff of

Durham and Sheffield Universities who contributed so much to the smooth-running arrangements and friendly atmosphere of the Symposium we are particularly grateful

to Dr,

University Mathematics Department, University Finance Department, Grey College, Douglas

Captain G.R.T.

Lund of the Durham Duffay, T.B.

the Bursar of Cruddis,

A.J.

we should like to record our gratitude to the London under whose auspices

and this book will appear,

have

M.O.

Sharpe.

Mathematical Society,

Council,

Mrs.

Woodward of the Durham

and our Sheffield colleagues Drs.

and D.W.

Finally,

L.M.

without whose

and the Science

financial

the Symposiiim took place and Engineering Research

support the Symposium could not

taken place. R.Y.

Sharp

ADDRESSES

OF CONTRIBUTORS

KAAN AKIN

and DAVID A.BUCHSBAUM,

Brandeis

University,

WINFRIED BRUNS,

Department of Mathematics,

Waltham,

Fachbereich

3,

Mass.

02154,

U.S.A.

Naturwissenschaften,

Mathematik,

Universitat Osnabrlick-Abteilung Vechta-Driverstrasse D-2848 Vechta, E.G.EVANS

and PHILLIP

GRIFFITH,

University of Illinois, HANS-BJ0RN FOXBY,

3-25-40,

Essen

1,

Ann Arbor,

Setagaya-ku,

Universitatsstrasse

D.G.NORTHCOTT,

48109,

upon Tyne NE1

7RU,

Hicks Building,

Exeter EX4

JAN-ERIK ROOS, Box 6701, JUDITH D.SALLY, Evanston,

Fachbereich

Postfach

103764,

D-4300

University of Michigan,

University of Newcastle upon Tyne, U.K.

Sheffield S3

4QE,

S-113

85 Stockholm,

and H.ZAKERI,

60201,

North Park

University of Stockholm,

Sweden. Northwestern University,

U.S.A.

Department of Pure Mathematics,

Hicks Building,

Ecole Normale Superieure, France.

U.K.

University of Exeter,

Department of Mathematics, Illinois

University of 7RH,

U.K.

Department of Mathematics,

of Sheffield,

Cedex 05,

3,

Japan.

U.S.A.

Department of Mathematics,

L.SZPIRO,

Tokyo,

Department of Pure Mathematics,

Sheffield,

Road,

Universitet, Denmark.

Nihon University,

Department of Mathematics, Michigan

School of Mathematics,

Newcastle

R.Y.SHARP

K0benhavns

U.S.A.

West Germany.

MELVIN HOCHSTER,

G.HORROCKS,

61801,

DK 2100 KszSbenhavn 0,

Universitat Essen-Gesamthochschule,

6-Mathematik,

D.REES,

Illinois

Department of Mathematics,

Sakurajosui J.HERZOG,

Department of Mathematics,

Urbana,

Matematisk Institut,

Universitetsparken 5, SHIRO GOTO,

22,

West Germany.

Sheffield S3 45,

Rue

7RH,

d'Ulm,

University

U.K.

75230 Paris

V-

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LIST OF PARTICIPANTS

Barnard, A.D.

(King's, London)

Macdonald,

Bartijn, J.

(Utrecht)

McLean,

I.G

K.R.

(Q.M.C.,

London)

(Liverpool)

Bijan-Zadeh, M.H. (Tehran)

MacRae, R.E.

(Boulder)

B(^gvad, R.

(Stockholm)

Massaza, C.

(Siena)

(Warsaw)

Merriman,

(Canterbury)

(Zurich)

Moore, D.J.

(Glasgow)

Brown, M.L.

(Coventry)

Nastold, H.-J.

(Munster)

Bruns, W.

(Osnabruck/Vechta) Northcott, D.G

Buchsbaum, D.A.

(Brandeis)

O'Carroll,

Chatters, A.W.

(Bristol)

Orbanz, U.

(K5ln)

Cruddis, T.B.

(Sheffield)

Orecchia, F.

(Napoli)

Douglas, A.J.

(Sheffield)

Porter, T.

(Bangor)

Evans,

(Urbana)

Pragacz, P.

(Torun)

(Munster)

Qureshi, M.A.

(Edinburgh)

(Osnabruck)

Ragusa, A.

(Catania)

Boratynski, Brodmann,

M.

M.

E.G.

Faltings, Flenner, Foxby,

G. H.

H.-B.

(Norman,

Oklahoma) Ratliff,

J.R.

L.

L.J.

(Sheffield) (Edinburgh)

(Riverside)

(Stockholm)

Rayner, F.J.

(Liverpool)

(Tokyo)

Rees, D.

(Exeter)

(Torino)

Rhodes, C.P.L.

(Cardiff)

Hajarnavis, C.R.

(Warwick)

Riley, A.M.

(Sheffield)

Herzog, J.

(Essen)

Robbiano,

(Genova)

Hochster, M.

(Ann Arbor)

Roberts, P.C.

(Salt Lake City)

Horrocks,

(Newcastle)

Roos,

(Stockholm)

(Ann Arbor)

Rotthaus, C.

(Munster)

(Tor\in)

Sally,

(Evanston)

(Southampton)

Schenzel,

Lascoux, A.

(Paris)

Sharp, R.Y.

(Sheffield)

Lech,

(Stockholm)

Sharpe, D.W.

(Sheffield)

Froberg, Goto,

R.

S.

Greco,

S.

Huneke,

C.

Jozefiak, Kirby,

G.

T.

D.

C.

L.

J.-E.

J.D. P.

(Halle)

xn Shimoda, Simig,

Y.

A.

Stafford, Strano,

R.

Strooker, Szpiro,

J.T.

L.

J.R.

W,V. (New Brunswick)

(Tokyo)

Vasconcelos,

(Rio de Janeiro)

Vetter,

U.

(Osnabruck/Vechta)

(Cambridge)

Weyman,

J.

(Torun)

(Catania)

Wilson,

P.M,H.

(Cambridge)

(Utrecht)

Wiseman,

A.N.

(Sheffield)

(Paris)

Woodcock,

Valla,

G.

(Genova)

Zakeri,

Vamos,

P,

(Sheffield)

71

H.

C.F.

(Canterbury) (Sheffield)

1

PART I

THE LOCAL HOMOLOGICAL CONJECTURES, COHEN-MACAULAY MODULES,

BIG

AND RELATED TOPICS

2 THE SYZYGY PROBLEM:

A NEW PROOF

AND HISTORICAL PERSPECTIVE

E.G.EVANS and PHILLIP GRIFFITH

This article is a brief survey of the results^ that led up to our solution of the

syzygy problem

[8]

as well as a discussion of

our solution of that problem as it was generalized during the Durham Symposium following conversations with Bruns,

Huneke, Roberts,

Foxby,

Hochster,

Szpiro and others.

From our view the syzygy problem began with three separate and unrelated events in 1969. (so far unpublished)

One was the siabmission of Hackman's

Ph.D. Thesis [11]

"Exterior powers and homology",

which contains on the penultimate page the statement of the problem. Using techniques of his thesis he proved that regular local rings of dimension three are unique factorization domains and writes as follows. "In order to prove the general UFD Theorem along the

one would need the following theorem:

same

lines,

If the projective dimension of

M is r and M admits a projective resolution O -> F

r

where the F^,

F

^ r-1

i < r,

F^ -> M O

are finitely generated free modules and F^ is an

admissible projective module, r . y (-1)^“ rk(F.) > k j=k ^ for all k < r-1;

O,

then

in other words,

the k-th syzygy module of the

resolution is of rank > k." A second event in 1969 was the appearance of Auslander and Bridger's monograph [1]

"Stable module theory" which contains among

many other results an explicit criterion for a module of finite projective dimension to be a k-th syzygy.

Because they were inter¬

ested in a different circle of ideas and were writing in more gene¬ rality than we need,

it is somewhat difficult to give a concise and

explicit reference to their connection with the syzygy problem.

Perhaps the best one is their Theorem 4.25 [1; p.127] which proves that,

if M has finite Gorenstein dimension,

concerning M are equivalent.

then seven statements

We need to remark that a module of

finite projective dimension has the same finite Gorenstein dimension Two of the seven equivaleht conditions are the following. (b) where

is

an exact sequence O ->■ M

the P^ are projective (f)

< k

There

For each prime

(that is,

ideal

^

M is

P every

^

a k-th syzygy). regular sequence of length

is Mp-regular. It is

interesting to note that this memoir is an outgrowth of

Bridger's Ph.D.

Thesis

Thirdly in

1969,

(Brandeis).

*

Peskine and Szpiro circulated a preprint of

what was to become the first two chapters of their remarkable article

[17]

which was famous

their

joint Ph.D.

[17;

for rings

This article established their

p.84])

containing a field.

is defined as

M a nonzero A-module of the

Thesis.

The

follows.

Let A be a local

for all

Peskine and Szpiro

if A is a

section property.

It is of

some

[17;

depth

>

i

(2)

depthH^{L.)

then all

for prime

prove

finitely

ideals P minimal indeed,

p.55] which states as

of finitely generated A-modules. (1)

p.86]

inter¬

interest to note that the weaker

Let A be a local ring and let

H. (L.) = O

Theorem 2.1,

is much easier to prove and,

their Lemme d'Acyclicite

Then

[l7;

in

finite projective dimension have the

inequality that depth Np < pd.M, SuppM n SuppN,

Then A has

is minimal

local ring containing a field,

generated A-modules of

ring and

finitely generated A-modules N,

one has dim Np < pd.M for each prime P which SuppM n SuppN.

intersection property

finite projective dimension.

intersection property if

that,

locale"

intersection property from which they settled many open

problems (cf.

"Dimension projective finie et cohomologie

o

->■

in

follows

from

follows.

^ O iie a complex

Suppose that

and = O or H^(L.)

= O.

for all i > 1.

1

The proof of the weaker inequality can be deduced lemma quite

easily.

First one replaces A by Ap,

from the

M by Mp and N by Np

4 where P

is a minimal prime in SuppMn

SuppN.

This

is harmless

since the projective dimension of M can only get smaller. is the only prime

in SuppMn SuppN.

a modules Tor^(M,N)

(including M ® N)

Oa-L

a-.

P

Therefore all of the homology have

finite length.

Let

->M->-Obea minimal projective resolution of M.

d

If

O

depthN> pd.M,

we apply the Lemme d'Acyclicite to the complex

Oa-L,®Na-. d

to conclude

O

that the complex is acyclic and,

standard depth counting argument for clude

Thus

Szpiro overcame was

length.

using a

long exact sequences,

that the complex is too short to have

nonzero and of finite

further,

Thus,

its

we con¬

zero-th homology

€he real difficulty Peskine and

to be able to establish the

stronger

inequality

one obtains by replacing depth by dimension. It is

interesting to recall

that all

three of

contributions were connected with Ph.D. T.heses.

these

early

It is

also note¬

worthy that the clear vision provided by hindsight has

shown that

the essential

ingredient in the final

solution came

understanding of the Lemme d'Acyclicite. remark that by

1970 Hartshorne

Perhaps we

from a better should also

[12] was developing a theme

in

algebraic geometry concerning questions on vector bundles of rank and complete

intersections which,

as

closely related to subsequent work on the Evans-Griffith

[15],

[4]

and Hartshorne

came

[16]

ideas

tried to extend Hackman's Briefly,

a module of rank k which was also a

A M and M would be free. esting results

syzygy problem

(cf.

in the mid

1970's.

in order to

Bruns-

Lebelt

settle the

ideal.

it to show that,

"large" Thus

syzygy,

it would follow

lines which provided the

if M was

then A M would that both

Lebelt managed to obtain several

along these

rather

he wanted to compare a projective reso-

lution of A M with that of M and then use

necessarily be a reflexive

was

[12],[13]).

The next collection of results

syzygy problem.

it turns out,

small

inter¬

first explicit

and affirmative results on the problem. In [11] was

1976 Bruns

[2]

established that the bound given by Hackman

the best possible

in a very general

sense.

He

showed that,

if R is a Cohen-Macaulay domain and if M is a k-th syzygy of

finite

5 projective dimension of rank exceeding k, submodule F

such that M/F

is

a k-th

then M contains a free

syzygy of rank exactly k.

proof consists of two parts.

One part uses

results of Eisenbud and Evans

[5]

exceeding k,

then there

The

the basic element

to show that,

if M has rank

is an x 6 M such that x is a minimal

generator of M note

for all prime ideals P of height k. One should P if k is less than dimR, then x can be taken in mM,

that,

where m denotes proof uses

the

the maximal

syzygy,

The second part of the

criterion of Auslander and Bridger

mentioned to show that, k-th

ideal of R.

then M/Rx

if x is

[l]

as described above and if M is

is again a k-th

syzygy.

Bruns'

cludes by descending induction on the rank of M. that,

if M is not free,

with k
L

L ^ O is a complex of free s o R-modules having finite length homology, then s is at least the dimension of R

(assuming the zero-th homology is nonzero).

This

gave added strength to the philosophy that many inequalities using depth as a bound might remain true when dimension is used. In 1974 Hochster [14] in his remarkable memoir "Topics in the homological theory of modules over commutative rings" validated the above philosophy. tained a field,

He constructed for any local ring R, which con¬

a maximal Cohen-Macaulay module

(not necessarily

finitely generated). These modules, by their existence,

allowed one

to replace depth by dimension in many existing inequalities. particular,

In

the earlier result of Peskine and Szpiro [17] and

Roberts [19] could be improved in this way. Now we shall examine our original proof of the syzygy problem [8].

Our first version contained various restrictions on the ring R

involved.

To be precise we needed that R contains a field in order

that factor rings of R have maximal Cohen-Macaulay modules cussed above).

(as dis¬

We needed that R is a domain so that rank is well

defined and we needed the Cohen-Macaulay property in order to apply the Auslander-Bridger criterion for a module to be a k-th syzygy. During discussions at the Symposium it became clear that the last two assumptions on the ring R could be dropped. Firstly, finite free resolution,

then one can define the rank of M to be the

alternating sum of the Betti numbers. Cohen-Macaulay,

if M has a

Secondly,

if R fails to be

then the depth of R is smaller and it becomes even

more difficult for a module to be a k-th syzygy of finite projective dimension.

It also became apparent during the course of the Symposium

that one could modify the proofs of Peskine and Szpiro [18] and Roberts [20] in order to provide a version of the Lemme d'AcyclicitI

suitable for one of the crucial steps in our proof. deviation in our new proof of the syzygy theorem below)

The final

(as presented

occurs in the use of our result on order ideals of minimal

generators [9].

There is a slight drawback here of a technical

nature in that we established our statement concerning the heights of order ideals of minimal generators under the assumption that the residue field is algebraically closed.

However we shall state a

slight modification of this result which is suitable for our needs here.

Except for this our proof is rather easier than our original

one in addition to being more general.

Def'Ln'it'ion. Let R be a local ring and let M be an R-module having a finite free resolution O ->•

F^

M -> O.

Then

the vank of M is defined by rank M =

^ (-l)^rankF.. Of course i=0 ^ this definition agrees with the usual one in case R is a domain or as generalized by Bruns [3].

Let

LEMMA.

(R,m,k)

he a oonTplete local ring and let n be a

fi-witely generated non-free R-module of f-intte progecttve dimension and having rank r.

Then there is a finite faithfully flat residue

field extension (R',m',k') of {R,m,k) and a minimal generator x of r'® m having order ideal of height less than or equal to r.

Proof,

The argument is essentially that given in [9].

a minimal generating set e^,...,e^ of M, is not free.

We fix

noting that t > r since M

Next we form the polynomial ring S = R[x^,...,X^] and

the S-module N = S ® M'.

v = Y X.e. of N i=1 ^ ^ we have from Bruns [3] that the height of the order ideal Oj^(v) does not exceed r = rank containing O

N

(v).

M = rank N. Let P be a prime ideal of height r R S As noted in [9], the height of the ideal P + mS

is at most r + dimR, maximal ideal Q

Considering the element

which is less than dim S .

(actually infinitely many)

Hence there is a

of S which contains

P + mS and which corresponds modulo m to a maximal ideal of k[x

,...,X ] other than

(X^,...,X ).

The remainder of our proof

is exactly that given in [9] provided Q is of the form (m,X. - a ,...,X - a ) — I 1 t t being the desired element. Q

=

finite extension of k,

^

where a. e R, the element x = Y ^•e. 1 j^_-| ■>However, this can be achieved after a

since only a finite number of equations are

8 involved. field

Since R is complete and local we may extend its residue

(finitely)

so that the resulting ring

(R',m',]c')

is complete

and local as well as being a faithfully flat extension of R.

Thus

we can achieve a maximal ideal Q of the desired form by passing to a suitable finite extension

(R',m',]t'). D

The more general solution to the syzygy problem THEOREM.

now follows.

Let (R,m,]£) be a local ri-ng oontain'ing a f-ield. Let

n be a finitely generated \-th syzygy of rank r. ^ If r is less than k,

then M is free. Proof.

We may assume that M is locally free on the punctured

spectrum of R since otherwise we may ‘localize to a ring of smaller dimension while keeping M a k-th syzygy of finite projective dimension.

We may also assume that R is complete and that M contains

a minimal generator x having order ideal

of height 5 r.

This

follows from the fact that the syzygy problem remains unchanged under faithfully flat change of base as well as the preceding lemma. Let O -> F

F -> M O be a minimal projective resos 1 O ^ lution of M. Then the complex F.® R/I, where 1=0 (x), has homology R ^ Tor^(M,R/I) of finite length for i > O, since M is locally free on the punctured spectrum of R. is nonzero,

Moreover,

the element x + IM in M/IM

since x is not even in mM, but generates a submodule of

finite length,

since x e IM on the free locus of M.

It follows that

the zero-th homology of F.® R/I, namely M/IM, has depth zero. It remains to show that s is at least dimR/l. s ^ dimR/I,

then pd.M = s > dimR/I> dim R - r,

rings are catenary.

For if

since complete local

Hence we obtain the inequality pd.M + r > dimR.

On the other hand one has that pd.M + k < dimR

which together with

the previous inequality gives that r > k as desired. inequality pd.M + k < dimR

actually can be improved to

pd.M + k < depth R < f, where f = min{dimR/P | P e Ass this last step as a separate lemma Intersection Conjecture).

The second

.

We isolate

(called by some the "New New"

Note that S plays the role of R/I in the

lemma. □ LEMMA. Let s be a local ring containing a field. °

Fq

O

o and h^(if.)

9

has a minimal generator which generates a nonzero submodule of finite length. Remark. ]F.

would be

Then

s

> dims.

Note that by the Lemme d'Acyclicite,


dim S. □ The

final

reasons. [8].

First,

More

not stated) IF.

has

argument here

the essential details

interestingly, by Roberts

finite

length,

analyzed what was our case.

This

and Szpiro)

is a bit brief.

the above

[20]

lemma was

> dimS.

for two

already proved

lemma.

(but

if the homology of

There Roberts

needed in a separate

theorem

is

are in our original version

in his proof that,

then s

This

carefully

That lemma covers

(and the nearly identical one of Peskine

really is obtained from a better understanding of the

1969 Lemme d'Acyclicite.

Thus,

of the argument was known for

in

some

sense,

the essential part

some time although the reduction of

the question to that case was not apparent

(to us)

original proof.

this result gives yet

Perhaps more

importantly,

another application of this circle of ideas theory. if we

In particular the proof of this

start with a Cohen-Macaulay

while some of the

to commutative ring

case

is made no simpler

even regular)

local ring

earlier applications of this technique were

already understood in such cases. more applications.

(or

until after our

Thus one

is

enticed to

look for

lo References 1.

M.Auslander and M.Bridger, Math.Soc.

2.

W.Bruns,

94

Stable module theory,

(American Mathematical

"'Jede'

endliche

Society,

3.

1969).

freie Auflosung is't freie Auflosung

eines von drei Elementen erzeugten Ideals", (1976),

Mem.Amer.

Providence,

J.Algebra,

39

429-439.

W.Bruns,

"The Eisenbud-Evans generalized principal

theorem and determinantal

ideals",

preprint.

ideal

University of

Osnabriick at Vechta. 4.

W.Bruns,

E.G.Evans and P.Griffith,

two and vector bundles", 5.

D.Eisenbud and E.G.Evans, theorems

"Syzygie'b,

J.Algebra,

67

ideals of height

(1980),

"Generating modules

from algebraic K-theory",

143-162. efficiently:

J.Algebra,

27

(1973),

278-305. 6.

7.

D.Eisenbud and E.G.Evans,

"A generalized principal

theorem",

62

Nagoya Math.J.,

E.G.Evans,

9.

(1981),

H.-B.

1976.

"The syzygy problem", Ann.

of Math.,

323-333.

E.G.Evans and P.Griffith, preprint.

10.

Collogue d'Algebre Commutative Rennes

E.G.Evans and P.Griffith, 114

ideal

41-53.

"Position generate et position specials en algebre

commutative".

8.

(1976),

"Order ideals of minimal generators",

University of Illinois

Foxby,

Math.Scand.,

at Urbana-Champaign.

"On the

in a minimal

41

19-44.

(1977),

injective resolution,

II",

11.

P.HacIcman, "Exterior powers and homology", Ph.D.Thesis, Univ¬ ersity of Stockholm, 1969.

12.

R.Hartshorne,

"Varieties of small codimension in projective

space". Bull.Amer.Math.Soc., 13.

R.Hartshorne,

M.Hochster,

Topology,

Topics

24

18

(1979),

in the homological

commutative rings, Mathematics

(1974),

1017-1032.

"Algebraic vector bundles on projective spaces:

a problem list". 14.

80

117-128. theory of modules over

C.B.M.S.Regional Conference Series

(American Mathematical Society,

in

Providence,

1976) . 15.

K.Lebelt, Moduln",

16.

17.

"Zur homologischen Dimension ausserer Potenzen von Arch.Math.

(Basel),

595-601.

"Freie Auflosungen ausser Potenzen",

Math.,

(1977) ,

23

C.Peskine and L.Szpiro,

"Dimension projective finie

Publications Mathematiques

C.Peskine and L.Szpiro, Sci.

Paris

Manuscripta

341-355.

Hautes Etudes Scientifiques,

.

(1975),

K.Lebelt,

mologie^locale",

18

26

Ser.A,

278

Paris,

1973),

42

et coho-

(Institut des

pp.47-119.

"Syzygies et multiplicites",C.R.Acad. (1974),

1421-1424.

P.Roberts, rings",

"Two applications of dualizing complexes over local

Ann.Sci.Ecole Norm.Sup.

P.Roberts,

(4),

(1976),

103-106.

"Cohen-Macaulay complexes and an analytic proof of

the new intersection conjecture",

Department of Mathematics, University of

Illinois,

Urbana, Illinois

9

61801,

U.S.A.

J.Algebra,

66

(1980),

220-225

12 THE THEORY OF HOMOLOGICAL DIMENSIONS OF COMPLEXES

HANS-BJ0RN FOXBY

The object of this article

is

to comment on the

homological dimensions

of bounded complexes

Noetherian

ring.

commutative

When

complex concentrated in degree

the theory of homological dimensions just to replace This

"modules" by

idea is not new.

of modules over a

a module

zero,

ttiis

theory of

is

thought of as

theory extends parts of

of modules;

thus

the

For example,

it is

essential,

"Theorie des

intersections et theoreme de Riemann-Roch"

f

"Residues

a-X

iT'f" *1

->...->X

S

the projective dimension of X;

fd X,

the

flat dimension of X;

id X,

the

injective

addition to these

title

1)

(

can define there

is ?■

the

following:

depth the ring must be

local.)

In

namely

(or could)

want to

It is possible

(That ({)

let me define

is

a

((j)) :

H

id^X.

For an integer n we

a quasi-isomorphism (():

injective modules

quasi-isomovphism

£

£ H

the

concepts.

a bounded complex of

phisms

are

is another important dimension,

< n if there exists

> n.

Thus

the Krull dimension of X.

As an example that

[SGA6].

dimension of X;

Let me point out some reasons why one would study these

and

the depth of X.

that we

dim X,

[Ha]

^O.

pd X,

(In order

and duality"

a bounded complex of modules over a ring A.

The homological dimensions mentioned in the

depth X,

is

and used

in the seminar notes

X = Oa-x

idea

"complexes of modules" whenever possible.

extensively,

Now let X be

a

such

means

that I that (J)

0

(X)

a- h

(Y)

in cohomology for all

£.)

say

X a- i where I =0

for all

induces

isomor-

13 This extends module M:

the definition of injective dimension of a

if M has

an injective

resolution

I

then the

inclusion

O M

I

induces

phism;

on

a morphism of complexes M ->■ I which is

the other hand if M ->■ I

is

a quasi-isomor¬

a quasi-isomorphism then

I H

(I)

Z * O,

= O for

injective modules

and so it is

sitting

easy to split off the

in negative degrees

in

I

irrelevant

and thus obtain

an injective resolution of M. The definitions of the homological dimensions and the Krull dimension can be

found in

tF^].

These dimensions behave very nicely (in fact, just as for

(2)

modules) Let me mention two examples. (2.1) functors

They

can be

Ext and Tor.

characterized by the vanishing of the

For example,

id X < n if and only

£ Ext

if

£ (M,X)

= O for all

hyperExt:

see

[Ha;

> n and modules M.

Chapter

I,

§6].

(Thus

Here Ext is the 5,-th £ Ext (M,X) is not the

£ complex obtained by applying the to the

additive

functor Ext

(M,

)

complex X.) (2.2)

Let the

ring A be

local

(and this

ment that it be

commutative

article possess

non-zero multiplicative

and Noetherian;

generated modules M and N there various

dimensions.

all rings

identities).

in this For finitely

(2.2.b)

id M = depth A if id M < °°.

(2.2.c)

sup{£

I

I

173,

Ext

are

(and oldest).

pd M + depth M = depth A if pd M

Theorems

the require¬

are many relations between the

(2.2.a)

[K;

includes

The next three examples of such relations

among the most well-known

See

(module)

i

(M,N)

214,

the proofs of these results

O}

217, is


Z

Hom(X,Y(8iZ) . For

(3.1.b,c)

complexes have to satisfy special requirements:

(2.2.a,b,c): [^2^

finitely generated) where k is

see

class

in the

sense

(as

a

(not necessarily

a subring).

it was proved that if X is

^ O for some £,

a bounded

then

a field

latter the The proof is that

(and is

local).

stronger than the

in the proof,

first one.

In the

theory of homological dimensions simpler than

[f^;(6.3)]

and [g]

the proof of the have become

that Hochster's big Cohen-Macaulay modules

role

directly,

then

The reason for the requirement that the is

(concerning

and tF^].

field,

a field

latter result is

was used.

[F^].

dim A < dim X + fd X,

provided that A contains

proof of the

and

dim A < dim M + fd M,

that Tor^(k,X)

(3.2.b)

The

[F^]

later paper [F^^

complex such

[Ha]

module over a local ring A such that M ® k ^ O

provided that A contains In the

see

give many results

it was proved that if M is

the residue

(3.2.a)

available.

to hold these

together with a very simple trick

computation of cohomology modules),

In

are

the

functorial isomorphisms: ^ Horn (X , Horn (Y, Z) ) ;

(3.2)

Chapter l] would

completely different methods

Hom(X(8)Y,Z)

including

some basic

and then generalize

(3.1.a)

isomorphisms,

for modules.

(2.2.a,b,c) one could develop a

Here X,Y and Z are bounded complexes.

These

for complexes

are only simpler when one knows

about complexes,

(3.1)

results

and these modules

are only

of complexes

first result

superfluous.

ring A contain a field from

[Ho^]

(so far)

play a key

available

in

15 this

case

(and some other cases).

(4)

The results get better

Sometimes holds

it is

also for complexes.

states

For example,

that if each module F = O^F

^F s

is

interesting to know that a particular result

in

< t +

£.

for all

F is not exact and the

local

complex F we have

< s

fd F

i,

then dim A < s

-

X.)

that A is

[PS],

[r],

t^Ho^]

certainly are

for example,

[Ha],

local with residue

that is,

that the dual

a field.

important tools.

[r],

[s],

to make

complexes

field k.

of a finitely generated module

up

[F^],

Dualizing complexes

for this,

but the price

these proceedings).

dimension

all

t,

can be

instead of

integers n (see

[b])

a

that

just modules. [F^l

Let me describe another one. finite

left finitistic projective a left R-module

Then Jensen has proved that left-pd m < “ for see

[j].

commutative

^ dim A Bass has and Bass

the disadvantage

are described in

left-pd M < t whenever M is

Let A be

are good for

to be paid is

that is,

flat modules M:

[F^].

thought of as one

ring R has

< «>.

[^2^

has

that the

left-pd M

Examples

in general is not

that has

to work with complexes of modules

Assume

with

Szpiro,

The useful Matlis

Some applications of dualizing complexes (in

(For this

or

duality with respect to E(k),

finitely generated.

one has

provided that

< t

To describe vaguely what dualizing

way

+ t,

Th-is theory is ideal for working with dualizing complexes

found in,

duality,

integer t such

and

see

Dualizing complexes

assume

an

the NEW INTERSECTION THEOREM of Peskine,

and Hochster:

(5)

can be

and if there exists

and assumptions.)

This result is Roberts,

(3.2.b)

complex

ring A contains

dim F = sup(dim H^(F) by definitions

case of

^^...^F ->0 s-1 O

finitely generated and free

that dim H^(F)

the

a special

Noetherian ring.

For all non-negative

constructed a module M with pd M = n

stated that he knew of no example where

16 dim A

< pd M < “. pd M < “

see

[GR].

Gruson and Raynaud settled this by proving that

=>

pd M < dim A:

(Note

that M is not supposed to be

finitely generated.) •»

The proof is not easy. (as most rings have) fd M < “ see

[f^].

=>

However,

if A has

a dualizing complex

then it is easy to prove even that

pd M < dim A:

From this

Gruson and Raynaud's

has a dualizing complex)

as well

result follows

as Jensen's

(in this

(when A

restricted

case).

Acknowledgement The author has been supported,

in part,

by the

Danish Natural

Science Research Council.

References [b]

H.

Bass,

Trans.

[F^]

H.-B.

"Injective dimension in Noetherian rings,

Amer.

Math.

Foxby,

Soc. ,

88

(1958),

II",

194-206.

"Isomorphisms between complexes with applications

to the homological

theory of modules".

Math.

Scand.,

40

(1977),

5-19.

CF2] [F3]

H.-B.

Foxby,

Math.

Scand.,

H.-B.

Foxby,

Algebra,

H.-B.

Foxby,

1982), P.

pp.

L.

[Ha]

[Ho^]

R.

14

(1979),

R.Y.

J.

Pure

Commutative

London Mathematical Society Lecture

Sharp,

Cambridge University Press,

Cambridge,

18-31.

J.

13

resolution,II",

149-172.

"Complexes of injective modules".

"A representation theorem for complete

Pure Appl.

Gruson and M.

Math.,

injective

19-44.

"Bounded complexes of flat modules",

Griffith,

rings",

in a minimal

(1977),

Durham 1981,

Notes 72(ed.

Cgr]

41

Appl.

algebra:

[G]

"On the

Raynaud,

(1971),

Hartshorne,

Algebra,

7

(1976),

local

303-315.

"Critires de platitude".

Invent.

1-89.

Residues and duality. Berlin,

M.

"The equicharacteristic case of some homological

conjectures on local rings".

Bull.

Heidelberg,

in

20

Hochster,

(Springer,

Lecture Notes

Mathematics

Amer.

Math.

New Yorlc,

Soc.,

80

1966).

(1974),

683-686. [Ho^]

M.

Hochster,

"Big Cohen-Macaulay modules and algebras

embeddability in rings

of Witt vectors".

conference on commutative Ontario,

1975

algebra.

Queen's University,

(Queen's University Papers on Pure

Mathematics No.

42,

1975), pp.

106-195.

and

Proceedings of

the Kingston,

and Applied

17 [j]

C.U. Jensen, "On the vanishing of lim^^^", J. Algebra, (1970), 151-166. ^

Ck]

I. Kaplansky, 1970).

[PS]

C. Peskine and L. Szpiro, "Syzygies et multiplicites", C. R. Acad. Sci. Paris Ser. A, 278 (1974), 1421-1424.

[r]

P.

Roberts,

rings", [S]

Ann.

P.

Schenzel,

J.

Algebra,

Commutative rings

Sci.

Ecole Norm. Sup. (4),

9

(1976),

103-106.

"Dualizing complexes and systems of parameters", 58

(1979),

495-501.

University of Oklahoma, Oklahoma 73019,

U.S.A.

and

Boston,

"Two applications of dualizing complexes over local

Department of Mathematics, Norman,

(Allyn and Bacon,

15

(from 1982)

Matematisk Institut, K(zibenhavns Universitet,

Universitetsparken 5, DK 2100 K(zSbenhavn 0, Denmark.

^

COMPLEXES OF INJECTIVE MODULES

HANS-BJ0RN FOXBY

The object of this article is to describe the modules in a bounded below complex of injective modules 1 = O

i

1

^ I

i+1

^ I

I

In this complete generality it is of course impossible to formulate reasonable assertions

(since the modules could be any

injective modules and all the differentials could be zero). have to impose restrictions. H

a

(I)

So we

We assume that the cohomology modules

vanish for sufficiently large 1.

For example,

I could be an

injective resolution of a module. However we will have to impose further restrictions. example,

take any exact complex J of injective modules.

For Then the

complex I©J has also only finitely many non-vanishing cohomology modules.

Therefore we shall restrict the study to the case where

I is mmi-ma'l in the sense that Ker(I module of I

£

for all £.

-»■ I

For example,

)

is an essenttaZ sub-

. .

I could be a mimmal

injective resolution of a module. The ring we are working over is denoted by A and is supposed to be oomnrutat'tve and Noethevian.

(The word "ring" always

incorporates the existence of a non-zero multiplicative identity.) Thus the injective module I

£

decomposes into the direct sum of

indecomposable injective modules,

I*' -

11

that is.

E(A/p)

pSSpec A for some cardinal numbers y^(p,I).

That is,

suffices to describe the numbers y^(p,I) £ A —

£ = tJ.

(p

A

—p

P

-

/I

p

),

-

to describe I

£

.

it

It turns out that

and so it suffices to assume that A is local

19 and seek information on y

(I)

= y

^

ideal in A.

(m,I)

where m denotes the maximal

^

The best results are obtained when H

genevated {f. g.)

for all £, because then y

(I)

a

(I)

,

is finitely

< «> for all 1.

Next follows a list of some of the known results about y The references are Bass [bJ, [ft],

and Foxby [f^],

Roberts [R^],

[f^],

I

(I).

Thorup

further comments on the results

will be provided in the subsequent sections of the article.

To

facilitate the presentation of these results here in the introduction let us assvune that I is a minimal injective resolution of a module M (although this restriction is not necessary). uniquely determined by M,

The complex I is

f

£

and so we write y^(M)

= y^(I).

There is a close connection between I and a minimal fvee resolution L of M when M is f.

g.

If 3,^^ (M)

denotes the rank of

then 3^(M)

I

=

y^(A)y^

(M)

pe/ for all £ if I is bounded, y^(M)

=

I

and

yP(A)B^_^(M)

peZ for all £ if L is bounded:

see

(2.6).

Let B be a flat A-algebra such that C = B/mB is non-trivial. Then yg(M0^B)

=

I

yP(M)yJ"^(C)

peZ for all £:

see

(3.1).

If p e Spec A and v = dim A/p, £:

see

then y^(p,M)


O. Here A A id M denotes the injective dimension of M (which might be infinite). ^

The opposite holds also if M is f.

g..

That is,

y

£

(M)

> O if

depth M < £ < id M (< “): see (6.2). In many cases even the A An following holds: y (M) > 2 if depth^M < £ < ((see (6.3)). As a consequence the ring A is Cohen-Macaulay

(and therefore

Gorenstein)

(6.4).

if y

(A)

= 1 when d = dim A:

Even if M is not f.

g. we can say something about the vanish£

0

ing of y

see

(M).

If £ > dim M and y

ni

(M)

= O then y

(M)

= O for m > £,

20 and if m > depth A and y

in

The small support,

(M)

> O then y

^

(M)

> O for £ > m:

see

(8.1).

supp M, of M is the set of prime ideals p

0

(p,M) >0 for some Z. This is a subset of Supp M, the usual A — 1. support of M, and supp M = Supp MifMisf. g.. In general supp M

with y

has nicer properties than Supp M. supp M n supp N =

see

For example,

U supp Tor i!.>0

(M,N):

(7.1) .

Notation and generalities

1.

The symbols X, Y and Z denote bounded complexes of A-modules such that H

z (X)

and H

a (Y)

(but not H

z (Z))

are f.

g.

for all Z.

The

The complex I is always assumed to be bounded below and consisting of injective A-modules

(as in the introduction).

L denote bounded above complexes of,

f. g. free modules. and Noetherian.

The symbols F and

respectively, flat modules and

The ring A is always supposed to be corrmutative

When A is supposed to be local, m denotes the

maximal ideal and k = A/m, the field of residue classes. 1.1

There exist complexes E and J such that

LEMMA.

the complex E is minimal^ and the complex J is exact,

I ~ E©J,

that

is, J is of the form 1^ 0

0 1 K _^ K

0 0

i+1

K _^ (Xh K

i+1

0 1 0 0

i+2

The complex E is uniquely determined up to isomorphism of complexes. Proof. I

Each I

= E^©K^

© K ^

Z

decomposes into the direct sum of submodules

in such a way that each differential

is of the form E (f)

Z

O

O

O

0

0

O

© K

0

E

Z-1

©

-z K

£.+ 1

© Z --K © -Z+1 K

->■

21 where tj;

Z

~i

:

K

^ K

i

is an isomorphism,

essential submodule of E

Z

.

and where Ker (})

and the inclusion E

projection I ->■ E are quasi-isomorphisms isomorphisms in cohomology).

If e'

Z

is

and then it is relatively easy to see that □

Definitions and Remarks.

resolution of

they induce

then the composite E -> I ->■ e'

is actually an isomorphism. 1.2

(that is,

I and the

is another minimal direct

summand of I with exact complement,

E -> e'

is an

(Use induction on 5,.)

The complex E is minimal,

a quasi-isomorphism,

Z

The complex I is an injective

if there exists a quasi-isomorphism Z

a minimal injective resolution of Z if, (as defined in the introduction). injective resolution

(see

[Ha;

in addition,

I,

and it is

it is minimal

The complex Z always has a minimal

Chapter 1])

resolution is unique up to isomorphism, by

and this minimal injective (1.1).

The complex F is a flat resolution of Z if there exists a quasi-isomorphism F ->■ Z.

The complex Z always has a flat resolution:

see [Ha]. When A is local, then L is a minimal free resolution of X if £-1 Z Z there exists a quasi-isomorphism L ->■ X and Im(L ->■ L ) £ nHlj for all Z.

Such always exists and is unique up to isomorphism. If H

Z

(I)

= O for Z large,

say for £ > s,

then I is the injective

resolution of the bounded complex

^ I

U = Oa-i

s-1

,

Ker (I

S

^ I

S+1

)

^

-^ O,

since the inclusion U ->■ I is a quasi-isomorphism. 1.3

Definitions and Remarks.

equivalent, and we write U

V,

The two complexes U and V are

if there exists a third complex W

and quasi-isomorphisms U ->■ W and V -> W. relation:

This is an equivalence

see [Ha; Chapter l1.

The supremum and infimum of a complex U are defined by s(U)

= sup{£

I

H^(U)

^ o}

i(U)

= inf{£

I

H^(U)

^ O}.

and

The complex U is equivalent to a non—zero module M if and only if s(U)

= O = i(U)

(and then M ~ H° (U)).

if it is equivalent to O,

that is,

The complex U is trivial

if it is exact.

22 Definitions and Remarks.

1.4

Hom(x,z)

denotes the equivalence

class of the complex Horn (X, I) whenever I is an injective resolution of Z.

This makes sense:

0 Ext

see [Ha;

Chapter l].

n (X,Z)

= H

(Hom(X,Z))

is then determined up to isomorphism.

This is the £--th hyperExt.

X ® Z denotes the equivalence class of the complex X n.

of X,

I

This

then

Nakayama's

2.2

implies

lemma LEMMA.

This

see

if L is

g.

extends

modules

(3.18)]. article

there

g.

module

-

2.5

THEOREM.

s(X)

is

(b)

2.7 (b)

this

o}.

5=

of X is

at

=0

free resolution

instead of the

complex X).

degree of

=

it) , and

X

^

^{t) . The

degree

of O is

-°°.)

= order of

for a f.

g.

module.

connection will not be used in important).

= order of

Pf^{t) . y

Y

— 1

it) = J ^{t)P/‘^{t

(a)

See

Theorems

COROLLARY.

.^^(t) =

(a)

4.1

-^^(t) =

J^^it)J^it~'^) if

COROLLARY.

(a)

depth A = id X Pi.

(c)

the

)

pd^X
O

regular, finite,

for all

The

about the

reduction

is defined if

automatic if pd M < °°;

regular.)

possible

and

and vanishes

condition if R is

^

ideas

= O as well

]. D

It is clear that the the

rigidity conjecture

closely akin

to

following. (1.2)

SERRE'S MULTIPLICITIES

CONJECTURE.

arbitrary regular local ring and let that

is

£(M®

N) R

is finite.

m,n

Let

R

be nonzero modules such

Then we have the following:

(O)

dim M + dim N < dim R;

(1)

if

dim M + dim N

dim R,

then

Xq(M,N)

= O;

(2)

if

dim M + dim N = dim R,

then

Xq(M»N)

> O.


O);

see

is part of the motivation means

"Cohen-Macaulay")

a ramified regular

of Serre's

also

to

= m and dim(R/P)

If

> O.

and also

dim N = dim R/Q,

so that e(M,N)

can

i

>

= £ (M (8> N)

+ dim(R/Q) that R/P has

assume

the

the proof of

that

a C.-M.

a C.-M.

module M with

module N with > O.

The point

>

O.

But M has

a prime

filtration

of R/P and other modules

e(R/P,R/Q)

small

>

(that is

are not known to exist even

invol¬ of lower

> O copies

of R/Q

from the bi-additivity of e

assumption we easily obtain that e(M,N)

Unfortunately,

=

o. finitely generated) in dimension

3,

C.-M.

even in the

case.

One can make Serre's by dropping the hypothesis that pd^M < oo.

= dim R,

such a filtration involving b

whence

equicharacteristic

and assume

1,

and other modules of lower dimension;

(ab)e(R/P,r/Q),

complete

show that e(R/P,R/Q)

ving a positive number a of copies

and the vanishing

that is

To

conditions

=0,

and N has

modules.

for primes P,Q with P + Q

that R/Q has

then one

that under these Tor^(M,N)

one knows

show,

To

for

local ring,

< dim R,

conjecture.

it suffices

dim M = dim R/P

modules

[H^],

then e(M,N)

= O whenever dim M + dim N

"vanishing part"

dimension,

[S],

a formal power

has been proved for dim R
-m'—»-M—is exact then

= Cm'] + Cm"] , where [m] denotes the class of M.)

Let d=dim R.

Is G generated by the classes [r/(x^,...,x^)], where x^,...,x^ is an 'R-sequence? Question

(1.3)

has an affirmative answer if dim R = 1:

is essentially a result of MacRae [m]. 2,

It is also true in dimension

at least after enlargement of the field: It turns out that if

hypersurfaces,

(1.3)

this

see [h^].

has an affirmative answer, even for

then the vanishing part of Serre's conjecture would

follow for all ramified regular local rings.

It would be enough if

the classes Cr/(x^,...,x^)] generated Q

where Q is the

G,

rational numbers. It may be worth mentioning that A.Weil [w] has challenged the generality in which Grothendieck has developed algebraic geometry because one cannot establish a notion of multiplicity and prove that it has the "right" properties in such generality. gives what is probably the right notion, Weil's point of view or not,

Serre's idea

and whether one accepts

it certainly provides additional moti¬

vation for proving that Serre's notion really does behave properly. Before leaving the subject of multiplicities I want to discuss some ideas connected with lifting. Let R be a complete local ring and let x be a non-zerodivisor. Let M be a finitely generated module over R/xR with The lifting question is as follows: does there exist a finitely

generated R-module

N

such that

not a zerodivisor on

(1)

X -is

(2)

N/xN = M.

N,

and

This is true when R and A = R/xR are both complete unramified

36 regular local rings,

for then R = aECx]] and one can take

N = M ® A[Cx]] . A It is an interesting question for if it were true when R is an unramified regular local ring and A = R/xR is ramified and regular,

it would establish Serre's conjecture:

of M from A to R it is easy to see that e

A

(M,N)

if m' = e

K

is a lifting

(m',N)

while

dim A - dim M - dim N = dim R - dim m' - dim N. Part of the motivation of Buchsbaum and Eisenbud in their study of the structure of free resolutions is the problem of lift¬ ing such resolutions our sense):

(which then automatically lift the module in

see [BE^] and [BE^l.

Peskine and Szpiro [PS^] gave a counterexample to lifting in the case where pd M is assumed finite but R is not necessarily regular.

In [h^] a counterexample to lifting is given when R is

regular and unramified, A = R/xR is ramified and regular,

and M is

a cyclic A-module. However,

there are still weakenings of the lifting question

which might have affirmative answers and which would give inform¬ ation about multiplicities. For example, hypersurface. pd^M < °°.

suppose that R is regular and A = R/xR is a

Let M be an A-module of finite length such that

We can then ask whether there exists a finitely generated

R-module N such that (1)

N/xN = M,

and

(2)

X is not nilpotent on N.

An affirmative answer would yield a proof of the vanishing part of Serre's conjecture for ramified regular local rings. While this question looks much weaker than lifting,

since we

are only asking that x not be nilpotent on N instead of that it be a non-zerodivisor, pd^M
O and,

for all equicharacteristic

This "new intersection theorem",

as it came to be

in its simplest form asserts that if R is local and = O —>-F,—^. . .——>-0 d O

is a complex of finitely generated free R-modules such that H,(F.) has finite length for all i and H. (F.)

^ O for some i,

7

then

dim R < d. The arguments given in the various proofs of the new inter¬ section theorem actually prove a slightly stronger result, which we shall examine in the next section,

and which, unfortunately, we

shall refer to as the "new new intersection theorem". theorem in the equicharacteristic case: a conjecture.)

(It is a

in the general case,

it is

This result is of particular interest because it

suffices to give a proof of the recent Evans-Griffith syzygy theorem (they use big C.-M. modules in their paper):

see [eg] .

It had been observed already in [h^] that the existence of big C.-M. modules implies the intersection conjecture and hence the zerodivisor conjecture and an affirmative answer to Bass'

question,

it is

not hard to use the same idea to prove the new or new new intersection

theorem.

(It is worth mentioning that while all three of the proofs

just discussed rest ultimately on reduction to characteristic p, Paul Roberts tproved the new intersection theorem in the analytic case by analytic techniques

(including the Grauert-

Riemenschneider vanishing theorem).) Thus,

the existence of big C.-M. modules seems to play a

central role in the study of these conjectures. that if R is regular local,

S is local and module-finite over R,

and S has a big C.-M. module, R-module

(see tH^])-

arbitrary S:

This,

Now it is known

then R is a direct summand of S as an

of course,

is conjectured to be true for

this is the "direct sumnjand conjecture".

The author

has been able to show that the direct summand conjecture itself is enough to imply the new new intersection conjecture: cuss this in the next section.

See

we'll dis¬

for more details.

Thus, most of the applications of the existence of big C.-M. modules would be obtainable if we could prove the direct summand conjecture, which is perhaps the least "homological" of all the homological conjectures.

A diagram of implications is given in the accompanying Figure 1.

41

Existence of small

Serre's conjecture on

C.-M.modules*

multiplicities for ramified regular local rings reduces to the vanishing part*

conjecture"'

Zerodivisor conjecture

* Known in the equicharacteristic case. Figure

1

Affirmative answer for Bass' question

42 2.The diveot summand oonjeoture and the new new 'tnterseotvon theorem The question of the existence of small C.-M.

modules for

complete local rings R immediately reduces to the case where R is a domain and then by Cohen’s structure theorems R is module-finite over a complete regular local ring A. R-module, M is a small depth M = dim R) A-module,

R—>-End

A

.

=■£ (A)

r must be an embedding, matrices over A

C.-M. module for R

The

action of R on M then gives

(r size matrices over A), which,

then

identified with the

it turns out,

scalar matrices

if there exists such an embedding,

exists a retraction R—>-A as A-modules,

map

a map

so that R is embedded as a subring of r size

(A C R is

Clearly,

summand of R as

(that is

if and only if M is non-zero and free as an r

say M s a

(A^)

(maximal)

If M is a finitely generated

an A-module.

In

then there

that is A is a direct

fact, .^^(A)

retracts to A:

simply

(a. .) >-> a,, (or a for any fixed t) . 1] 11 tt This is one way of seeing how the existence of small C.-M.

modules implies the direct summand conjecture. also suffice:

this is shown directly in

Big C.-M. modules but one can also use

the following beautiful result of Phil Griffith [g]

(which we shall

not prove here). (2.1)

THEOREM

(P.Griffith). Let A be a complete regular local

ring and let R be a domain module-finite over A. C.-M. module, A-free.

If R has a big

then it has one which is countably generated and

□

[it is tempting to try to get from this result to the exist¬ ence of small C.-M. modules: motivation.

I believe this was part of Griffith's

No one has succeeded.]

In any case,

the existence of big C.-M. modules would imply

the following. (2.2)

CONJECTURE.

Let R be a regular Noetherian ring and let

S be a module-finite extension of R.

Then R is a direct summand of

S as an R-module. We make some elementary remarks about this problem. homomorphisms r_>s—>T,

if R.^T splits then R—splits.

Given Hence,

43 in Conjecture

(2.2)

we are free to kill a minimal prime of S which

meets R trivially and so assume that S is a domain. splits if and only if Horn

{S,R) —^Hom

(R,R)

Now R—

is onto.

This proves

that the problem is local and that we may assume that R is a regular local ring.

Moreover,

this argument also shows that we can replace

R by a faithfully flat local extension such as,

for example,

R

(which is still regular),

(although we may have to kill another mini¬

mal prime to get the new S to be a domain again).

By this trick we

may assume that R has an algebraically closed residue field. We next observe that when R contains the field of nationals the direct summand conjecture is triv:^ally true - even if we only assume that R is normal instead of regular.

For if L,L'

are the

fraction fields of R,S respectively and [l' : L] = d then

gives the required retraction. The case of characteristic p > O is handled using the Frobenius endomorphism F.

We may assume that R = K[[x^,...,X^]], where

K is an algebraically closed field of characteristic p > O. choose an R-linear map (f)

:

S—R such that (}) (1)

^ O.

First

(Because S is

7

a torsion-free R-module it can be embedded in a free R-module.) 0

Then choose e so large that (J) (1) Let q = p^. A = F^ (R) . (j) (1)

^ mP

, where m =

Let A = K[[X^,...,X^]] C R.

Since K is perfect,

R is a free A-module and since (p(l)

is part of an A-free basis for R.

A-linear map 4^

:

R—A such that 4^ (4(1))

(X^ ,. . . , X^) R.

^

(X*^,. . . , X^) R = ^R,

It follows that there is an = 1-

Thus

^ °

4 is an

4

0

A-linear retraction of S to A, A-linear retraction of F^(S) tative diagram F® e S -^3-^ F (S)

u

and its restriction to F

to A = F^(R).

(S)

is an

But we have a commu¬

so that the A Ch.

inclusion R

F®(S).

S

It follows

This proves

the

is

isomorphic to

that there

is

the

inclusion

an R-linear retraction S—^R.

direct summand conjecture

in

characteristic

p > O. Our next objective intersection conjecture cteristic p

> O.

in

this

section

to prove

the new new

from the direct summand conjecture

Of course,

this proves

we don't need to say

"conjecture".

necessary to get the

arguments

case,

is

it in

Later,

to work in

we

the

in

absolute

indicate

the

chara¬

sense: changes

the mixed characteristic

where we do not know whether the direct summand conjecture

holds,

but the

implication

direct summand

> new new intersection

remains valid. A key point in the a "funny" which is

arguments

kind of Koszul

complex,

follow is

greater restrictions

in

the

construction of

to which the usual one maps,

acyclic under very mild hypotheses.

be made very generally in

these

to

characteristic p

This

but

construction

can

> O and under somewhat

the mixed characteristic

case:

we discuss

later. First recall

usual Koszul

that if x,,...,x 1 n

complex K. (x^ ,. . . ,x^,-R)

are non-zerodivisors (or,

briefly,

in R,

K. (^;R)

),

the

which

is n

^i (O

may be

»-R

>-R

»-0) ,

identified with n (O —>- x.R C.> R->'0) . 1=1

(Here,

1

when n =

1,

K^(x^,R)

is

identified with R and K^(x^;R)

with

x^R.) Now let R be R

00

=

lim

a ring of characteristic p

F (R —^ R

where every map is

F

> O and let

F F 5- R—.—>-R—>...)

the Frobenius.

R°° may be

thought of as R

e with all p -th roots y 6 R

°°

we may write F

00

adjoined. -1

(y)

= y

F is

an

automorphism of R

and

for

1 /d

Let x^,...,x^ be non-zerodivisors

in R.

For simplicity

assume

45 that R is

reduced,

so

that R c R

.

Then x

,... ,x

oo

divisors

in R

1

.

n

Let

= U

U)

are non-zero-

r“

e=1 00

when X

is

a non-zerodivispr

in R

oo

.

Then

(x

a flat ideal

is

)

in

00

R

. ^

00

Now let K.(x

,...,x

;R

CO

)

oo

(or K.(x;R

In

n CO ® (O-^ (x. ) C_^ i=1 1 Of)

(When n Note

n

=

00

and

(xj 1

(2.3)

(x

We shall on n.

In

If

THEOREM.

characteristic

R of

=

(x^).)

the

p

n

)

s

(x

1

Then K.

> O,

inductive

) . . . (x

is

a flat complex.

then

n

)

=

.X ) )

((x.. 1

n

are non-zerodivisors in the ring

x^,...,x^

is acyclic.

K?(x^,. . . ,x^;R°°)

sketch a proof of

OO

step we

this

theorem.

assume

the

We use

induction

acyclicity of

00

K.(x.,...,x ,;R 1 n-1 (x.) 1

oo

),

OO

J =

denote

that K

is

oo

= R

or simply K.)

CO R-^O) .

CO

1,

CO

)

—

which makes

it a flat resolution of R /J,

00

+...+

(x

oo

.). n-1

Let I •

=

a flat resolution of R /I.

these

flat resolutions

total

complex,

(x

oo

n

oo

) ,

SO

where

oo

that O —>*1 ——^R /I —^ O

We want to show that if we tensor

(without their

augmentations)

then we get an acyclic complex.

and take

But this

the

simply says

that 00

Tor^(R°°/I ,R°°/J)

=0,

i

>

1.

1

Since

I has

a flat resolution of

this when i = will are

suffice

in J. Note

by

to show that IJ =

(u^^^) ^ Thus,

it suffices

But that particular Tor is

stable under F

and u = as

1.

length one,

1 1

u g IJ,

Suppose We as

I

The

that u e

can think of reguired.

that the usual Koszul

tensoring the diagrams

n j.

simply

I

to

n J/IJ and it

crucial point is

I M J. as

Then in

I

show

that I,J

u

e I

n J,

and

(u^^^)^

□

complex maps

to our

"funny"

one:

46 (x, )

O

O

R

1

U R

x.R

-5-

O

1

id

O,

R we get a map K.(x;R)

^ K.{x;R

). 00

The reader is referred to [h^]

for more

Notice

at once

that from

(2.3)

we have

infoirmation the

«

(2.4)

is at most

following.

OO

The flat dimension of B.

COROLLARY.

about K. .

00

00

/((x^)

+...+

(x^) )

□

n.

Let A = k[[x,,...,x ]], 1 n field of characteristic p

where K is

> O.

Let T be

an algebraically closed the

integral

closure of A 00

in an algebraic closure of its

fraction field.

Then T = T

00

the be

theory described above the maximal

ideal of T:

algebraically closed). If T were Noetherian, course,

here,

applies,

T is

call

Thus,

and

(x^)

it m . —T

K has

+...+

(x^)

Now K = T/rn —T

finite

,

so that

OO

turns out to

(since K was

flat dimension over T.

this would imply that T is regular.

a huge ring which is nothing

like

Of

regular. OO

We now want to show how to use the new

(or even new new)

that if R is

the

acyclicity of K.

intersection theorem.

local and 0—>-F

—^O is

then dim R ^ s. The is

"new"

It is easy to reduce

version relaxes

assumed that H^(F.)

minimal generator of H

In either version it is complete

the hypothesis

has

O

to the

finite

(F.)

local domain and,

finite

complex of

length homology

slightly in this

which is killed by to the

for simplicity,

^

case where Hq(F.)

length, i SI,

easy to reduce

The theorem asserts

a finite

finitely generated free modules with non-zero

to prove

case:

and that there

O. it

is

a

a power of m = m . R case where R is

a

we henceforth assume

this. The

slightly improved new version

needed to prove

turns out to be

the Evans-Griffith syzygy theorem:

Assume now that the

theorem is

false

see

and that s

just what is [eg].

< dim R = n.

Let M = H^(F). write K.(xt;R) shows

O

O

O

R

which, be

be

for

of the

large

To give

t there

R chosen

is

M

so that d^cj)^ (1)

is

R/(x^,.

enlarging t if necessary, t+t'

One then constructs

;R) —>-K. (jt

inductive step one has

t'.

large

= Ker d.,

1

However,

show that one

contained in m^ F^ Ira d. i+r

the reader is

the

1

standard map of

the

referred to

-)-

can

—

fill

in the

trivial

that m Ker d,

and so,

arrow

d)! ^ ^1+1

C Im d

1

;R)

^

for

1+1

6! ^1+1

^

large N.

in F^ will be

by the Artin-Rees

construction of

for suffi-

if we knew that Im d.

N

image of K._^^(x^'''^

O Ker d^

this permits ^

recursively

a diagram

—

t'

etc.,

will

t+t' K, (X^ ^ ;R)

This would be

but all we know is

for large

(j)^,

(|>q(1)

i+1

K.^l(x""" ;R)-

ciently

O,

;R) .

K. (x^;R) X —

to

Now f =

making repeated use of the t

,x^)n

the minimal generator v of M

is killed by a power of m.

F

O

(l)

a free generator of F^.

and one wants

first

a map of complexes

R

by hypothesis,

In the

We

form

(s^

complexes K. (x

for R.

the proof one

F.

(|)^ is

Koszul

a system of parameters

(x^,...,x^;R).

that for sufficiently

K. (jc^jR) —F. ,

where

Let

lemma,

in

For more details,

[H^]. oo

But now since F.

is

free

and K.

is

acyclic we

can construct a

00

map of

complexes F. —»-K.

in such a way that the

free generator 00

f =

(1)

of F

discussed earlier maps

to the element t

Composition yields

a map of complexes K.

1

OO

;R)

in R

.

OO

—^K. (J^/'R

)

of which

00

the

degree O piece

is

the

inclusion map R

R

while

the degree n

48 piece is O because it factors through n > s).

- O

(we're assuming that

On the other hand there is a "standard" map between these

complexes constructed earlier, which behaves in the same way in degree O but in degree n maps the free generator 1 of R = K^(}{^;R) to

t X

t ...X

in

((x,...X )

).

Inin

acyclic,

t

Since K.(x ;R) —

these two maps are homotopic,

OO

is free while K.

is

so that their difference

00

i

; R ^ ((x^...x ) 1 n

)

factors via a map h

(see diagram)

oo

{(x ...X ) ) ^ ^ K \h \ s

R

R

through d, whose matrix has entries t X

t ...X

In

(x

6

t ,...,x )((x 1 n 1

±x,,..., ±x . 1 n 0°

t

...X

n

)

It follows that

),

so that for some sufficiently large integer e there exist elements

1

,...,v e R n t t X ^X ^

1

n

Let N = tp

e

such that = I X%. (x

. . .X

.111

and y,

1 /p = x.

n

)

1/P

R contains a regular ring A with

,y

as a system of parameters. Let B = A[v,,...,v ]. n In The above equation can be rewritten as N

/

r V

N

1

or as /

^N-1

V

N

(y,-i yiyin 1

In the case of characteristic p > O we can now apply a retrac¬ tion 0

:

B—>-A and get the equation

,

(Yi---yn)

N-1

V 1-

^

=IVi

(b. 1

= 0(v.)) 1

holding in the regular ring A, where it is easily seen to be imposs¬ ible. We next discuss how to get the idea of this proof to work in mixed characteristic.

It turns out that we can construct an acyclic

49 complex to play the part of K.:

the difficulty is that we cannot

prove the direct summand conjecture. Specifically, cteristic p > O,

let R be a complete local domain of mixed chara¬

let x^ =

be a system of parameters and

PjX^,...,x

let R be a domain integral over R such that 1 /p 00 s e R and (2) R is integrally closed.

(1)

if s e R

then

00

For example we can take R

to be the integral closure of R in

an algebraic closure of its fraction field. OO

flat ideals K.

= (8) i

(2.5)

1 /p

(x.)

= IJ

(O

We can still define

CO

(x.

)R

(x. )

and as before let R

O)

1

Under the hypotheses above, the complex kT is

THEOREM.

acyclic. For details we refer the reader to [h^]. similar to that of the earlier result: number of x,.

(i ^ j)

acyclicity.

one uses induction on the

There are a couple of differences, however.

n = 2 one uses the fact that R 1/ ^ Xj

The proof is

”

is integrally closed,

is an R-sequence for all e and f,

When

1 / p^ so that x^ ,

to establish

In the inductive step the key point is still to show

that 00

f (x J

OO

+_+

(x

1

OO

J) n-1

n

(x ) n

CO

=

OO

((x.)

(x

1

OO

J ) (x ) , n > 3, n-i n

OO

The idea is to work modulo

(x^):

the earlier result for charactistic

p > O then yields that OO

((xj 1

OO

+...+

(x

CO

n-1

))

n

(x ) n

co

C

co

(x ) 1

+ [ (x ) 2

CO

+...+

(x

n- I

CO

)](x ). n

If z is an element of the left hand side, we have 1/P® , Z = X^ V + y,

and then z - y =

1

/d^

v e

°°

R-sequence argument,

it

{x°°) , and the desired result holds. □ n The reader has probably noticed that instead of the direct

follows that V e

summand conjecture,

one could use the "fact"

system of parameters of a local ring R, t t , , t+1 t+1 X,...X $ (x, ,...,x )R. Ini n

then

that if x^,...,x^ is a

50 This assertion, not surprisingly, to the direct summand conjecture.

turns out to be equivalent

In fact,

statements are given in Proposition

(2.10)

a number of equivalent below.

Before giving

that result, we shall discuss briefly the canonical element conjec¬ ture studied in [H^j. Let R be a local ring of dimension n with maximal ideal m.

We

use

(M) to denote the i-th local cohomology module of the R-module m M with support in m: one definition is (M)

m —

= lim Ext^(R/m^,M). —>— t

A finitely generated R-module Q is called a oanon'Lcat module for R if Horn

(f2,E(K)) = H^(R) , where E (K) is the injective hull of the R in residue class field K = R/m. The module 9. is determined up to non-unique isomorphism,

if it exists.

of a Gorenstein ring and,

(If R is a homomorphic image

in particular,

if R is complete,

then such

a module 9 always exists.) Notice that for every module M we have a natural map ExtJ^(K,M) —>-H^(M) 9^.

(using the definition above), which we denote by

If we choose a projective resolution of K over R we get an

exact sequence n ■ syz K

O

Pq—"K

’ n-1

which, under the Yoneda definition of Ext, of Ext

R

(K,syz K) . We refer to ^ ^

element in H^(syz K). is a

(non-unique)

R

=

6

—O

represents an element e the oanon-icaZ

nT^(^) syz^^K

Given a different choice of resolution there

map between the resolutions which induces a map

from the original module syz K to the new n-th module of syzygies (syz K)'.

This in turn induces a map

(syz'^K)-^

( (syz^K) ')

which turns out to take the canonical element in H^(syz^K) to the n n , El one in H^((syz K) ) independently of the choices made: moreover, when restricted to the cyclic modules generated by the two canonical elements,

this map is an isomorphism.

Thus,

the canonical element

is well defined and unique in a certain sense. whether

In particular,

^ O does not depend on the choices made.

7

The canonical

element conjecture asserts that for every local ring R,

n

^ O.

7

We

R

now quote without proof some results from [H^] which show how this conjecture relates to some of the others. (2.7)

PROPOSITION.

T'he foUowing conditions on a local ring R

are equivalent: (1) 9„ M

(2)

e

^ O for some module M;

7

^ O;

syz“K

f O.

(3)

If R has a canonical module 9, th^ the following fourth con¬ dition is also equivalent to the above three: # o. □

(4) (2.8)

module

PROPOSITION.

then Q (2.9)

M

If a local ring R has a big Cohen-Macaulay

4 o and, hence, n

PROPOSITION.

7^ o.

If R is local,

finite over a regular local ring A, R.

R

r\

□

R

4 O, and R is module-

then A is a direct summand of

□ Thus,

the canonical element conjecture implies the direct

summand conjecture.

But the converse is also true.

Before stating

the result which contains this fact, we make the following notational convention:

if A is a complete local domain,

then T^ denotes the

integral closure of A in an algebraic closure of the fraction field of A. (2.10) (1)

PROPOSITION.

The following statements are equivalent:

the direct summand conjecture holds for all regular local

rings A; (2)

the direct summand conjecture holds for complete unrami¬

fied regular local rings with algebraically closed residue class fields; (3)

if A is a complete unramified regular local ring, then A

is a direct summand of T^; (4) Hom^(T^,A) A A (5)

if A is a complete unramified regular local ring,

then

4 O; if A is a complete unramified regular local ring with

52 maximal ideal m,

—^

then H (T ) / O; m

A

(6)

for every local ring R, we have

/ O;

(7)

for every complete local domain

we have

(8)

if

^ O;

.. . ,x^ is a system of parameters of a local ring R

then there do not exist integers b > a > O and elements such that (x,...x)^= ’

"

n y i-i

R

y.x..D "

"

Several remarks should be made.

All of the statements are

known in the equicharacteristic case.

Thus, we might as well con¬

sider only the mixed characteristic situation.

We could have fixed

the residual characteristic p and/or the dimension n of the rings A and R discussed: (8)

the statements are equivalent for fixed p,n.

it would suffice to prove the impossibility of the case where

a= t, b = t+1. Moreover, x^ = p. (3)

In

it would suffice to do the case where

Many of the implications are trivial or easy

=> (2),

(3)

=> (4) o (5),

are more subtle.

(6)

(7))

but some

((2)

{(1) (1),

=> (2) , (4)

(1))

The reader is referred to [h^].

We should note that one needs infinitely many cases of the direct summand conjecture to prove that n

/ O for one local ring R. R is that it behaves functorially

One reason for studying n

under various kinds of change of rings. The statement these conjectures.

(8)

seems to be the most down-to-earth form of

For many years the author has been pointing out

that even the case where n=3,

a=2,b=3 is open.

has now eliminated this possibility, he has shown [H^] that for n > 3,

= I

at least when x^

The author = p.

In fact

if

y^x.,

i=1 then a/b > 2/n.

This is rather weak for large n,

what one really wants to show is that a/b >

considering that

1.

The question of whether the equation above can hold when n= 3, a = 3, b = 4 remains open,

so far as the author knows.

We pointed out above that,

in connection with Proposition

(2.10), we might as well consider only the mixed characteristic

situation. from [h^],

The following corollary,

again quoted without proof

is concerned with that situation.

(2.11)

COROLLARY [h^;

Corollary

(5.4)].

Let R he an n-dimen-

sionat loQal ring wh.'loh is a homomorphic image of a q-dimensional Gorenstein Zocal ring S; say R = S/i.

Assume that R is of mixed

characteristic p > o, and that p is not a zerodivisor in R. If p is also not a zerodivisor on Q' = Ext^ ^'^'*(R,S), R

. □

then

S

0

It should be mentioned that the Matlis dual of

' is R

m

\r) ,

so that p is not a zerodivisor on H ' if and only if ^ (R) is R m p-divisible. However, the latter condition is not always satisfied.

References [a ]

M.Auslander, "Modules over unramified regular local rings", Illinois J. Math., 5 (1961), 631-645.

[a^]

M.Auslander, "Modules over unramified regular local rings", Proceedings of the International Congress of Mathematicians, 15-22 August 1962 (Institute Mittag-Leffler, Djursholm, 1963), pp.230-233.

[b]

H.Bass, (1963),

"On the ubiquity of Gorenstein rings".

Math.

Z.,

82

8-28.

[be^] D.A.Buchsbaum and D.Eisenbud, "Lifting modules and a theorem on finite free resolutions". Ring Theory (Academic Press, New York, 1972), pp.63-74. [be

] D.A.Buchsbaum and D.Eisenbud, "Some structure theorems for finite free resolutions”, Adv. in Math., 12 (1974), 84-139.

[d^]

S.P.Dutta,

[D2]

S.P.Dutta, "Weak linking and multiplicities", versity of Pennsylvania, 1981.

[d^]

S.P.Dutta, "Generalized intersection multiplicities of modules", Trans. Amer. Math. Soc., to appear.

[d^]

S.P.Dutta, "Frobenius and multiplicities", versity of Pennsylvania, 1981.

[eg]

E.G.Evans

(2), [gJ

114

University of Michigan,

and P.Griffith,

(1981),

P.Griffith, rings",

[H^]

Thesis,

J.

"The

Ann Arbor,

1981.

preprint.

preprint.

syzygy problem",

Ann.

Uni¬

Uni¬

of Math.

323-333.

"A representation Pure Appl.

Algebra,

theorem

7

for complete

(1976),

local

303-315.

M.Hochster, "Cohen-Macaulay modules". Conference on commuta¬ tive algebra. Lecture Notes in Mathematics 311 (eds. J.W. Brewer and E.A.Rutter, Springer, Berlin, Heidelberg, New York, 1973),

pp.120-152.

54 [H ] ^

M.Hochster,

"Contracted ideals from integral extensions of

regular rings", Nagoya Math.

J.,

51

(1973),

25-43.

[h^]

M.Hochster, Topics in the homological theory of modules over commutative rings, C.B.M.S. Regional Conference Series in Mathematics 24 (American Mathematical Society, Providence, 1975) .

[H ]

M.Hochster, "An obstruction to lifting cyclic modules", fic J. Math., 61 (1975), 457-463.

[H^]

M.Hochster, "Associated graded rings derived from integrally closed ideals and the local homological con-jectures", Colloq. d'algebre, Universite de Rennes I, 1980 (Universite de Rennes I, 1981), pp.1-27.

[Hg]

M.Hochster, "Euler characteristics over unramified regular local rings", Illinois J. Math., to appear.

[h^]

M.Hochster, "Canonical elements in local cohomology modules and the direct summand conjecture", preprint. University of Michigan, Ann Arbor, 1982.

[hm]

M.Hochster and J.McLaughlin, "Quadratic extensions of regular local rings", Illinois J.Math., to appear.

[l]

S .Lichtenbaum, "On the vanishing of Tor in regular local rings", Illinois J. Math., 10 (1966), 220-226.

[m]

R.E.MacRae, "On an application of the Fitting invariants", Algebra, 2 (1965), 153-169.

[mb]

M.-P.Malliavin-Brameret, "Une remarque sur les anneaux locaux reguliers", Seminaire Dubreil-Pisot, 24§me annee, 1970-71, exposd 13.

Paci-

J.

[PS^] C.Peskine and L.Szpiro, "Dimension projective finie et cohomologie locale". Publications Mathdmatiques 42 (Institut des Hautes Etudes Scientifiques, Paris, 1973), pp.47-119. [PS2] C.Peskine and L.Szpiro, Acad. Sci. Paris Sdr.A,

"Syzygies et multiplicites", 278 (1974), 1421-1424.

C. R.

[^^1^

P"Roberts, Two applications of dualizing complexes over local rings", Ann. Sci. Ecole Norm. Sup. (4), 9 (1976), 103-106.

^^2^

P*Roberts, Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture", J. Algebra, 66 (1980), 220225. ‘^-“P-Serre, Algebre locale: multiplicitds, Lecture Notes in Mathematics 11 (Springer, Berlin, Heidelberg, New York, 1965).

[w]

A.Weil, Foundations of algebraic geometry, American Mathemat¬ ical Society Colloquium Publications 29 (American Mathematical Society, Providence, 1962). Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109,

U.S.A.

55 THE RANK OF A MODULE

G. HORROCKS

Two simple invariants of a module are its rank and the codi¬ mension of the set of primes at which it fails to be locally free. In general these invariants are unrelated as can be seen by taking direct sums of ideals.

However in the geometric context of extend¬

ing a locally free sheaf given over an open set to its closure the modules that arise are reflexive and this na'ive example fails. Restricting to regular rings and localizing at the non-free set of primes leads to the following problem:

for a regular local ring A

determine the ranks of those non-free reflexive A-modules which are locally free except at the maximal ideal. Denote by m the maximal ideal of A, by k its residue field A/m,

and by X the spectrum of A punctured at m.

in question X-bundles.

Call the modules

The possible ranks of non-free X-bundles

have been determined only when the dimension d of A is at most 5, and for these dimensions there are indecomposable X-bundles of all ranks.

In the first section I review briefly some of the methods

used for constructing X-bundles and in the second describe an approach to the problem of finding restrictions on the ranks especially in terms of their cohomology.

Here the Syzygy Theorem

of Evans and Griffith has interesting consequences [1,3].

1.

Constructions

Reflexive modules are free for d < 2.

Assume that d > 2.

Any syzygy with level at least two is reflexive module itself), les.

(level O is the

and those arising from artinian modules are X-bund¬

Moreover local duality [5] shows that indecomposable artinian

modules give indecomposable X-bundles.

Second syzygies of cyclic

artinian modules provide easy examples of indecomposable X-bundles

56 of all ranks greater than d-2. The p-th syzygies

(2 < p < d)

of k serve as building-

blocks for all X-bundles in the sense that any X-bundle can be obtained as the complement of a free direct summand of a bundle with a filtration whose quotients are the syzygies T.

The size of

the free direct summand can be found from the following.

Assume that

LEMMA A.

e,f

are x-bundles without free direct

summands and that o^E-^M->-F->0'is exact. resolutions o

p

E*,

Choose minimal free

R ^ F* for the duals o/ E,F and let

->■ m' ^ f' -y o be the cokemei of the given exact sequence

e'

when it is embedded in P* © R*. The rank of the free direct summand o o of M is equal to the rank of the connecting homomorphism Tor^ (f' ,k)

e' (g) k.

With this criterion it is easy to find the ranks of the free direct summands of extensions where E = T^, is even, p=d-2 and q = rank at least d - 1. 1

T

Ext'‘(T-^,T

2

)

1

F = t'^.

Except when d

the resulting non-free complements have

In the exceptional case the module O

O

is the second exterior power A^(m/m^)

•!»

and provided

the extension is a non-singular element the resulting bundle is a null-correlation bundle of rank d-2 been made independently by Moore [8]

[9].

(These calculations have

from a different viewpoint.

He has also considered three-fold extensions.) One bundle arising from the foregoing construction which may have special interest corresponds to d=7, p=4, q = 2. Ext^(T^,T^)

is A3(m/m^)* .

The module

The stabilizer of a general element of

this space is the exceptional group

and for a local ring A with

a coefficient field the resulting X-bundle lifts from a vector bundle on IP^ with a G^-action,

1 + 3h^ - 28h^

rank 9,

and chern polynomial

(where h is a generator of cohomology).

A non-free X-bundle of rank d-2 for any d has been constructed by Vetter [12] as the kernel of a mapping writes down a quite explicit matrix. les of the same rank on IP

(2d-3) A ->■ dA for which he

Tango [10]

finds vector bund¬

by factoring out sufficiently many

general sections of the second exterior power of the tangent bundle. The dual of Vetter's bundle comes from a special choice of these

57 sections.

To construct these examples

free module F of rank d, 5

=

1

The

1

take a base

choose generators

5^,...,^^

a.,...,a

1

for a

for m and put —

d

cokernel of

5:F ^ A^f is

a bundle G of rank

3- rank

1

elements

all

in

e

space A^(k

k

).

geometric

),

and it

case a^f is

is

the product of X and

sufficient to choose

infinite

In

an N-dimens-

this

choice

It is

is valid for any k

g

is possible provided

this way a bundle of rank d-2

construction

lay ring A).

The rank of G/ImV is

codimension of the grassmanian of lines

ambient space. Vetter's

mod p for any p e X.

containing no non-zero elements a A

When k is

less than the

its

^

d

P d subspace of A^(k )

lonal

N is

In the

0

/ F A

of A^F determines

summand V of rank N whose non-zero

satisfy this condition.

(d-1) (d-2)/2 - N. a vector

a direct

a

An element

sub—bundle of G provided that cr

Suppose that a^f has

(a,B

(d-1)(d-2)/2.

in IP^

^

is obtained.

(and any Cohen-Macau-

equivalent to choosing V to be the

sub-module

of A^F spanned by ^ In

[9]

these


O

in which each unnamed homomorphism

epimorphism,

is multiplication by

is multiplication by

By

,b _^JM n+1 —n+1

O

and A-homomorphisms

obvious natural

and

)M

0

A-modules

0

M/b M —n+1

—n

the

remaining homomorphisms

follows.

[9; 2.2],

k.a.M C b.M for

i = n,n+1,

and so we

can

define

1—1-1

Y

:

M/a M—>-M/b.M by Y

-1

1

-1

(ni)

= k.m for

1

of an element m of M inM/_a.M (respectively m) .) = k

,m = n+1

iHlm for

Next,

t

note

.k = n+1 n

all m 6 M.

(The natural

image

1

Also,

6

(respectively M/^,M)

^ n+1

:

is

M/a M—>-M/b M is —n —n

denoted by m

defined by

0

, (m) n+1

all m G M.

that

11

.h

nn

y

h

,

,,s.+h , ,s n+li 1 n+1n+1 n+1

(1=1 so

that t

t

(m) n+1

the

GAk

,+b by[9;2.2]. n+1 -n

,(Ak + b )M C n+1 n —n —

and this a

,k n+1 n

enables =

t m n+1

(Ak

. n+1

thus

to define

a

for

all m G M

(and here

n+1

:

M/(k

that

,b )M—^M/(k .,b )M by n —n n+1 —n

the natural _

are

see

+ b )M, —n

us

appropriate modules

We

denoted by m

and rh) .

images

of m in

66 With the dbove notation,

3.5 PROPOSITION. D{s;t;H)

the diagram

is commutative and has exact rows and columns.

Proof.

It is

clear that all

from the upper left one

commute;

the squares

moreover,

in

the

diagram apart

for m e M, r n

m = k

hence

t n+1

square

as well.

columns of D(s;t;H)

Proof.

the

., n+1

Theorem,

then t .me n+1

3.7

(Ak

. n+1

in conjunction with the from Peskine's

Remark.

if b is

morphism of A-moduies zerodivisor on X, In the

and so the upper left that the

rows

and

(ker = O.

= OIf m e M is

+ b )M c (Ak —n — n

+ b )M, —n

and so

in the proof of the Exactness

following remark,

and Szpiro's

f is

idea for

d'Acyclicitd"

an ideal of A and f

for which bker

then

"Lemme

the

:

f = O but b

X —^ Y is contains

[5;1.8].

a homo¬ a non-

a monomorphism.

course of the proof of the Exactness Theorem,

need to use some

such

□

last lemma will be used,

which comes

/

1

□

clear that b (ker a ^) —n n+1

claim follows. This

2.2],

It is easy to see

are exact.

It is

that m e ker a

[9;

n+li

With the above notaiiion,

3.6 LEMMA. _

n^^n+l '^n+ln+l^n+l^™

k m - k ,s m e b M by n n+1 n+1 —n

commutes

s.

(t

local cohomology theory.

For i ^ O,

we

shall

we use

to d.

denote L

3.

the i-th right derived functor of the

with respect to

make

use of the 3.8

an ideal

if

Let X be

s =

(s

1

and only if

Notation.

,...,s

that Hs = t,

n+1

)

Let n be

functor

we shall

(X ) Z

a positive

and let p e Spec(A).

O.

integer.

and t=(t.,...,t Jeu ^ 1 n+1 n+1

we shall be

but also in its

In particular,

an A-module,

£^p 3.9

cohomology

following.

Observation.

Then p 6 Ass(X)

aofA[7;2.1].

local

extension

interested not only

in

For

and H ~

e D

n+1

fA)

such

the diagram D(s;t;H)

67

o

O

O

n+1

o

->-M/a M

-^M/a M —n

^n

-^M/b —n+1

—n

n+1

M/(k

-»-M/(k ,b )M n —n

M/(k

,,b )M n+1 —n

0

)M

->-0

O

shall denote by E(s;t;H).

By

3.5,

,

E(s;t;H)

or otherwise of each of its

on whether or not the

•O

n+1 -+1+1

0

but the exactness depends

'n+1

->-M/b M

~n

which we

-^O

n+1

-^M/b M

O

,M

—n+1

®n+1 T

O

-^M/a

—n

rows

is

commutative,

and columns

appropriate named homomorphism is

injective. The proof of one induction,

implication in

and the next two

the Exactness Theorem uses

results provide

a basis

for that induc¬

tion. 3.10 LEMMA,

\3

(i)

If

is exact at

M,

then every member of

is a poor H-sequence. '

_1

If

(ii)

is exact at

c(‘^^,M)

M

and

M,

then every member of

is a poor n-sequence. Proof. with

(ii).

This,

in

(i)

Suppose

u m e u M. 2 1 is

Part

Then d

is

a routine exercise,

that

, U'^lM

—1 exact at U, 'm, 1

=

e

--r = O by (u-|,U2)

(i),

shows

.] i]

that

(u^,U2)

is

Thus,

since

Hence m G u M. 1 a poor

Assume that c(‘i4',M) is exact at M and

Then, for each choice of s = [h

[9; 3.3].

such that

□

3.11 PROPOSITION.

H = ~

and m e M are

m m^_ , -n = —for some m e M. (u^) (1)

conjunction with

M-sequence.

(u^,U2)

and so we only deal

G D.,(A) 2

rows and colvcmns.

for which

Hs = t, ~~ ~

,

t

=

the diagram

^

M.

^2

E{s;t;H) ~

has exact

68 Proof, Y

,0

1

In the notation of and a

,T

2

2

2

2

monomorphic.

are

3.9

and 3.4,

we have

all monomorphisms.

By

to show that

3.10,

and

are

2 To see

that

is

such that h^^m e b^M = t^M. tm=h sm=s,tm'. 111111

a monomorphism,

Thus h^^m = t^m'

Hence,

by

3.10,

suppose m e M is

for some m'e M,

m = sm'e_aM. 11

and so

Thus y

is

a

'

monomorphism. Next,

we

show that

is

a monomorphism.

that ^1^1^22”* ^ -l'^ " ^1^' hii(t2-h2iSi)m G t^M. (t

,t )

is

'^1l’^22®2’^

But h^^s^

a poor M-sequence,

=

by

t^,

such

^1^'

and so h^^t^m G

3.10,

12

Let m G M be

and so h

t^M.

m G t M.

Now Hence,

.Ml

since y

is

1

a monomorphism,

m G _a M.

It follows

that

0

that m G M is

such

that

1

is

z

a mono-

morphism. We now

consider y^.

'^1l'^22'^ 6 b^M =

Suppose

(At^+At^)!^.

'^11^22'^ + ^2™"^

there exists m'G M such

that

Therefore

"1^11^22“ ^

S'"'

so that t2(s^m')

G t^M.

3.10(ii),

and so

s^m'G t^M;

Hence,

3.l0(i),

by

Thus

But

(t^,t2) thus

is

s^m'

a poor

M-sequence by

= h^^s^m" for some m"e M.

m'= h^^m", and so

^11^22"^ ^ Vii“"^ Therefore *^i-)^22^2™

^2^2'^11^”^

so that

^1l'V^21^1^'" ^ =2Vll“''^ S'"Hence

t2^'^1l'"

®2'^1l'^”^ ^ ^l"^'

for some m'" GM.

But s^h^^

so m + s m" = s m'" ;

2

1

that h^^m + s^h^^m" = s^h^^m'"

= t^

is

a non-zerodivisor on M,

hence m G a M;

It follows

-2

that y

is

and a

monomorphism. It is now immediate a monomorphism.

from [3;

Exactness Theorem uses

3.12

at

M,

h'g

THEOREM.

''m, ...,u^^M,

(a)

the proof of one

an inductive

in a separate

and that, for every and

Theorem 5]

that

is

□

As mentioned earlier,

inductive step

Chapter 4,

Let

n G K

argument,

with

i=1,...,n-1

exact rows and columns.

and we

the

isolate

the

theorem. n

>

1.

,4ssiOTe

that every element of

for which

implication in

h's'=

that

C(y^,M)

is a poor H-sequenae,

and for each choice of s' t',

is exact

the diagram

E(s';t',-H')

^

'^■+1

has

69 Then every element of choice

o/s=(s,...,s

I

e

Jeu

),t=(t^,...,t

n+1

Hs =

is a poor n-sequence and^ for each 1

n+1

the diagram

t,

, and

n+1

= [h .]

H

~

i;]

has exact rows and

E(s;t;H)

columns. Proof. must be theses

We

show

first that each sequence

a poor M-sequence. that

m £ M is

(u^,...,u^)

such that u

n+1

It

is

follows

(u,,...,u ) 1 n+i

from 3.1(ii)

a poor M-sequence.

me

So we

(Au^ +...+ Au )M. In

Hence,

e u n+1

and the hypo¬ suppose by

that

[9;3.3],

we

have u

,n (u,,...,u ) 1 n Hence,

since

C(‘^ n,

{(u.,...,u

jin

a

restriction

conditions

for all

i

therefore

,1,...,1)

3.1(i)-(iv),

from

3.14

A

G

:

y (M)

(u^,...,U

)

G U

Inn

},

‘'U =

(U.) .

^satisfies

and,

by hypothesis,

it is

the

in

is

family

1

IGIN

a poor M-sequence.

conjunction with ; V

the

—>■ U ^M, n

case

The

fact

that

result

[9;3.2]

for which

-s

m

m

'"i.V for

for

the

that the natural A-homomorphism ij{M) (

to A^;

n

Then

G IN, each member of U. follows

of U

let

triangular subset of A^.

the

[9;3.6]

For each

all m G M and

(v

, . . . , V

1

(v.,...,v ) 1 n

Before we move on

)

n g V,

is

an isomorphism.

to applications

of

□

the Exactness Theorem to

74 big Cohen-Macaulay modules

in Section

observation which should be made is

contained in

of which

are

3.16

the

4,

there

about the

second of the

one

further

.

complex

following

straightforward and left to

is

two

the

lemmas,

the proofs

reader.

Let V be a triangular subset of

LEMMA.

This

Then there is

an A-isomorphism b ^

:

V

-n

■V

A ® M A

which is such that, for

(v

1

,...,v

n

M

a e A,

® m

)

(v

m e M

1

and

, . ..,V

.n

)

V, we have

(v

. □

There is an isomorphism of complexes of A-modules

3.17 LEMMA.

and A-homomorphisms

:

such that

IN,

n G

A ®^M

—^-M is the natural isomorphism and, for each

4)’^ : u ’^A ®^M —u ^M is the isomorphism given by 3. 16.0

4. Applications to big Cohen-Macaulay modules Throughout this and local,

that m is

and that M is

's.s.o.p.'

we

shall

It will be

's.o.p.'

a s.o.p.

Hochster,

such

reader is

as

[2],

for A.

Furthermore,

for details

Macaulay A-module to every s.o.p.

we

if it is

for A:

see

say

as

n

allows

us

to

1) ,

the

follow¬

'system of parameters',

say

x^,...,x^ the work

of the

that M is

a

that M is

a

big

if x^,...,x^ is and writings

in

an

of

relationship between

conjectures

this

commutative

balanced big Cohen-

a big Cohen-Macaulay module with respect [8].

The Exactness Theorem of Section 3.17,

We

referred to

concept and the various homological algebra.

that dim A = n

'subset of a system of parameters'.

Cohen-Macaulay module with respect to The

that A is Noetherian

convenient to use

will stand for

will stand for

Let x^,...,x^ be

M-sequence.

assume

the maximal ideal of A,

an A-module.

ing abbreviations: while

section,

3,

used in

conjunction with

characterize balanced big Cohen-Macaulay A-modules

follows.

(>

4.1

THEOREM.

(A

1).)

For each

i

is local and has maximal ideal g

'M , we set

m

and dimension

\ ~ {' • - •6

: there exists

with

j

such that

0

In fact, M. is a big Cohen-Macaulay module with respect

to Cohen-Macaulay

H

n

[9;3.2])

for all

6 IN

3.1(i)-(iv)

.

conjunction with [9;3.9]

for A.

and dimension

is interpreted as

x^

is exact and Of

m

:

Then the family itions of

in

he a s.o.p. for the local ring A

x^,...,x^

be the expansion {see

u(x)^

{(x^

of

results

a specified s.o.p.

{which has maximal ideal let

same

□

^

M

those A-modules which are big Cohen-Macaulay modules

with respect to 4.2

exact and

3.17) is

m

and dimension

n

{>

Let

1)).

x^,...,x^

be a

Then

, ,-n-1 = U{x) A, n+1

where

a U(x)

=

{ (X

n+1

-X

, 1)

: there exists that a.

if k is

This

a field,

result has t

•

•

.,X

are n

a^,...,a^

^ =...= a

3+1

Remark.

j

with

0< j tn

e ]N

and

such

n

1

n

= o].

connections with the known independent indeterminates

fact that, and

B = k[[X^,...,X regular

]],

and we use

local ring B,

(E

(B/ B

r

■

'

r to denote

used in

that there

is

ideal of the

[4;

3].

then

—

—1

=)

kCX^

[9;3.11

],

n

I

the B-module of "inverse polynomials": arguments

the maximal

and 3.2]

see

Theorem

As

the

can easily be modified to show

a B-isomorphism

. where we X

are using

,...,X

1

for B,

the notation of 4.3

it follows

_■]

zation of the

Proof. is exact. an

fact that H We use

We

r

(B)

= k[x

,...,X In

for each i

automorphism of U(x).^A.

the

s.o.p.

a generali-

_1

the notation of 4.2:

show that,

to

that 4.3 may be viewed as

n

vides

and X refers

3.

by that theorem,

e IN,

C(‘1i^(x),A)

multiplication by x^ pro¬

Since,

for a G A and

a (X

. ,X.

1

)

G U(x) .,

1

we have

1

.X. a 1

X

1'

a. +1 1

a^ + 1

,x. r

/X.

we

just need to show that x.

is

a non-zerodivisor on U(x).

1

is

immediate

from

A;

this

1

3.15 because each member of U{x)^

is

a poor

A-sequence. This done,

H^(U(x),^A) m 1 The

we

can now deduce

= O

for all

j

that,

ences which can be obtained from the

5.

1,...,n,

> O.

result can now be established by use

in mind that U(x).^A = O 1

for all i =

for all

i

of the n short exact sequ¬

,A) :

exact sequence

bear

> n+1. □

Applications to the minimal injective resolution of a Goren

stein ring Throughout this section, we

shall only

assume

we

shall

assume

that A is Noetherian;

that A is Gorenstein when this

is

explicitly

stated. We begin with 5.1

LEMMA.

a general

lemma.

Let u be a triangular subset of pP',

let m be an

77 element of the A-module M and let

(u,,...,u ) 1 n

u m

If

(i)

(ii)

(u

1/ (u

,

O

(u

Proof,

= o in u

,. ..,u ) I n

1

n-i

,1)

O

(i)

m

then

m

: (U

, . . .,U

1

There exist

= O.

(u,,...,u ) 1 n

also, then

e u

, . ..,u , 1) 1 n-1

e U.

wU

n-1

(wt,...,w ) ' n

6 U

)

n

and H = ~

[h..] n

e D (A) n

rn-1 such

that Hu = w and — ~

Hu m e n

M.

I

Hence

i=1 n-1

n-1

I h ni U .

w

Li ,^ ^

n

r

m e

) Aw, M. ■ 1 1 [i=1

1

1=1

n-1 Therefore,

by

[9;2.2],

^ n-1n-1

11

I

m g n

[9,-3.3(11)], h

. . .h

11

(w

n-1n-1

,...,w

' in U

M.

It

which

the

desired conclusion

(ii)

This

is

an easy

Notation.

has height

follows

some

of A^:

from

since Hu = w.

consequence

of

□

(i) .

notation which will be

We

adopt the

For each i

e IN,

= {(u,,...,u ) e A^ ill

:

in

an elementary exercise in

(i)-(iv)

fact, of

For (finite)

the

3.1,

(u

to

=

family

and so we may

,...,u ) e U , Inn

convention whereby we

force

through¬

=

the

ideal A of A

set

ht(Au, 1 j

It is

by

section.

5.2

U

V, u u h^ ^..h , .h m ^ 11 n-1n-1 nn that —;-- = O, (w , ...,w ,w ) 1 n-1 n

from this

We now introduce out the

hence,

w m n _ ^

.,w2) n-1 n

^ 1 follows

M;

1=1

+...+ Au,)

j

>

j

for all

1, . . .,i}.

check

that

is

a triangular subset

(U.) .

satisfies

the

form the

complex C("//,A)

conditions as

in

we shall denote by P(u^,...,u ) In

3.2. the

set

{p

e Spec (A)

Remark.

5.3 which A is =

:

P

3

Au^

+. . . + Au^

and ht p_ = n}.

It should be noted that,

Cohen-Macaulay, {(u^,...,u^)

we have,

6 A^

:

for all

u^,...,u^

in i

the

special

case

in

e 3SI,

form a poor A-sequence},

78 so that, is

in view of the Exactness

actually exact in

5.4 Remai'k.

this

Theorem

3.3,

the

complex

case.

is immediate from C9;3.3{ii)] that, for each

It

i 6 ]N, Supp(U^^A) C {p 6 Spec(A) In

fact,

we

Ass(U^^A)

and

u,

(u^,...,u^)

:

6 Ass(U.^A),

(O : a)

ation of 5.2,

6

^ {p £ Spec(A)

Proof. Let p Now p = —

ht p > i-l}.

can say rather more.

For each i

5.5 LEMMA.

:

for some

G U^.

ht p = i-l}. so that,

by

a =

Suppose that ht p ^

i.

£ ^

1,

-i

A,

where

a e A

1

Then,

and interpreting P(u^,...,u^

ht p = O} when i =

ht p ^ i-l.

5.4,

• a -r (u^,...,u^)

as

the

using the not¬ set {peSpec(A):

we have

U

£,

q e P (u^, .. .

so

that we may select

U

G p \

Then

q G P(u^,

)

•fU.

1-1

(u^,...,u^_^,v^)

O

P =

:

e U^,

(u

and,

,...,u, 1

O

by

(5.1) (ii),

O

,u,)

1-1

(n.,...,u.

1

1

(8)

.,1)

1-1

: (u 1

,...,U. ,V.) 1-1 1

V. a But V.

e p,

1

and so .,1)

1 dieting

(8).

Suppose that

a G A.

1

1-1

The result follows.

5.6 COROLLARY.

(u^,...,u.

Let

(u

1

G u. {where 11

O

:

has height

(u^,...,u^)

5.7 THEOREM.

Let

i

contra-

i

G n) and let

—r f o in u.^A. Then each associated

(u.,...,u.)

1

prime of

= O,

1

□

,...,u.)

a

,

-,v.)

1-1

G IN.

1

i-1.

□

Since, in view of

5.6,

a given element

of U^^A has annihilator which is contained in at most finitely many prime ideals of height 6

:

there is an A-homomorphism

i-1,

U.^A

(u/a) 1 P

1

£ G Spec(A) htp=i-1

79 which is such that, for a e

and

the component of

is

in

0(a)

of height

p e Spec(A)

i-1,

y.

The map Q is an isomorphism. Proof. p =

(O : a)

Suppose

for some a e ker 0.

have ht p > shows

that ker 0^0,

that

i; 0

but 5.5

shows

By

and let p e Ass (ker 0) . 5.4

and the definition of

that ht p =

i-1.

This

Thus 0,

we

contradiction

is monomorphic.

Let p^

be

a prime

0^6= (6

ideal of A of height i-1.

© p ht£=i-1

Spec (A)

Let

(u/a) i

htp=i-1 where O

if P 7^ P^,

6

a_ P

..,v.) 1

if P = P^;

t here,

a e A,

(v^,...,v^)

order to show

that 0

Set a =

the

O

:

(v

e

is

and t 6 A \ p^.

surjective,

,

, . ..,V, ) * 1 /

associated primes

a

of

j

=

1,...,k,

be

to show that 6

have height i-1;

Let a = -

+...+ Av.

^ p^.

Also,

e im 0.

thus

By 5.6,

see

,

q.,

-J

c a c p^

we may

all

that p ^

D

with r(q.)

and ht p,

—k that V.

we

in

k = p.

the minimal primary decomposition

Since Av,

enough,

a proper ideal of A.

“ must be one of these.

It is

choose

s

j=2

for i-1,

it follows

-I

n qj

6

=

for

\ El'

Then

^

sa 6

=

(v

El

,...,v.

1

,i;

_ stv, 1

Also,

use

of

5.1

f

enables us

to see

sa

O

with

P(v^,...,v,

=

Astv.

{at

:

the notation of 5.2 {pe Spec(A)

+ q, ^

s) = q ^. (9)

^'^i.Vr'^i^

('^l.^-I'"') Next,

that

U p e P(v^,...

:

(and the understanding that

ht p = O} when i =

P,

1),

we note

that

8C for if this were not the

case

for some prime

ideal p’

and Astv^ c p'

would lead,

ments

that p^ v'

= p'

= bstv.

1

and

of height i-1,

P-j

and the

respectively, ^ P^ •

U

E. E

inclusions

to the

Thus we may

+ c e A \

1

C p

contradictory state¬

choose

)

pGP(v^,...,V^_^) j

with b G A and c G q^. 5.1

it would follow that Astv^

Note

that

, . . . ,v,

,v^), G

and that,

by

(9),

O

sa

:

O

sa

:

(v^ / • - -

)

(j)) (T)

In the

is

standard mod Gl

the subset of these elements forms a basis one

for

obtains

a description

of standard tableaux

(which agrees

§1

we

said that the

image of the boundary map

.p-1 A F in the Koszul complex was

p L .F or, q+1

Recall that the map

,p-1 S F ® A F -5-SF-*-.. F - A F ® S

aP

A F ® F

9 .p-1 F ->- A F ®

S

q

^F

q+1

S

^F ® F

F,

q+1

p-1 where

3. 1

multiplies q copies of F to S F q

interchanging S^_^^F and A^

"'f)

and 3^ 2

diagonalizes

A^

(in addition to

"'f to F ®

...

® F.

p-1 Since that

3^ Im

is

a surjection and

9 ~

also see

then L-, ,F = A^F

A = (A',

3^

and we have

is

an

injection,

we

see

that if we

let A'

surjection

S F ® L, ,F ^ L,F where q A A

the

be

immediately

the partition p

1,...,1).

q In general, SF®L-F^L,-

and A =

for any partition

-

^,Fas

A,

we may define the

follows.

(X^,...,X^),then (A,1,...,1)

If we

let A =

surjection

(A^ , . . . , A, )

(A^ + q,A^,••.,A^).

=

Thus

S F ® L,F is the image of

q

A

1®d:

SF®A

fi8...8A

F —^SF®S.^

Sr

F ® Sr F

S*’

Sr F ®

where u

multiplies

\

S~ F 8 S F onto

Sr

1

1)

A.J

q

8 d

(or d A

8 A

F 8

A^+q

F.

...

>2

S F -^ Sr

\

q

Since u^

1

is

S F ^

t

F ® Sr F

A^+q

8 f onto

®...®S^F

and u

A^

multiplies ^

surjective,

the

image of

clearly gets mapped surjectively onto the

image of

loo d

, ^ (A,1 ,. . . 1)

and it is

this

surjection of

S F ® L,F onto L , . F q X (X,1 , ■ .. , 1)

q

^

which generalizes

the surjection of S F ® q

A^F onto L, . „,F, (p, 1, • ■ . , 1)

and

q that we

shall

To

consider in

this

facilitate matters,

partition and

is

we

(A

+£,...,A

(£,...,£)

9:

S

which is

r

integer,

there

F ® L, F A+£

S

r+l

is

a

itself,

®

...

is

a

A + £ length

F ® L

and let

A be

a partition

of

A+(£-1)

F

® A

F ® A

F

F

S

easy to check,

of the

1,

F

® A ^

V ® A

V

A +£—1

+£

S F ® A ^ r

gram of A+£ is

^F ® A r+1

induced by the Koszul map A

9:

left

the

is

canonical map

A

is

where k is

A

induced by the map

S

This

shaPl denote by

+£)/(£,...,£),

integer,

For every r,

S F ® A

which,

we

If

is of length k-1.

Now let £ be a positive length k.

some notation.

K

I

and

introduce

a non-negative

the skew partition of A

section.

^F ® A r+1

F.

particularly when one

the diagram of

last row of

A

augmented by

realizes £ boxes

that the dia¬ added to the

A:

r—- .. 1 ik-1

Thus,

A

the

Schur map

F®...®A

^k-1

is

f®A

the

composition

F—>A

^1

f

®

A

^k-1

F ® A

5

F ® A F

(*) -> S~ F

S~ F ® A F A

"a

t where one

the

first map diagonalizes

diagonalizes

£

A F to F

-\-ji

A

...

A

k

® f.

F to A

,£ F ® A F and the

last

101 THEOREM

length

If F is a free R-module,

3.1.

and q, is a positive integer,

k,

s

„F ® L. „F q-2 X+2

S

\ is a partition of

then L, ^F X+1

.F q-1

S F

q

{**)

o

^(X,1.1)^ is an exact sequence. Proof.

Notice

X that when k=1,LF=A

1

X^+5, F

and L

F = A

F,

A

so

that

(**)

reduces

to the

Koszul complex in that case.

The proof proceeds by case

in which q =

1,

first consider some integer

I,

we have g

induction on q.

as well

as

additional

the map is

(*)

Z

The

is

step,

we must

For every positive

ct

injection j ,

and we

3 1-

see

is

clear from the

that,

on the generators,

diagonalization map

^k-1 ^k"^^ .® A ^ 'f ® A F The map a

inductive

the

A F ® L,F —>■ L,, „ ^F and an inX (X,1)+£-1

of the map

induced from the

the

canonical maps.

a surjection

jection L, .F —+ A F ® L,F. X+£ X factorization

to handle

In order to treat

6

A

^1 'f

the map induced from the

® A '^F ® A^F.

identity map

X 8 A '^F

A F

®

,

® A

'f

\

£we know that we may regard A F ® L F as

From 2.1

1

A

the

A

and L

cokernel of □

as the

that □

=

1

® □,

+

3.

£ F ® A F.

-i\.

cokernel of ^^d we observe

We therefore have proved the

follow-

ing. LEMMA £,

For every partition

3.2.

and every positive integer

the sequence

° is exact. Lemma the

X,

special

® V - hx.D+E-f ^ °

^ □ 3.2

proves Theorem

case

Assuming now that consider the map

of

3.2

3.1

in which

the Theorem is

£

for

the

=

gives

true

1

case us

in which q = (**)

1,

when q =

for q and setting y

=

for

1.

(X,1),

we

of complexes

,3 3®1 .2 3®1 3®1 „ ...->■3 F®A F®L F -^ S ^F®A F®L,F ->- S F®F®L,F -3 .F q-2 X q-1 X q X q+1

. . .„F q-2

L

p+2

1®a

1®a

1®a -> 3

.F q-1

L

..F i+1

3 F

q

L F U

^(U,1.1)^

■o.

102 The reader can

check

the horizontal maps sored with the plex

(**)

^

is

indeed a commutative diagram,

of the ordinary Koszul

and the horizontal maps

replaced by y.

and the

two complexes

argument completes

4.

those

the modules

hypothesis on q, that the

3®1

identity,

with A

by Lemma 3.2,

that this

The kernels

S F®L, q-r A+r+1

9

above

are

the proof that

those of the

for r > O.

exact, (**)

complex ten-

of the maps

acyclicity of the Koszul

with

1®a

com¬ are,

The induction complex,

tell us

and a simple homological is

exact for q+1.

D

Skew hooks

As we have

already said,

the modules L,

^

^

F were

de-

(P,1,••■,1)

q-1 noted by L^F in as

Es].

These modules,

the diagram of the partition

or shapes,

(p,1,...,1)

are

looks

called

hooks,

like

P

Prompted by the discussion in consider skew shapes of the

which we naturally call functors, the

which we

[4;

pp.558-559],

led to

form

skeW-hooks,

again

we were

and the

call skew-hooks.

corresponding

skew-Schur

More technically,

we make

following definition.

Definition

and set

~

4.1.

^

~

Let P.j /.. . ,Pj^,q.|,. . . ,q^ be positive

(k-j).

Consider

the partitions

i=j

^k-r''

V'’ A and

.. . ,a^,a^,. . . ,a^,. .. ,a^,. . . ,a 2' 3 'lo-l

^

- • • '^2-1'^T^,...,a^

v"* , 1,. . . ,1; '^k-r''

1,..., a^^-1,

.. . , a^^-1) .

integers.

103 The

skew-shape

X/y

is

(P-j / • • -q,j / • • •

will be

>

and the

denoted by L

For k =

1,

we

skew-hook of type

called a

corresponding skew-Schur functor

F.

see easily that we have our original hooks,

and

P....Pk that

for k

>

1

the modules

L

F are

generally not irreducible,

q-,--.qk

even in

characteristic zero.

A few elementary quite easily,

and we

facts about thes€

state them in the

PROPOSITION 4.2.

Let

skew-hooks

can be

seen

following proposition.

p.j»...,Pj^f

q.|»---fqj^

be positive inte¬

gers. 1/ q^

(i)

D,

■kx. _ r —

=

1

P..---P +P

I

T Jj

’r

i

M' J/ P^

(ii)

for some

=

1

i

=

.,“1 p

i+^

then

1,...,k-1,

^---P

i+2

•qp-.-q^

for some

i =

then

2,...,k,

P^...Pi...Pk

^l-'-Pk F = L

F.

Lj

qv Pl--.P (iii)

i/ p.,

Because

•^k

F,

=

1

of 4.2,

that q.

>

= q, ,

then

we may as well

1

for i

=

k

^k^k-1'

kp « L ^

L

assume,

1,...,k-1,

'

F.

□

when studying

thatp.

>

1

for i

= 2,...,k,

.qk and that p^q Ik In

which

is

[4]

the

>

1.

we defined a map

action of the

an element of SF ® AF* ponds

to the

Hom(F,F)

(C^

from L

Pq

1

' .F q^-i

trace element C is

L

P, F to L

^q ^2 F

e

F ® F*

In

fact,

considered as

the element of F ® F* which corres¬

identity map of F ->■ F under the natural

F®F*) .

F

SF ® AF

is

isomorphism

an SF ® AF* module

and the

action of C on SF ® AF is precisely the Koszul complex boundary map 2 F (since C =0, one automatically obtains a complex). Since L^F is F q the

image

(or kernel)

of the

action of C

r

on

a suitable homogeneous

104 strand of SF ® Af,

E L^F is an SF ® AF-module

so that we have

the

q Pi action of C

factor,

on L

F

and the

P2 ^F ® L

F,

q,-1

AF*

action on

considering the

the

second.

P1P2 L

qiq2

It

is easy to check

P., F is

the

image of this

first

that

P2-I

action

in L

P

P2tq.,-2

q^

SF action on the

F ® L

F and since the

q2

complex

Pi+P2+qi-2 O

L

5 F —>

F ® L

^2

^ C

IS exact,

Pi L

we

see

that L

P2-'' =F Pi F ® L F —^ L

h

P.,“1

—>- L

qi

L

^

^1^2

F®L

L

^

F ® L

C

p

P

-2

F—F®L^

q2

q^+1

„F ®

F is

F

L

(*)

F

q2

^

F —> O

P2-2 F.

Moreover,

we see

that E

L

^2

is

also

SF ® AF*-module.

we

can define L

P P 12

_ F ® L

^^1^2

F as

P1P2

above procedure,

SF ® AF*-module,

operate by C

, ^

P3-I F ® L

q.|q2

L

an

iterating the

'^3

and get a map to L

easily seen to be

PiP, '^F P1P2

Thus,

P ^3

^F '^2

also the kernel of the map

F ® L

% an

F ^2

P1P2P3

F.

The

image of this

action

is

q3

F.

Thus, by iteration,

we have a well-

'^l'^2‘^3

defined action of C

^l""'^A ^A+1 a ^ ® ^rr

to that in

[4]

ing theorem.

on L

F ® L

"^"’’^k rr ^ whose

which proves

image

is

^F to

^1*"’^k L ' ^F.

the exactness of

(*)

gives

A proof similar

us

the

follow¬

105 THEOREM 4.3.

The complex

C, F F

C

F

L

F -O

is exact, where 6 is the obvious diagonalization, and 8 is the ob¬ vious surjection (generalizing the maps 6 and "h in Remark.. as

Just as,

a sequence

size of its

classically,

of nested hooks,

Durfee

square

(see

sequence of nested skew-hooks, to the

formulas. found in

section we

to the

the number of which,

again,

is

end of

§2 we

we have

A

of standard tableaux. for F and T is

standard
L) (1.3)

sup{i

< °° then I

Tor^(M,L)

=)= O} = pd M - depth L.

(This is easily established by induction on depth L.) now follows,

since it is also well known that £(A/det A)

for any injective linear map A p > 1. F^ ->■ M, of the

:

A ^A

:

Choose a free module F^,

induced map F /aF

0

F /aF O O

that pd K = p - 1,

M

0

^ M.

It

since pd(F^/aF^)

follows

= 1.

since x( “

are additive on short exact sequences, COROLLARY.

see for example

(2.6).

Let K denote the kernel from the exact sequence

O

from the inductive hypothesis,

(1.4)

= £(Coker A)

a surjective linear map

and an element a e ann M - z(A)Up.

O -> K

The equality

Now the identity follows ^^d £(A/G( - )A)

the latter by

both

(0.1)(c).

If A is regular and dim A = 4,

then (1)

□

124 and (2) hold. This was first proved by Hochster [5;

Corollary 2.11]

(by

different methods).

Proof. or from

Directly from

(1.1),

(1.1)

it follows that

dim M = 2

and dim N = 2,

be proved by standard methods.

(Namely,

M = A/p and N = A/q where p,q e Spec A. 2

holds.

it follows that the only remaining case in

when dim A = 4,

A = Ext

(1)

(N,A) .

Then depth fi = 2,

and in this case

(2)

Directly, is that

(2)

can

it suffices to assume that 2 Write Q = Ext^(M,A) and

and hence pd Q = 4 - 2 = 2.

(If

t

B = A/(a^,a2) where

^ ^

A-regular sequence,

then

U ~ Horn fact

(M,B), a second syzygy B-module.) Also depth A = 2, so the B (1.3) shows that = f(f^'5?>A) >0. On the other hand,

since Q

(respectively A)

r^ e Spec A and ^ 2 P nx(M,N)

1).)

= x(^f^)

has a filtration with factors A/r_ with

(respectively ^2 1)

for some non-negative integer n

(1)

already holds,

(which actually is

□ (1.5)

Remark.

if A is regular and dim A = 5,

shows that the only remaining case in dim N = 2.

(1)

It is possible to prove that

(using some techniques from [2]). a

since

then

(1.1)

is that when dim M = 2 and (1)

also holds in this case

However this is already Icnown;

(different) proof should be in Dutta's thesis [3].

2.

A new descript-Lon of the MacRae ideal

First recall that there is an exact sequence of abelian groups, the so-called localization sequence (2.1)

(A) -> K^(s"’’a)

K^(A;S) ->

(A)

Kq(s"’’a).

Here and in all that follows S denotes a multiplicatively closed

subset of A with S A).

z(A)

n

= 0

(that is,

S contains no zerodivisor on

For the localization sequence consult [1;§l0].

definitions can be found in Cl;§§1, 4 is denoted by ion of K^(A;S)

(.'9'^^ (A) ^) .

and IO], where the group However,

the following descript¬

is the one that will be needed later.

be found in Possum,

Foxby,

The basic

Iversen [2]

(see,

Details can

in particular. Propos¬

ition 4.8, where the group is denoted by K^(Hot(P(A),S))

).

125 Description of

K^(A;S).

presented by generators

The

[P.],

abelian group K^(A;S)

only depending on the

class of the bounded complex P.

is

isomorphism

of projective modules with S

P.

«

exact,

subject to

relation

[P.]

=

the

[Pl]

relation +

[P.]

= O if P.

[P." ] whenever there

is

is

exact,

and to the

an exact sequence of

bounded complexes

O —P.' —> P. —> p."

—> O

of projective modules with S

Definitions. (j)

:

-1

P.'

,

S

-1

P.

then

[cfi]

6 O —> P^ ^ P

1

S

-1

P."

exact.

If P^ and P^ are projective A-modules and

P, P is A-linear and such that S 10

isomorphism,

and

denotes

the class

-1

d)

;

S

-1

P. 1

in K^(A;S)

->■ S

-1

P

is

an

O of the

complex

—^ O

o

concentrated in degrees

O and 1.

If A is an n X n-matrix with entries in A such that the matrix S

-1

A

class

(with entries in K^(A;S)

in

:

-1

A)

A ->■ A

if s

e

see

(2.2)

in

A

[1]

and

"'a])

S

then

linear map A

then

[s]

S

3

of the

denotes

. . .

-1

(e K^(A;S)).

Let U(A;S)

= U(S

-1

element of U(A;S)

will be

thought of as

A generated by a unit v of S

(S“\)

det_ A

U(A)

class

in K^(A;S)

sequence

(3)

can be

-1

□ where

for any ring R

group of units.

An

a cyclic A-submodule Av of

A.

There is a commutative diagram with exact rows

LEMMA. -^

the

n-matrix with entries

A)/U(A),

the multiplicative

K^(A)

the

K is invertible then

denotes

(2.3)

denotes

A^ ^ a'^.

localization

the notation U(R)

S

[A]

[2].

= [A] - n[s]

Defvmtvon.

-1

:

If s e S and h is an n

LEMMA.

and such that 3([s

invertible,

(multiplication by s) .

Now the homomorphism described:

is

induced by the

In particular, induced by s

S

--^ U(S

-^ K^(A;S)

-a- o

-s- U(A;S)

-^ O.

det

V)

126 Proof.

The homomorphisms det

and det

are just the usual S

deterroinants

(see [1;§1])

A

and so the left rectangle is certainly

commutative. Thus to define det^ it suffices to prove that 9 is surjective, and this follows easily from the sequence

-1

phism

injective.

Here,

fact that A is local has been used: injective homomorphism p

R

:

Z ->■ K

O

(2.1),

since the homomor-

for the first time,

for any ring R there is an

(R)

given by p

R

(1)

= [r];

each projective A-module is free p^ i’s an isomorphism.

morphism K

(A)

K

(S

-1

A)

the

-1

is the composite P

P, S

»

since

The homo-

3-nd thus

A

□

injective.

-I

(2.4)

= det^[F.] ^f S

G(M)

PROPOSITION.

M =0 whenever F.

is a fin-ite free resotutvon of M,

Proof.

This is by induction on p = pd M, but first notice

that det [f.] = det [P.] if P. M:

see [2;

is another finite free resolution of

Proposition 3.5].

If p = 1 the identity follows directly from If p > 1 choose a, F^

K as in the proof of

and

(1.2)

(2.2).

and use

□

(0.1) (c) , (d) .

Remark.

(2.5)

and

(0.1)(b)

The above result gives an alternative method _i

of computation of G(M).

Namely,

possible to choose linear maps s. 1

all the maps a 4 z(A)).

since S :

F.

^ F,

1

O Z =

1

d

©

3

©

F^

©

Then G (M) 11. (det a 1

^

is exact it is for all i such that

1+1

= s, d. + d s. are injective 1” I 1 1+1 1

1

Define a matrix Z as follows.

s

F.

2i

)^ = (det

©

(namely with det a

i

This the

follows

from [2;

Theorem 2.1

and Theorem 4.4].

convention whereby the determinant of a O x (2.6)

Remark.

Let dim A =

non-zerodivisors on A. det _

:

K

(S

-1

A)

1

^ U(S

A)

is

0-matrix is

and S = A -

Then the ring S

-1

A

is

(We adopt

z(A),

1.)

the set of

semilocal and so

an isomorphism:

see

[1;

Corollary

S~ A Now let A

(2.8)]. det A e

s

[det A]

= [A] in

:

a”^ ^ A^ be an injective

(McCoy's Theorem), (S

defined by x ([?•])

=

and it follows

A).

linear map.

from the

The homomorphism

^ (-1)

(H, (P.) ) .

above

K^(A;S)

It follows

Then that

^ Z is

that

1

£(A/(det A)) This

= xO([det A]))

is however known:

proof is

see

[4;

= xO([A]))

Lemme

(Coker A).

=

(21.10.17.3),p.298]

(where the

somewhat different).

Acknowledgement The author has been supported,

in part,

by the Danish Natural

Science Research Council.

References 1.

H.

Bass,

C.B.M.S.

Introduction to some methods of algebraic K-theory, Regional Conference

(American Mathematical 2.

R.

Fossum,

H.-B.

Series

Society,

Foxby and B.

in Mathematics

Providence,

20

1974).

Iversen,"A characteristic class

in algebraic K-theory", to appear. 3.

S.

Dutta,

Thesis,

4.

A.

Grothendieck,

University of Michigan, Elements de geometric

P\iblications Mathematiques Scientifiques, 5.

M.

Hochster,

ive algebra. and E.A. pp. 6.

7.

8.

Paris,

32

algebrique,

1981.

IV,

(Institut des Hautes Etudes

1967).

"Cohen-Macaulay modules". Lecture Notes

Rutter,

Ann Arbor,

Springer,

Conference on commutat¬

in Mathematics Berlin,

311

(eds.

Heidelberg,

J.W.Brewer

New York,

1973)

120-152.

R.E.

MacRae,

"On an application of the Fitting invariants",

J.

Algebra, 2 (1965),

C.

Peskine

and L.

Acad.

Sci.

J.-P.

Serre,

Mathematics

Paris

Szpiro, Ser.

Algebre 11

153-169.

A,

"Syzygies 278

locale:

(Springer,

(1974),

et multiplicites",

multiplicites.

Berlin,

C.

R.

Lecture Notes

in

1421-1424.

Heidelberg,

New York,

1965).

Department of Mathematics, University of Oklahoma, Norman,

Oklahoma 73019,

U.S,A.

and

(from 1982)

Matematisk

Institut,

Kszibenhavns Universitet, Universitetsparken

5,

DK 2100 K?5benhavn 0,

Denmark.

129 FINITE FREE RESOLUTIONS AITO SOME BASIC CONCEPTS OF COMMUTATIVE ALGEBRA

■

D.G.NORTHCOTT

This paper deals with certain developments finite

free resolutions.

interest not only for

their own sake,

led to a reappraisal of It appears

The developments

in the theory of

in question are of

but also because they have

some basic concepts of commutative algebra.

that in the absence of Noetherian conditions,

very familiar and

fundamental

concepts

certain

let us down rather badly,

but happily there have been found ways of dealing with the resulting problems.

This

than a passing

1.

in itself,

the author,

may have more

Modules with Euler characterist-ia zero

element,

Thus

to

interest.

Suppose that R

lution,

it seems

and let of

F

finite

is

a commutative ring with a non-zero identity

denote

length,

the class of modules by means of

an R-module E belongs

to

F

that have a reso¬

free R-modules of

finite rank.

precisely when there exists an

exact sequence 0->F^F n n-1 where each F.

is

a

->...->F^-F->Ea-0, 10

(1.1)

free module with a finite base.

The aim of the

1

theory of

finite

free resolutions

their resolutions)

is

to

study such modules

(and

preferably without imposing any unnecessary

conditions on the ring R. Let E belong to resolution of E.

The

F

and

suppose

that

(1.1)

Euler oharaateristic,

is

Char

a

(E),

finite

free

of E is

K.

defined by Char

n R

where rank is

an

=

y

(-l)^rank

v=0 (F

R This

(E)

)

denotes

R

(F

), V

the number of elements

V

'Lnvar'Lant

in a base of F

. V

of E,

that is

to

say it does not depend on the

130 chosen resolution,

and it has many properties.

property that is relevant here states of course,

The particular

that Char^(E)[

> O.

plenty of modules whose Euler characteristic

There are, is

zero.

If

therefore we put F^={E 6 then

]F^

F :

Char^(E)

is a subclass of

= O},

F

which is

likely to be especially

interesting.

Noethevian

It is known that when R is ing characterizations of

F

u

.

These may be

•

there are two contrast¬ stated as

in

(A)

and

(B)

below.

For E in T' we have

(A) Ann

of

(E),

F^

if and only if the annihilator

E e

F

if and only if

is non-zero.

E,

For E in r we have

(B)

E e

Ann

(E)

contains

a non-zerodivisor. However the effect of dropping the Noetherian condition is discon¬ certing for, whereas out,

(B)

(B)

as was becomes

shown by W.V.Vasconcelos, false.

remark to become clear.

r 6

suppose that I

time for the full In order

a non-zerodivisor for

I

in rCx].

involved,

an indeterminate.

then r e

(B)

If

IR[x] and it remains

IRCx] to

to be composed of zerodivisors and yet for

We can then rescue

let

On the other hand it is perfectly

Let us agree to

non-zerodivisor whenever

say that

I

contains a

IRCx] contains a non-zerodivisor.

by changing it to read as

For E in IE we have

(B) '

significance of the last

is an ideal of R and X is

contain a non-zerodivisor.

latent

true

as Vasconcelos pointed

to explain what is

I and is a non-zerodivisor in R,

possible

remains

is not altogether false.

It has taken some

us

Nevertheless,

(A)

E e

F

0

follows.

if and only if

Ann

R

(E)

contains a latent non-zerodivisor. All

this

suggests

that we should modify our attitude to non-

zerodivisors and to concepts which involve them.

One obvious

concept which comes into this

grade.

the grade of an ideal sequences r

I

,r

z

,...,r

I

s

is in I

category is

that of

the upper bound of the such that,

zerodivisor on R/{r^,....r.

for

each i,

Classically

lengths of all r.

1

is a non-

This number will be denoted

131 t>y

order to take account of possible latent non-zero-

divisors,

we

introduce an infinite

indeterminates

,X2,X^,... of different

sequence

and put t

Gr

(The

(I)

— lim n-xo

v"!^ IRCx KLX. / > • • /X J 1 1 n

n

]).

limit exists because the right hand side increases with n.)

Let us

call

Gr

(I)

tvuc

the

grade of

^le ring R there exists an ideal that gr

(I)

= O and Gr

K

(I)

= n.

I,

I,

If n >

2,

then in a suit-

generated by n elements,

Thus

the two grades

such

can differ

K

considerably although it is not difficult to see that they always coincide when R is Noetherian. Let us module E,

return to

in

F,

(B).

belongs

In its original

to

F

when and only when gr

O

and we have already noted that this may be Noetherian.

form it asserts

However we now have

the

R

(Ann

R

that a

(E))

> O,

false when R is not

following theorem which holds

without any such condition on R.

Let

THEOREM. Gr

R

(Ann

R

(E))

E

belong to

Then

E e

seems

than classical grade,

to be

I believe,

that we

should use true grade rather

and pres\imably this

not just in the context of

the theory of

should apply generally and finite free resolutions.

M.Hochster who first questioned the appropriate¬

ness of the traditional definition of grade.) show presently, for

the consequences

the moment let us Denote by P

finite Thus

length,

where

11^

is

a

P

O

n-1

projective

10

^ E

F

Gr

R

(Ann

of

modules.

R

(E))

The

class

P

is

and our previous discussion now

to identify the subclass

P :

but

o,

finitely generated projective module.

= {E e

try to

there must exist an exact sequence

P^

this by putting P

shall

subject of resolutions.

finitely generated

somewhat larger than the class enables us

I

the class of modules which have resolutions,

by means of

-> n

n

As

are unexpectedly far-reaching,

stay with the

for E to belong to

o ^ IT

if and only if

F

> O.

The moral

(It was,

F.

> O}.

that generalizes

F^.

We do

132 So far we have been operating get the feel of what is happening case where R = Z.

For this

in something of a vacuum.

let us

/-modules,

see that P

is

the class of

F

For E in

the finite abelian group. O -> e'->- E ->■ e"

P^

Then 0(E)

This

suggests

and

denote the

multiplicative,

order

of

that is

if

then

that for a general R there may

be a similar multiplicative invariant associated with when R is Noetherian,

= P^ .

is "'the subclass of

O is an exact sequence in P^,

= 0(e')0(e").

P

finitely generated

let 0(E) is

=

finitely generated

with this interpretation P^

finite abelian groups.

0(E)

and so

that is to say it is the class of

abelian groups,

look at the

situation there is no distinction

between free and projective modules, It is easy to

take a quick

To

R.E.MacRae has

identified this

P^.

Indeed,

invariant.

But when we try to drop the Noetherian condition we immediately encounter the kind of difficulty I have been discussing.

In this

instance we are not yet equipped to deal with the problems arise,

so

that

let roe prepare the way by following up a previously

mentioned clue.

2. The theory of attached prime ideals It was

stated earlier that the principle of replacing classical

grade by true grade has now be

some far-reaching consequences.

illustrated by means of the

As

is well known,

is

said to be

set of such prime ideals There example,

is

Ass

for some (III) (E

S

is denoted by Ass

)

statements

is

R R

a prime

(Re)

P e Ass

for some e

:

then P

e E.

(E). concept.

E =

then

For the

o.

zerodivisor on

E

if and only if

(E).

If s is a multiplicatively closed subset of

= {PR„ S

ideals.

all hold.

r e r ts a R

ideal,

finitely generated,

is empty if and only

(E)

An element

(II)

Ass

if P = Ann

is

an extensive theory surrounding this

following three

r G P

with E

if R is Noetherian and E

(I)

theory of associated prime

if E is an R-module and P

associated

This will

P e Ass

R

(E)

and

P

n s

=

0}.

R,

then

The

133 However when the

finiteness

three

can be

statements

conditions on R and E are removed all

false.

Since the

ularly inconvenient,

an example

may be of interest.

Suppose then that

minates,

let R = Q[X^ ,X2 ,X^,...]

(X^,X2,X^,. . .) . check

that Ass

Then R/I (R/I)

is

is

First we note

that if gr

I on a module E,

can define the

(I)

is partic¬

show how easily it can happen are

and let I be

indeter-

the R-ideal

a non-zero R-module and it is

easy to

empty.

It is here that our

ideal

to

failure of

earlier remarks about grade can help. (I;E)

then,

denotes

the classical grade of an

using the

true grade Gr

(I;E)

of

same device as before, I on E.

This

said,

we

assume

R

for the moment that R is Noetherian and E Then a prime R-ideal Rp-ideal PR^ is

P

gr^ R

p

is

a zerodivisor on E^.

(PR^;E^) P P

Consequently

(E)

=

now clear.

{P G Spec(R):Gr

For arbitrary R and E we now put (PR

ideals

P

-E

)

in Att

= O},

(E)

are

R

this definition we have Ass Strict.

Every

assertions we use

R

(E)

C Att

R

(E)

attached

and the

finitely generated attached prime

be an associated prime Noetherian.

if and only if

= O.

^ say that the prime

and we

and this happens

the

when and only when

The way ahead is Att

finitely generated.

associated with E if and only if

associated with E^,

every element of PR^ P G Ass^(E)

is

is

ideal,and

so Ass

R

(E)

R

(E)

(II)

and

(III)

turns out to whenever R is

But we have gained considerably because, (I),

With

inclusion may be ideal

= Att

to E.

for

example,

are now true quite generally provided

attached rather than associated prime

ideals.

The theory of attached prime ideals has been well developed largely through the efforts of P.Dutton. of associated prime seems

ideals

to provide all

Almost all

the properties

generalize and the more general

theory

that one could reasonably expect.

3. QfAas'i-dnvert'ib'le ideals There

is

one more concept that needs

can continue with

the

an invertible ideal.

to be modified before we

theory of resolutions. Let

E be

the

This

full ring of

is

the notion of

fractions of R so

Y.

that a typical member of

has as numerator an element of R and as

demoninator a non-zerodivisor of R. of E,

If M and N are R-submodules

then we can form their product MN just as we

of two ideals. with R as its

The submodules

then form a commutative semi-group

neutral element.

The

ideals of R are members of this

semi-group

(because R is a subring of

invertible

if it has a

But what has suppose that I

form the product

E)

and an ideal

is called

semi-group inverse-

this

to do with grade?

is an ideal of R.

To answer this question

Then there

is a result which

•*

states

that for

I

to be invertible

that

I be projective

Thus

I

is

(as

it is

a module)

necessary and sufficient

and contain a non-zerodivisor.

invertible if and only if

I

is projective and gr

(I)

> O.

K,

This prompts

the

following.

Definition.

The ideal

is projective and Gr

R

(I)

Quasi-invertible invertible Again if

ideals.

I,J are

is

ideals

ideals,

said to be

quasi-invertible

if

I

>0. enjoy many of the properties of

For example,

I = JA for a unique of two ideals

I

they are always

finitely generated

I C J and J is quasi-invertible,

ideal A.

then

Finally we mention that the product

is quasi-invertible when and only when both the

factors are quasi-invertible.

4. The MacRae invariant We are now ready to resume the discussion of free and projec¬ tive resolutions.

Let E e

Then,

finitely presented and therefore G

^

F -> E

represented by a p x q matrix A,

initial

R

(E)

easily seen,

let us

is

say ranks p and q respect¬

for each of F and G.

of E.

(E)

This

is

is closely allied to Ann

is

that,

(E) R

they coincide when E is

fact here

is

the well-known

R and in particular

Then f

and the p x p minors of A generate

that depends only on E.

Fitting invariant

The vital

E

there exists an exact sequence

free modules of

Suppose we choose a base

an ideal

is

O,

where F and G are ively.

as

oyclia.

because E is

shown that there is a smallest quasi-invertible

in

P^,

it can be

ideal containing

135 .

This quasi-invertible ideal will be denoted by

It

is essentially MacRae's generalization of the order of a finite group, but now we have removed the restriction that R has to be f

Noetherian.

Among the properties of the new invariant we select

two for mention because they are striking in themselves and partic¬ ularly relevant here.

The properties in question are enshrined in

the next two theorems. THEOREM. If o ^ E'-> E

then

R

(E)

e"-»-o is an exact sequence tn

=5^ (e')?^ (e"). R R

,

rf'

THEOREM. If E heZongs to F

, then ^ (E) is a principal ideal

generated by a non-zerodivisor. For applications of MacRae's invariant it is often important to know when

R

the whole ring.

(E)

is a proper ideal,

that is to say different from

This happens,

equivalently when Att

R

(R/^ R

of course, when R/S^ (E) O or R (E)) is not empty. Now it is known that

Att

(R/^(E)) = {P 6 Att (E):Gr„ (PR„) R R Rp P suspect that this can be simplified to Att

R

{R/??(E))

= {P e Att

R

(E) :Gr

R

(P)

= 1} and indeed I

= 1},

although I have not been able to find a proof.

In fact this seems

to be connected with an open problem concerned with the phenomenon of grade stability,

and,

as the

problem may be unfamiliar,

it

perhaps merits a digression. Suppose that P is a prime ideal.

It is easy to see that

Gr

(P) < Gr^ (PRt,) • R Rp R Should it happen that Gr^(P)

= Gr^

(PRp),

P then P is said to be grade stable.

Problem. in Att^(E)

is it true that whenever E e P

all the prime ideals

are grade stable?

It seems very likely that the answer to the question posed here is

'Yes'.

For example,

rated member of Att observed by MacRae, Noetherian.

R

(E)

if E e P

then every finitely gene¬

is grade stable.

Consequently,

as was

the answer is affirmative whenever R is

Of course, because we are striving for full generality,

this doesn't help here.

However there is one additional piece of

136 evidence.

It can be shown that if E 6 P

one prime ideal in Att

(E)

and E ^ O,

is grade stable.

then at least

Fortunately this is

sufficient for the applications described below, but an affirmative answer,

if correct, would add a finishing touch to what is already

an elegant theory.

5. Applications We conclude by indicating what can be achieved by way of applications.

First we have the fol^Lowing.

An ideal I, of R, can he generated by a non-zero-

THEOREM.

divisor if and only if (i)

I e F ,

and

(P) = 1 for all P in Att (R/I). R R For Noetherian domains this too goes back to MacRae. (ii)

Gr

Note

that in the form just stated, where there are no extra conditions on R at all,

there is a satisfying economy in the hypotheses.

It

is also worthwhile noting in passing that the theorem serves to pinpoint the origin of the connection between unique factorization and finite homological dimension. As the proof illustrates the interconnections between the various topics described above it will be given in outline.

How¬

ever the interesting part of the demonstration consists in showing that conditions zerodivisor,

(i)

and

(ii)

imply that I is generated by a non-

and so only this aspect will be considered.

Condition deduced that Gr

(i) (Ann

ensures that R/I e F (R/l))

and from

is greater than zero.

(ii)

it is easily

Consequently

R/I e F

. Thus (R/I) is defined and it is moreover an ideal O R generated by a non-zerodivisor. We also have I = Ann

K

(R/I)

whence, because

R

= .?^„(R/I) K

(R/I)

C

—

R

,

is a principal ideal,

I = f^^(R/I)Ann^(?^^(R/I)/I) . Put E = 5^ (R/I)/I. then we shall have I = ^

(5.1)

It will suffice to show that E = 0,

(R/l)

for

and it has already been noted that

the latter is generated by a non-zerodivisor.

In any event

137 t and hence E as well, is cyclic and therefore = Ann^(E) Thus

(5.1)

= Ann^

(R/l)/l) .

can be rewritten as

I = .^^(E)^^(R/I) .

'

(5.2)

We also have ^ (R/I) e F because ^ (R/I) is R R a free module of rank one. Consequently E = (R/I)/I belongs to F; R indeed E 6 F because Ann (E) contains I and so is not zero. Thus D R By

^ (E) R

(i),

I e F.

is defined and now I C ^ (E)?^ (R/I) K R

(5.2)

yields

C ^^(R/I) R

(E) c ^ i'E) . But, by definition, ^ (R/l) R R R quasi-invertible ideal containing (R/1) = I, and because

is the smallest (E)(R/I) ,

because it is the product of two quasi-invertible ideals,

(E)!^ (R/I) = ^ (R/I) and thereR R R = R by another of the properties of quasi-invertible

quasi-invertible. fore

(E)

is itself

It follows that

ideals.

We have now shown that ^ (E) is an improper ideal and this, R as we saw earlier, means that theve aan be no prime ideal P in Att

R

(E)

for which Gr.^ (PR ) = 1.

■'

R P p On the other hand E = ^

R

(R/I)/I is a submodule of R/l from

which it follows that Att

(E) C Att (R/I). R R Accordingly, by condition (ii), Now we know that if E / O, Att

R

(E)

(P) = 1 for every P in Att (E). R R then there will be at least one P in

which is grade stable,

Gr

and for such a P we would have

Gr

(PR ) = Gr (P) = 1. p But we have just established that this situation cannot arise. Consequently E = O,

as we wished to prove. □

Let us return to the statement of the last theorem.

An ideal

which can be generated by a non-zerodivisor is the same as a non¬ zero free ideal.

It is therefore to be expected that there will be

a related result concerning projective ideals. better to state the companion theorem

In fact it is

in terms of Picard modules.

An R-module M will be called a Picard module if there exists a second module N such that M

is isomorphic to R.

denotes the isomorphism class of M,

If [m]

then the isomorphism classes

138 of Picard modules form the Picard group, Pic(R),

of R when compos¬

ition is defined by [M^] + [M^] = [M^ The Picard modules are just the rank one projective modules, but the definition just given is more succinct. We now have sufficient terminology to state a final result. THEOREM. Let K be a submodule of a Picard'ynodule M.

Then K is

also a Picard module if and only if (i)

K e P , and «

(ii)

Gr

(P)

= 1 for all P e Att

(M/K). R R To obtain from this a result concerning projective ideals all that is necessary is to replace the Picard module M by the ring R.

References 1. P.Dutton, "Prime ideals attached to a module". Oxford (2), 29 (1978), 403-413.

Quart.J.Math.

2. R.E.MacRae, "On the homological dimension of certain ideals", Proc.Amer.Math.Soc., 14 (1963), 746-750. 3. R.E.MacRae, "On an application of the Fitting invariants", J.Algebra, 2 (1965), 153-169. 4. D.G.Northcott, Finite free resolutions, Cambridge Tracts in Mathematics 71

(Can±iridge University Press, Cambridge,

5. D.G.Northcott, "Projective ideals and MacRae's invariant", J.London Math.Soc. (2), 24 (1981), 211-226. Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, U.K.

1976).

PART III

MULTIPLICITY THEORY, HILBERT AND POINCARE SERIES, ASSOCIATED GRADED RINGS, AND RELATED TOPICS

140 BLOWING-UP OF BUCHSBAUM RINGS

SHIRO GOTO

1. Introduction The purpose of this paper is to give a characterization of «

Buchsbaum rings in terms of blowing-up. Let A be a Noetherian local ring with maximal ideal m and Then A is called Buchsbaum if the difference

dim A = d. 1(A)

= «-^(A/q)

- e

(A)

is an invariant which does not depend on the choice of the parameter ideal q of A.

(Here Z

(A/q) and e (A) denote, respectively, the A q^ length of A/q and the multiplicity of q.) This is equivalent to

the condition that every system a^,a2,...,a^ of parameters for A is a weak sequence, (a

1

that is the equality

,...,a,) r

:

a

1+1

holds for every O < i < d

=

(a ,...,a.) 1 1 (c.f.

[21;

: m —

Satz

10]).

Macaulay ring A is a Buchsbaum ring with 1(A)

Thus a Cohen-

= O and vice versa.

In this sense the concept of Buchsbaum ring is an extension of that of Cohen-Macaulay ring and the theory of Buchsbaum rings has started from an answer of Vogel [24] to a problem of Buchsbaum [3; p. Let q = R(q)

=

(a^,a^,..•,a^)

© q’^. n^O

228].

be a parameter ideal of A and put

Then the canonical morphism f :

Proj R(q)

-+ Spec A

is said to be the btowing-up of Spec A with centre Spec A/q.

Recall

that d Proj R(q)

= ^U^Spec A[x/a^

1

^ ^

and that the fibre of f over Spec A/q is given by Proj G(q) G(q)

=

® n>Cr-

—

Let 8^(.)

denote the i-th local cohomology functor.

where

With this notation the main result of this paper is stated as follows. THEOREM

Suppose that dim A > O.

(1.1).

Then the following

$

conditions are equivalent: is a Buchsbaum ring-,

(1)

A/H°(A)

(2)

Proj R(q)

is a locally Cohen-Macaulay scheme for every

parameter ideal q of h. The theory of Buchsbaum singularities is now developing very rapidly

(see [5,6,7,8,9,10,17,18,22])

enjoy pretty good properties. baum.

For example suppose that A is Buchs¬

Then for every prime ideal p of A such that p

ring A

is a Cohen-Macaulay ring with dim A

E

i

over the local cohomology modules H that is m.H

(A)

= O,

d-1 1(A)

=

i

m

(A)

[23])

Satz 2] where,

m

= d - dim A/p. More-

£ (A)

(i / d)

are vector spaces,

.h^(A)

for each i, h^(A)

as a vector space over A/m. —

denotes the dimension of

It was shown in [17]

(see also

that Buchsbaum rings may be characterized in terms of Koszul

homology relative to systems of parameters. [22]

m the local

and we have

d-1

I

i=0 [16;

and it is known that they

and [29]

criterion,

a very powerful criterion,

There was given in

the so-called surjectivity

for Buchsbaum rings in terms of local cohomology,

subsequently, using this,

a lot of examples of non-Cohen-Macaulay

Buchsbaum and normal rings were discovered [5] pointed out in [6]

.

It was also

and [10] that certain Buchsbaum rings are char¬

acterized by the behaviour of the Rees algebras R(q) ideals q of them.

and

of parameter

Nevertheless in spite of the importance of the

theory of Buchsbaum rings there has been established no definitive characterization which really clarifies them as singularities. From this point of view our theorem As a consequence of COROLLARY

(1.2).

(1.1)

(1.1)

one has the following.

Suppose that dim A > 0.

conditions are equivalent-. (1)

A/H^(A) m

may have some interest.

is a Gorenstein ring-,

Then the following

142 Proj R(q) is looally Gorenstein- for every parameter ideal

(2) q of A.

We shall prove

(1.2)

in Section 4.

A similar characterization

of complete intersections may be found also in that section. (Further applications of paper.)

Theorem

(1.1)

(1.1)

will be discussed in a subsequent

itself will be proved in Section 3.

Section

2 is devoted to some remarks on rings with finite local cohomology which we shall often need in the proof of Finally the author wishes

to

(1.1).

thank Y.Shimoda for helpful «

discussion during this research.

Lemma

(3.4)

was suggested to the

author by him. Throughout this paper A always ring with maximal

denotes

ideal m and dim A = d.

a Noetherian

Also

— the i-th local cohomology functor.

m

(.)

will

local stand for

2. Preliminaries We

say that A has

logy modules h^(A) £

A

(H^ (A) ) m

are

are

finite)

finite

cohomology if the

finitely generated for all

First of all we note

PROPOSITION

local

(2.1)

i

the

[19,

(that is

the

local

cohomo¬

lengths

d. following.

22]. The following conditions are

equivalent: has finite local cohomology;

(1)

A

(2)

there exists an ^primary ideal i of A such that for

every system

of parameters contained in

every integer

o < i < d we

(a^,...,a^)

:

a.^^

=

(a^.a.)

I.H^(a)

=

:

I.

(O)

is a Cohen-Macaulay local ring with dim A

for every i. ^ d and A = d - dim A/p for every

IP p of a

Proof. Let J be J =

such that

See [19].

and for

have

When this is the case,

prime ideal

l

p / m.

See also [22;

Lemma 3]. □

an ideal of A and let

D J(p) peAss A/J

denote a primary decomposition of J in A.

We put

143 Assh A/J = {p 6 Ass A/J

I

dim A/p = dim A/J}

and

As every element of Assh A/J is a minimal prime divisor of J this definition of U(J) position of J.

does not depend on the choice of primary decom¬

Notice that Ass A/U(J)

= Assh A/J and dim A/U(J)

=

dim A/J. For the rest of this section we assume that our ring A has finite local cohomology and we let I be an ideal of A obtained by

(2.1) .

Let a^/a^/.-.ja^ be a fixed system of parameters for A We put

contained in I. LEMMA

(2.2).

Proof. H

O

(A)

=

(O)

=

H°(A) m

= U(0)

:

(2.1)),

depth A/U(0) H^(A) m

=

As I is m-primary,

~

I since I.H

O

mm

(c.f.

(a^,...,a^)

U(0)

(A)

(O)

:

H^(A)

= O.

—

(O < i < d). I =

(O)

3 (O)

:

:

a,. 1

I and so we have that

Because Assh A = Ass A \

has finite length, whence H

m

(A)

D U(0).

> O we get the opposite inclusion H*^(A) m

= U(0).

The equality

(O)

:

a^ = 1

(O)

:

{m} —

Since

c U(0).

Thus

I follows from the

□

choice of I.

COROLLARY

U(qi) = q^

(2.3).

^ =^i

a^_^^

for every

□

O < i < d.

THEOREM

integers. .

(2.4)

(c.f.

[10;

Let 0■ O

and split it into the two short exact sequences O

(O)

O Apply the

aA

:

a

A

i

O,

aA

(a)

A/aA ->■ O.

functors H

(.)

to

(a)

(b)

and obtain isomorphisms

5. H^(A) m

~ H^(aA) ~ m

(i >

1)

and a short exact sequence O -»■

(O)

:

a

because the length of exact sequence

> H (O)

O, m

^

(A)

f

—>■ H

O,(aA), m

a is finite.

O

(c)

Similarly we get a long

o

—>-H°(aA) -^H°(A) -»^H°(A/aA) in mm 1 -(aA) m

i

»-H

1 m

(A)

»-H

1 m

(A/aA) -.

(d)

«

-►H^(aA) ^^H^(A) m 51

>-H^(A/aA) H.

.

of local cohomology modules which comes from the sequence

(b).

Therefore,

replacing H^(aA) in the sequence (d) by H^(A) for i > 1 m m and combining the sequence (c) with the resulting one, we obtain

the required exact sequence O-^ (O)

:

a-^h‘^(A) m ->-H

*1 m

-^H^(A/aA) m

m *1

(A)

m

*1 (A) -(A/aA)m

->-H^(A) -^H^(A) -^H^(A/aA) -. m m m (Note that the triangle H^(aA) --->-H^(A) m m

m is commutative for every i.)

The last assertion

(3)

follows from

this sequence. □ COROLLARY

Let J = (O) (a)

:

Suppose that d = 2 and that depth A > o.

(2.7) .

h”* (a) m b = (a) :

.

Then :

J

for every system a,b of parameters contained in J. take J to be an ideal l obtained from Proof.

By the sequence of

(Hence one may

(2.1)(2).)

(2.6) (2)

we have that H*^(A/aA)

T

H

m

=

—

On the other hand because I

(A).

(2.6)(1) that is

we get that

(a)

trivial.

>

:

b/(a)

: b is contained in

((a)

:

b/(a))

C H^(A/aA);

(a)

m : J.

is finite by

hence J.[(a) :

b] C

(a)

The opposite inclusion is

□ (2.8).

COROLLARY

that d

(a)

A

3.

Let & be a regular element of

Then A is a Cohen-Macaulay ring if

H^(A/aA)

=

(O)

A

and assume

146 for every

1

< i < d - 2.

Proof.

Considering the exact sequence in

that the homomorphism

(A)

(A)

IE'

(2,.6) (2)

we find

is onto and that a is

a non-

E.

zerodivisor on

(A) for all 2 ^ i ^ d - 1. Because H (A) has m m finite length for i 7^ d by our standard assumption, these facts

yield that ring.

m

(A)

=

(O)

for all i 7^ d-

Thus A is a Cohen-Macaulay

□

3. Proof of Theorem

(1.1)

In this section assume that d = dim A > O and let q =

(a^,a^,•-.,be a parameter ideal of A.

We shall maintain

the following notation: R =

© q^, naO—

the Rees

algebra of q; —

I the associated graded ring of q;

G = M = mR + R^,

the unique graded maximal

N = mG + G^,

the unique graded maximal ideal of G.

Also, we shall denote by H^(R) M

cohomology modules of R

ideal of R;

(respectively H^(G)) N

(respectively G)

the local

relative to M

(respect¬

ively N) . We note the following. LEMMA

(3.1).

ring S =

Let P be a prime ideal of a Noetherian graded

tained in P.

Then P* is again a prime ideal of S and

Macaulay (respectively Gorenstein) Proof.

See [11;

PROPOSITION

Then the length for all i 7^ d. Proof,

G

(1.1.3) ].

is a Cohen-

local ring if and only if

is.

□

(3.2).

Suppose that Proj G is Cohen-Macaulay,

(H^(G))

of the local cohomology module is finite

N

Moreover A has finite local cohomology. First of all notice that the local ring Gp is Cohen-

Macaulay for every prime ideal P of G such that P 7^ N. follows, by Macaulay.)

s con¬

end let p* denote the largest graded ideal of

(3.1),

(This

from our assumption that Proj G is Cohen-

Then we get that the

length

£

G

(H^(G)) N

is

finite

i 7^ d because mG is a unique minimal prime ideal of G and dim G/mG =d

(c.f.

[19;

(2.5)

and

(3.8)];

recall that

for

147

O

f

< “

(t^)

)

denotes the multiplicity of the ideal

f '^)

d ' • • • »f j ) d

iri G.

On the other hand we have that n

G

n

.f, ) d

1

^ £

A

(A/(a

n a/)) d

1

and that

n, (f ^

(G)

=

n n. .e (G) i=l 1 (f^,...,f^)

=

n n. . e (A) i=1 1 q

f = e

(A)

n .^d

for all integers n^jn^/.-./n^ > O.

Hence by the inequality

(/)

see that

7

sup n ,...,n > O 1 d which yields,

.^

again by

(3.3)

^

of [19],


O. and X

(2.5) e

(a

1 X

2

Because

((O)

a^) n (a^,.

we actually get the equation

,...,a

.) k+1

:

(^)

Jc+I

in A.

)

=

(O)

by

(2.2)

Therefore

a? and hence we may write 1

^2^2 \+1^k+1 n-1 with y^ G q , because ( (^2' ' ' ' ' \+1 ^ by

a

1

(2.4).

n-1

9. n a^)n q = (a^,-

Now consider both the equations

{^)

and

Then we

(7^^)

find that ■ ‘Fk+i’

2

^

k

which allows us to write ^ , , with

e q

n-1

w

^ ^2^2 ,, since

.,a^)

n q

k+r

n-1

(a.

■Va

Thus

in B, whence f e

(a

,a /a

,...,a /a )B.

1^1

iC

baum ring.

+.

n-1

••+ V^rVh Therefore

I

^ B-regular sequence. □

^1'^2^^1'’’‘'^d^^l PROPOSITION

n-1

+

' = ^-^'k+Z^

(3.5) . Let k = A/m and suppose that A is a Buahs-

Let Tc/k he an extension of fields.

Then there exists a

Buchsbaum local A-algebra A with maximal ideal m such that A-flat,

(b)

Proof.

m = mA and

Passing to the completion of A we may assume that A is

Noetherian local R-algebra S such that R ively S)) exist:

'K is

(c) )? = A/m as y.-algebras.

a homomorphic image of a regular local ring R,

(here m

(a)

(a)

say A = R/I.

S is R-flat,

(b)

Choose a m

= m S S R (respect-

(respectively m ) denotes the maximal ideal of R S and (c) k = S/m^ as k-algebras. (Such an R-algebra S must

see,

for example,

[12;

Chapter O,

(10.3.1)].)

Then because A is Buchsbaum we see by Satz 1 of [20] canonical homomorphisms Ext^(R/m^,A)

->

(A)

Let A = S ® that the

R

A.

149 are surjective for all i ^ d; Ext^(S/m ,A) t —S

hence so also are the homomorphisms

= S ® Ext^(R/m ,A) —(A) R R —R

= S ® R

for all i 5^ d.

(A)

m

—S

—R

Therefore we get by Theorem 1 of [22] that A is a

Buchsbaum ring and it is clear that A satisfies all the require¬ ments

(a),

(b)

and

□

(c).

Proof of Theorem (1.1)

((1)

=> (2)).

may assume that A is a Buchsbaum ring.

Passing to A/H*^(A)

ra

we

Let k denote the algebraic

closure of k = A/m and choose a Buchsbaum overring A of A so that A satisfies the conditions in the induced

(3.5).

Let R = A ®

morphism Proj R —^Proj R is flat,

R. Then because A to prove that

Proj R is Cohen-Macaulay it is enough to show that Proj R is CohenMacaulay.

Therefore we may assume that the residue class field k

of A is algebraically closed.

As

d

[J

Proj R =

i=1

Spec A[x/a. ^

it suffices

to prove

for every


■ O

(O < i < d - 2) ;

imply that (H^(A/U(a A) ) ) Am d

for every

1

= H

This

(A).

^ i

< d -

£^(H^(A)) Am

2.

completes

It is

+ £

(H^^hA))

Am

clear that

O U(a A)/a,A = H (A/a A) d d m d

the proof of all the

assertions

in Claim 2.D

155 CLAIM 3.

Proof. q'n U(a^A) X G

Let q'= —

C a^A.

2

(a^.../a ) 2 d-1

ring by Claim with y G

(a

= a^A.

,...,a

d

).

It suffices

Let x e q'n U(a^A).

whence x g

clearly,

we may,

q n U(a^A)

1.)

Then

n U(aA) 1

(5^)

by

_

(2.4).

(Recall

Thus x G a A + U(a^A). Id

a A and z I

G U(a A). d

by Claim 2,

to show that

Then

as

that A is a Buchsbaum

Let us write x = y + z

z = x - y G U(a^A) D U(a,A) Id

a z = a^u and a^z = a^v with u,v g a. Id d 1

write

Thus

^

= “I"' whence u G

U(a^A)

C

(a^)

(a^)

:

Therefore = y +

X

:

a^.

Recalling that

m by Claim 2 we

= a^w and so

z is

(a^)

=

c U(a2A)

find that a^u = a^w

z = a^w as

a^

for some w G A.

is A-regular.

in a^A and hence we get that q'n U(a^A)

and that

Thus

c a^A as

□

required.

Now let us First of all

finish

the proof of the

implication

(2)

=> (1).

consider the exact sequence

O->- U (a^A) /a^A-A/q -*- A/q->- O

which follows note

the

from the

symmetry between a^

£^(A/q)

= £-(A/q)

1 as U(a^A)/a^A = H (A) d d m know

from Claim X.-(A/q) I(A)

e

= e_(A), q

(A)

“

denotes

«-,(A/q) A —

we -

e

by Claim 2.

we

does not depend on

the

On the other hand we already

a Buchsbaum ring.

Hence

Recalling that

that the difference

= 1(A)

+

(h\a)) Am

choice of q because

d-2, (A) i=1 by Satz

2 of

(H^(A)) Am :i

< i


O and let

(^1'^2'*’*'^d^

a parameter ideal of A.

Section 3

We shall preserve

the notation of

and identify the Rees algebra R = R(q)

algebra A[a^X,a2X,...,a^x]

with the A-sub-

of aCx] where X denotes

an indeterminate

over A. First of all We

let us recall

shall give a proof since we

PROPOSITION

the

following

shall use

(Icnown)

fact

[15].

this proof once more.

Suppose that A -is a Cohen-Macaulay ring.

(4.1).

Then A is a Gorenstein ring if and only if the ring B = A[x/a^

I

X

6 q]

is Gorenstein. Proof. let f 1


- O

far we have

Let

THEOREM. j*

:

exact.

now ready

nology we used so

separated,

induced filtration;

> gr (N) ■

gr (M)

and let

f

:

convenient characterization of

We are

filtered modules

n F .N

Provided the

if the

I.

if

= f (M)

:

for

generators of

that a homomorphism of

strict

called

and not on the presentation of

that

strict.

corresponding to

(^,g;P) j*

The

~

"1

(^;P)

j* ffi id^^•

j*

same argument applies

To

see

Let j^/j^

i*

and the composed sequence

© F where F is

Thus

I.

does

is

a

free B-module,

strict if and only if

for

j*,

and the

j*

assertion

follows,

Remark f

,...,f I

is

.

2.

Suppose

It is

that

immediate

I

is generated by homogeneous

that j* with respect to this

forms

sequence

JC

a strict homomorphism.

Furthermore

it

is clear that in this

case gr ThB/k,B) Thus

if

I

is

~Thgr(B)/k,

gr(B)).

generated by homogeneous

forms,

our

theorem asserts

1 that B denotes is

is

strict if T

(gr(B)/k ,gr (B))^ = O

the v-th homogeneous component of

exactly

the result in

[3].

for v


■ gr(B)).

leading forms

Let

P^ -^P-»-B ->0 be

the

corresponding presentation of B,

filtrations.

Then,

by

the

choice

of "the

equipped with f^,

the natural

the associated complex

gr (p) ^gr (p) ->-gr(B) ->-0 is exact. In a later step of find

the proof we

,...,e J such that for i = (a)

f.

= F.

1

1

deg F,

(b)

= deg

show

that we

can

A corresponding to

the

1,...,£ we have

f

. 1

£ 0 Let Q —yQ -—>-A->-0 be

the presentation of

F^,...,F^ of J

commutative

to

mod I,

1

generators

are going

satisfying

(a)

and

(b).

We

then obtain a

diagram

yQ-^A-

i yp of

i

(2)

—^B —

filtered modules,

reduction modulo I. gr(Q)

i

£

i

the

induces

the

correspond to

commutative

diagram

i

(3)

We

claim that

sequence

gr(Q) ^ exact,

(2)

arrows

->-gr(P) ->-gr{B) ->0

filtered modules. (i)

is

Now

the vertical

-^gr(Q) -^gr(A) ->-0

J

gr(P) of

where

Ugr(Q) -*-gr(A) —O

and

(ii)

the

induced homomorphism ijj

:

Ker gr() —>-Ker gr()

is

surjective. If ;t* (i)

and

(ii)

denotes we have

the

sequence of

leading forms

of

t_,

then by

167 O = Coker This means

~

that ^ is

Thus

satisfying

(i) (a)

To prove

and

1.

of

sequence.

nomial

theorem will be

and once we have

we assume

The general

G,

case

is

As before G*

and G denotes

that x = A*.T*,

its

G Q with deg B

>

that B

G

are

satisfying and C = B.

Since we (a) It

(AT^-B) +C G J,

and

follows and

induction on the

denotes

leading

the

a- gr(A)).

Im gr($).

can

that J is

find C G J

that C-B = D.T^,

(A-D)* = A*. G J.

form of

G J.

a poly¬

is

It

assertion

can

follows

generated by

such that deg C = deg B

and hence

(A-D)T^

Thus we may assume

Since T^

the

length

Hence we

regular modulo J,

follows by

=

from the it follows

Hence we have x = A*.T* with A*GKer(gr(R)

Since deg A*

that the

In

(O)

that v{x^)

that

(x^,...,x

1

infinite). =...= v(x^)

Then and,

x /x ,.,.,x ./x in K are Id d-1 d V

(3

)

is

if secondly,

alge-

Then

:

coefficients d{q,v)

are uniquely determined by the

equation d{q,x)

'l

=

d{q,v)v{x) .

V

Now we come to

the relation with Teissier's paper.

suppose that q^,...,q^ are d m-primary

ideals,

Then e(q^ —1

function in the

This

...

follows

q^) '-M

is

a

"multilinear"

where d = dimQ.

= ^(a, I ■ • • laj ■■■ laa’ +

sense

d(q,x)

1

=

the

that

... k: j... Ig^^).

from the result quoted from Lech's paper

Next the uniqueness of

First

coefficients d(q,v)

earlier.

in the degree

formula

d(q^,v)v(x)

leads very easily to a proof that d(q,L..q ®,v)

—1

is a homogeneous

—s

polynomial K1

IkI=d-1

1

:...k

s

’,v) n. I

:

and we may similarly define a

Kc

. . .n

"multilinear"

s

function

d (q

Now Teissier the

and Risler proved a result which we

can write

in

form e (q

d(q

1%^

provided x is

a

d expressions

for e(q

-1

1%) =

e (3_,

1

where v(q.)

sufficiently general |...|q

and the numbers

d(q

this

d(q.

xeqj

a number of relations

and with

):

1.d.

are non-negative~integers. yields

I will

and this yields

I

= min v(x),

—1

element of q^,

finish.

The for

.

q, lq,i-1'^i+1

fact that this holds the numbers

for each i

178 References [C]

C.Chevalley,

44 [l]

[Sa]

[Se]

(1943),

C.Lech,

"On the

theory of

local rings",

Ann.of Math.,

690-708.

"On the associativity formula

(1956),

for multiplicities".

Ark.Mat.,

3

301-314.

P.Samuel,

"La notion de multiplicite en algebre et en

geometrie algebrique", J.Math.Pures Appl.,

30

J.-P.Serre,

Cours au College de

Algibre

locale:

France,

1957-1958, Lecture

Berlin,

1965).

multiplicit^s, Notes

(1951),

in Mathematics

11

159-274.

(Springer,

«

[t^]

B.Teissier,

"Cycles

^vanescents,

conditions de Whitney",

[t^]

Astirisque,

7-8

B.Teissier,

"Sur une

sections planes,

Singularites a Cargese

et

1972,

(1973). in^galit^ a

la Minkowski pour

les

multiplicit^s", (Appendix to a paper by D.Eisenbud and H.I. Levine), Ann.of Math., 106 (1977), 38-44. Department of Mathematics, University of Exeter, North Park Road,

Exeter EX4 4QE, U.K.

179 FINITENESS CONDITIONS

IN COMMUTATIVE ALGEBRA AND

SOLUTION OF A PROBLEM OF VASCONCELOS

JAN-ERIK ROOS

Introduction It is algebra is

fair to a

say that a major part of current commutative

theory about noetherian

rings.

However,

tries hard to restrict oneself to noetherian rings, inevitably a simple

led to

example

Let R be field of Akizuki

study non-noetherian (it will be

a

commutative

dealt with in more

even if one

one

is

rings.

detail

Here

in

§6) .

commutative noetherian domain and let K be

fractions.

It is well known

and Cohen)]

and [10])

that the

(cf.

[13;

is

Theorem 93

following assertions

its (Krull, are

equivalent: (i) R c S

every ring S between R and K, c

K,

is noetherian; (ii) Thus

the Krull dimension of R is

if dim R >

2,

there is

always




[17;

R

be the

Which finiteness

Appendix,

Example

5]

3.)

generalization of noetherian

rings are

coherent

rings.

Definition (left)

coherent

1.

if each

finitely presented generators

and a

A

(not necessarily commutative) finitely generated

(that is

left

ideal

can be presented by a

finite number of relations).

ring S of S

is

called

is

finite number of

180 Thus

if a is an

ideal

as in Definition

1,

then we have an ■>

exact sequence of left S-modules ^ where

the

are

Here are (a)

ring in any number of variables

over a noetherian

where also

coherent is mentioned (c)

Here

is

the

interesting

coherent.

G is polycyclic This

OT

a non-commutative example.

an

is proved in

[6],

not enough to describe

those

must satisfy. there

we

Papick has

is

shall prove

Definition 2.

is not

is

in

/[g]

coherent if and only if a polycyclic base

terminology is

explained.

it is however known that coherence finiteness

conditions

Papick has

a non-coherent

(cf. 2

the

The group ring

Cl].

Indeed,

always

A-dimension

where

see

Returning to Problem 1,

dim R > 2,

group G

ascending HNN-group over

For a partial generalization,

Problem 1

S[x]

fact that

(Soublin).

finitely generated solvaible

However,

ring is

(not

S =

p.90],

group.

S-raodules.

some examples of coherent rings.

finite!)

coherent.

of a

free

n■ F

free,

O

-^ U ->- O

(4)

finitely generated S-module

(t-1)-presentation.

and U'

has a

S functor Tor^(S/J, . )

If we apply the

to

finite (4),

we obtain an exact sequence of left S/J-modules O —^ Tor^(S/j,U)

—^ S/J« U'

I

—> S/J® F

O

O

—> S/J® U —> O (J

(5)

kD

and an isomorphism Tor^(S/J,U)->■ Tor^ It

^ follows

from the

(t-l-i)-presented

^(S/J,U'),

induction hypothesis

for O
• O,

as

TT

P -^ U --y Coker

Consider the R-module K = (M® P) O R This module is mapped onto U by the map p = (f (Id ® tt) ,7r) . M R define

f.

a natural map M® K ^ R O

M® P and the R

O -^ which gives

by taking the Then

R a M-module and we have

R q; M-modules We

>

zero map elsewhere.

generated free

(K,iJ;).

O

('3,f),

(K ,f ) 0 0

rise

-^

We

identity map on is

a natural

a

finitely

epimorphism of

whose kernel we shall denote by

therefore have an exact sequence (K,t|;)

© P.

^

to an exact and commutative

of R a ^ diagram

M-modules

189

M® Coker ib

->

K

-> Coker

R

-y o

ip

Id ® X M R -

m p R

(M® P) R

Id

f,

that

->

O

(12)

U

and ij; turns

defined by

-> P

M R

M® Coker f R

Here

e P

are

-> Coker

the natural maps

out to be the natural

the

diagram.

Apply the

O

f

induced by

inclusion in snake

lemma

f,

and

[note

(12)]

and x

is

to

(12)1

This

gives

an exact sequence

O

apply

Ker f

Put

H =

the

functor Tor

M® Ker

R

and,

for

i

>

Im

T,

-> P

Coker ij; break up

(13)

(M, . )1

into

We

O.

f

(13)

two short exact sequences and

easily obtain

two exact sequences

-> Tor.| (M,Coker

Kerdd ® x) M R

f -

-i- Coker

f)

(14)

O

1,

R , Tor, (M,Ker f)

Tor, (M, Coker t[i)

Tor^^.| (M,Coker

f)

1

(15)

->■ Tor,

.(M,Ker f)

-

1-1

[Note that Tor,

Tor.(M,H)

„ (M,Coker f)

1+1

Now, is

since

(U,f)

Ker ii =

using

(13),

Kerdd ® x)

morphism]

M R

and using

LEMMA.

Let

>

1.]

is n-finitely presented if and only

(n-1)-finitely presented,

induction,

if i

1

(14),

[which the

(R,m)

we

can easily deduce Theorem

(15),

follows

using the from

(12),

following general

if 3 by

fact that since

f

C

is

a mono-

lemma.

be a local noetherian ring,

let M be an

artinian ^-module, and let L he an ^-module of finite length. Tor^(M,L)

(K,^;)

Then

is of finite length for all i > O.

Proof.

Let I(k)

be the injective envelope of k = R/m,

and

190 let P^(L) Horn

R

(Tor^(M,L),I(k)) i

=H^(Hom

(P

R

(L),Hora

R

=

(Horn

where R is

the

so

an R-

(L),I(k)))

[and R-

Horn (M,I(k)) R

is-a

] module of

finite

R that Horn (Tor,(M,L),I(k)) R 1

that Tor^(M,L)

(M® iv

= Exti(L,Horn (M,I(k))), R K

completion of R,

R-module and L is

K

(M,I(k))))

= Ext^(L,Hom^(M,I(k))) R R

follows

We have

be an R-free resolution of L.

finitely generated length.

'*> an R-module of finite

is

is an R-module of finite

length.

It length,

This proves

the

1

lemma and Theorem COROLLARY

and let

M

3 is

Let

1.

□

completely proved. (R,m)

be a local oormrutative noetherian ring

be an artinian 'R-module.

Then

and only

A-dim R q M < a

if the following condition is satisfied. For all finitely generated R-modules R Tor.(M,v) ^

is of finite length for R

.

.

1

v


1.

It

Corollary follows

if,

for every R q

Ker

“ f of finite

1

< i

Tor

a

< a -

1

is

clearly

from Theorem

M-module

length,

[this

(M,Coker f)

1

(U,f)

3

true

that

if a = O.

A-dim R q

with Coker

R and Tor^(M,Coker

f)

Assume now that

M < a

f finitely generated, of finite

length

is an empty condition if a = 111,

has finite length,

if and only

Therefore

(C)

a

for

we have

implies

that

that

A-dim R q M < a. Assume now conversely finitely generated R-module for

1

< i

< 0

-

1.

that A-dim R q such

M ^ a,

that Tor,(M,V)

Construct an R q M-module

and let V be

is of (U,f)

finite

any

length

by taking

U =

(M® V) © V and by defining M® U U to be the identity on R R M® V and zero elsewhere. Then Coker f = V, Ker f = O, so that (U,f) R is 0-finitely presented, since

A-dim R q M < a.

length, proved.

^

so that

and therefore

(a +

1)-finitely presented.

But this gives that Tor

is verified and Corollary

{M,V) is

is of finite

completely

□

COROLLARY 2.

Let

(R,m)

be a local commutative noetherian ring,

let N be a finitely generated R-module and let Then

1

o

A-dim R q

M < a

M = Horn

R

(N,i(k)).

if and only if, for all finitely generated

191

v such that

R-modutes

we have that

Ext

Proof,

R

follows

R (Tor. (Horn

K

1

COROLLARY

R

from Theorem

3 and the natural

(N,I (k)) ,V) ,I (k))

Let

3.

is artinian.

~

Ext

i

R

I(k)


- O

the beginning of a

Dualize

(17)!

(17)

finitely generated projective

We obtain

an exact sequence,

resolution of M.

which defines

a module

D (M) : r *

O -s- M* -P* (Of course the

D (M)

depends on

image of - Ext'(D(M),R) R

The

P* ->- D(M)

functor Ext*( .

1

Let us

>

that M is

o

-M

2

M** ->- Ext^(D(M),R) R

said to be

torsionless modules

are

torsionless

exactly

generated projective modules.

the

If M

if a

submodules

is

is of

torsionless,

—O.

(19)

a monomorphism. finitely

then

(19)

can be

written -] O -^ M -^ M** -Ext where V,

being a submodule of P*

torsionless. module V, maps

Repeating

choosing a

onto M*

K

the

(a

first syzygy of D(M))

argument above

this map and

^

is

(18)

also

for the torsionless P^

that

to get

^ °

the beginning of a projective

an exact sequence

(V,R) -O

finitely generated projective module

and using

P2 -^ as

R

(cf.

[12]

resolution of V,

or the papers

we easily obtain

cited above)

1

O ->■ V -^ V** -^ Ext since M turns out to be can be works

any

(M,R)

in a non-commutative where we

All

setting).

suppose

torsionless module,

-^ O,

(20)

first syzygy of D(V).

torsionless R-module.

of Theorem 5,

any

a

R

this

is quite

Let us

that X-dim R a

and if V is

Note

that in

general

now return I(k)

to

3.

□

In the next section we shall develop a theory that,

in

particular, will show that the A-dimension can be 3 in the preceding Corollary.

5.

A duality dimension for local rings

In [2o] several examples of local rings which there is a universal constant n(R)

(R,m)

are given for

such that if V is a finitely

generated R-module with Ext^(V,R) = o, 1 ^ i ^ n(R), then i ^2 Ext (V,R) = O for i > 1. In particular, if m =0, then n{R) = 2 R works; if R is Gorenstein it is evident that n(R) = dim R works, etc. This leads us to the following definition.

Definition.

Let

(R,m)

be a local ring.

Put

for each finitely generated R-module V with a(R)

= inf • Ext^ (V, R)=0, R

1->-

similar to the

Taking a non-zero element in

S/YS = k[X,X ,X

2

sub¬

ring

aii isomorphism.

representations

~ S and we use

ring!

the

Cl9]

A-dim S.

identity map on R and by X^

having smallest possible n^ easily gives have

proved in

to study briefly

and let s'

,

Then the

= r[x„,...,X ,...]/(Xy-x^,Xy-xX,...,X ..y-xX ,...). 2 n 2 J 2 n+1 n

S'

(1)

in

n

,...]/(X^,XX„,XX

just the

which is in

We

(B,T)

3

....,XX

n

,...).

trivial extension

easily seen

§2).

used for

A-dim S < 2,

z

to be

coherent

can now apply on

the

for

(cf.

(S,Y)

the

top of p.

93 of

[24].

and here we have equality,

since

S

Remark same

is not

coherent.

PROBLEM 6.

Let R be a noetherian local domain, with field of

fractions K, and Integral closure R. more generally, all

When Is It true that R, or,

rings between R and K, have X-dlmenslon < n

(for some n > 2)?

7.

Some open questions and generalizations

In the over a

long paper

[3]

(cf.

also

(not necessarily commutative)

[23])

three

noetherian

classes

of modules

ring R were

studied,

namely

(a) for

1

< i

(b)

the n-torsionfree modules M, < n

(here

D(M)

is

the

defined by Ext^(D(M),R) R

"transpose"

the n-reflexive modules M,

of M;

cf.

=

§4);

defined by requiring that a

o

201 map M

>-

Ext^( . ,R)

(D^)

(M)

induces

(cf.

[3;

p.3])';

(c) is

those modules

for which

there

are

O -M -^ P

with P^ In

the

^11

commutative

these notions

->■

that

. . .

-h p

-)- M -o O

the

a (R)

=

(to which we

coincide if R is

remain

is of the

that

form D(M),

[

is

we

syzygetic dimension

several problems

Since every

with M ct(R)

finitely generated, of

§5

can be

introduce y(R)

as

free module 1

the

J

reflexive dimension

3(R)

and

follows:

f.g.

t-reflexive module

y(R) =inf{ t I

every

f.g.

t-th

syzygy

is

a

is

(t+1)-reflexive};

(t+1)-st syzygy}.

What are the relations hetiveen the preceding three

7.

a(R),

follows

free

every

integers

it

interpreted as

6(R) =inf{ t I

PROBLEM

about

finitely generated

finitely generated t-torsion

(t+1)-torsion

In an analogous way,

restrict ourselves now),

if and only if R is a

not Gorenstein

"duality dimension"

inf •

shall

(for all n)

unanswered.

every

3 (R)

and y(R)?

In particular, are they all finite

for a commutative noetherian local ring The preceding problem is deciding whether X-dim R g that,

of syzygies,

exact sequences

^ n-1

case

Govenstein ring.

the

M that are n-th modules

finitely generated and projective.

these notions

module

an isomorphism after application of

(R,m)?

closely related to the problem of

1(k)


O

(in the

ation

(4)

is

-F. -^ F. ^ ->i i-1

that does not mention such that,

the

in a represent¬

. . .

thivd

for i

>

1.

(F^ = R),

(Note

a general notation. are

then all

that Im d^

equivalent reformulation of

interest for Question I will now be

the Tor^(M,k)

-^ F^ ->- k ->- O O the maps

1

*

are monoraorphisms

[26]

R ->- Tor_^ (Im d.,k)

1

*

if,

we always have

, d^

d.

R Tor^dm d. ,k)

M Pj^(Z)

(3)

It then follows more generally

is a minimal R-free resolution of k

true

(ii)

that R should be

[25,26].

that if

A

close as possible

->- Tor^(m,k)

monomorphism

. . .

as

the natural map

Tor^(ra,k) is a

comes

> O) .

sequence

(3),

(4)

situation

An equivalent reformulation of spectral

vanish";

r ^ . p-r,q+r-1

and

2

(K)

but also

that the differentials

r

p,q

of positive degree),

(ii)

= m.) that is

formulated.

of special

First we

introduce

If M is a module over a local ring R such

finite-dimensional vector spaces over k

for example,

M is

finitely generated),

(this

that is

then we introduce

by Mr R i “ L dim Tor.(M,k).Z .

Note that P_(Z) K series

are

is

called

ations with the we

atways

have

the P_(Z) R

(1)

(it is

Poinaavi-Betti series).

spectral (

of

(5)

sequence

(4)

it is

E

are zero for

r

5(dim m/m

>

2

(7) □

- prof R).

in order here.

2 1.

In the E

term of

(7),

we take

the Tor

's of the P

(graded)

Tor^(R,k)-modules k

graded and this

grading gives

Remark 2. algebra

and

and k.

For any

a coalgebra

are

naturally

the extra index q.

local

(the

These Tory's

ring

dual

(R,m),

Tor^(k,k)

algebra is

is both an

- more details

are given

★

in are

§2

-

Ext

(k,k)

related so

with

the Yoneda product)

that Tor^(k,k)

becomes

and these two structures

a graded Hopf algebra

[24,17].

2 There

is

also a graded version of this

result,

so

that E^

^ becomes

a bigraded Hopf algebra.

Remark

3.

R Tor^(k,k)

The natural map R ->■ Tor^(k,k)

(a monomorphism)

(8)

210 is not only a map of algebras,

but also a map of, coalgebras,

and

oo

thus

a map of Hopf algebras.

right in

(7)

is

Remavk

The quotient // in

the Hopf algebra cokernel of

4.

The differentials

in

(7)

(8).

are derivations

compatible with the Hopf algebra structure. assertions about the differentials in

the E -term on the

and are

These and the other

are made explicit and are proved

[5].

Remavk

5.

The spectral

sequence

(7)

is

the

"local algebra

«

version"

of the Eilenberg-Moore

topology

[5].

spectral sequence

Arguments with differential graded Tor, in algebraic topology,

give the

inspired by arguments

following important corollary

Let

COROLLARY TO THEOREM A.

in algebraic

he US in

R ->• R

that the finite minimal ^-module resolution

of

R

and assume

(3)

admits a

multiplicative structure {.associative, commutative (graded) compatible with the differential in the usual way. spectral sequence Remark situations

(8)

7.

in

commutative

algebra where R ->■ R satisfies (cf.

Cover,...

"relative"

Remark 9 below).

setting;

see

for example,

that P R

-

one

to conjectures

can prove

[14]

Remark 8.

(Z)/P~(Z) R

Remark

finite-

Reversing the preceding

that in some

(cf.

[8],

cases Y^

(R,m)

There are

§3 below. (7))

is

counterexamples

a Golod ring or a complete (7)

was

degenerates

relative versions (r^

of all

[5].

the preceding

not necessarily regular)

first treated by Avramov in [7],

is Tor^

intersection,

The relative Avramov spectral sequence

*1

(that

does not have a nice

where Avramov gives

Relative Golod maps R^ ->■ r^

(generalizing

maps

in these cases,

of Buchsbaum and Eisenbud).

if

9.

are treated in

small

Therefore

only depends on the

can show that the spectral sequence

results.

where results of

thereby using methods originally invented for attacking

multiplicative structure

one

[7],

the

are treated in an even more general

dimensional graded algebra Tor^(R,k).

Question I

Then the

Note that there are several explicit well-known

Buchsbaum and Eisenbud,

arguments -

,

□

degenerates.

assumptions of the preceding Corollary

we have,

[7].

for so-called

o

(k,k)

->- Tor^

(k,k)

is

a monomorphism).

211 but Avramov has

recently

found a more complicated version

general maps.

However,

the multiplicative

more

complicated than

[18]

have

in

first thought

[9].

structure Lofwall

for

involved is

[30]

and Jacobsson

central extensions of Hopf algebras

found that

[9]

come up here

a natural way.

Finitistia global dimension,

2.

X-dimension and extensions of

Hopf algebras; the Yoneda ^ytc-algebra of a Golod ring For

any ring R and R-modules L,

M,

N,

the natural pairing

(composition of maps) Horn

(M,N)

X Horn

R

(L,M)

->- Horn

R

(L,N) R

extends naturally to a pairing of Ext, Ext^(M,N) R This

is

the

X Ext^(L,M) R

-Ext^’*’^ (L,N) . R

so-called Yoneda product.

In particular,

if

(R,ni)

is

a

★

local

ring.

Ext

R

(k,k)

becomes

a graded associative

★

k = R/m,

and Ext

similarly Ext

R

*

(L,k)

(k,N)

O ->- ^ is the

becomes

is

is

algebra

If

map

~ with R regular and ^

(which is

~2 m ,

it follows

dual to the map of Tor's

in

(9)

is

and if we

§1)

"extension"

(9)

also a map of coalgebras,

let K denote

an induced Hopf of Hopf

the

(9),

then K

It follows that we have

R

k

(k,k)

-Ext~(k,k) R

cases we even have

->- k.

(lO) ^

that the Hopf algebras Ext

★

and Ext~(k,k) R

are

an

algebras k

in these

even of Hopf algebras

coalgebra kernel of

algebra structure.

k ->■ K -^ Ext

g and g,

that

■ - -> Ext~(k,k) R

(k,k) But

Moreover,

and

k

onto.

[24,17], has

R

(k,k)-module,

R ->■ R -O

•k

Ext

a graded left Ext

a graded right Ext (k,k)-module. R

“ a representation of R, induced

algebra over

the enveloping

respectively,

algebras

and that K is

graded Lie

algebra kernel k of

Therefore,

underlying

(10)

is

the

of graded Lie

shown by Avramov

enveloping algebra of the

a natural Lie

algebra map g ->- g.

an extension of graded Lie

where R is

a Golod ring,

it

[7]

and Lofwall

follows

(k,k)

algebras

algebras

O ->- k ——> g -g -O. Now it was

R

(11) [28]

that the K of

that, (10)

in is

those a

free

cases

212 graded algebra space

(it is even the

{Exti””* (R,k) }

R 1.

It follows

that

free is

it has global homological dimension

i-2 from (10)

(or

(R,m)

several

of a Gtolod ring

algebra over, the graded vector

■*

has

related to those of a free

(ID)

that the Ext-algebra Extp,(k,lc)

interesting properties,

algebra.

Indeed,

closely

it is known that

it

Ext~(k,k) R the

for R a regular ring is

following general THEOREM

1.

an exterior algebra.

Now we have

result.

Let

H be a

graded^ oonnected (aocommutative) Hopf

algebra that is an extension of a finitely generated exterior algebra E

by means of a free algebra

thus we have

F:

k ->- F -H -E -^ k.

Then the finitistio global dimension of

(a)

that is

H,

a (left) positively graded '^-module

M hd M H

f.gldim H = sup

of finite homological dimension

(hd)

is < (b)

H

is graded coherent,

left ideals are finitely presented Proof.

We

start with case

following even more general

that is all finitely generated (only graded ideals are (a),

which will

result that we

studied).

follow from the

shall need later on in

§4.

Let

THEOREM 2.

k -^ H^ -^ H^ -► H^ -^ k

be an extension of graded connected Hopf algebras.

Then we have the

following inequality for the corresponding finitistio global dimensions: f.gldim H^

Proof.

^

f.gldim H^

f.gldim H^.

(12)

We may of course assume that t^

that t^ = f.gldim H^ H -module

+


- F^

where

the F^

are H^-free

--

...

-V F,^-^ F^ -M -»- O,

(or projective

-

this

is

equivalent

(13)

[20]).

213 Now to

the ring map

spectral

sequence

^

any

(T is

there

2

= Tor

P^q I

""s

(k,Tor

^2

p

q

claim that the

module N of

(H.,T)) 3

Indeed,

=> Tor

hd

^2

(k,T).

(14)

n (14) < 0°,

M

degenerates when T is and since H

^2 algebra of that H hd

,

is

2

it

free

an H -module.

M < oo,

and since

f.gldim H.

= t^,

gives

that N is

H^-free.

Therefore

(13)

Ter

for q

""2 q

> O,

Tor

(H

3

,N)

^11

~ Tor

so that

""2

{k,N)

q

- Tor

'^3 n

the

(k®,^ H^,N) 2

does

(14)

n By hypothesis,

'^2

sufficient to connected H;

see,

H2

M < t^

so that

1

^ (k,N)

= O

q

into an isomorphism

(15)

(15)

N has

is

(15)

is

zero

also zero for

finite homological

for

large n.

large n,

< t^, ,

since

and this

[20;

Appendix]),

f.gldim

=

combined with

t^.

so that

dimension

(it is

for a

and this homological Thus,

(13)

But

gives

by

(15)

again,

that

+ t^. 3

1

is

Now Theorem exterior algebra

now proved. 1(a)

follows,

□ since

f.gldim E = O if E is

(an exterior algebra being an

2 )'s),

M < t^,

that

^

Theorem 2

k[x]/(X

degenerate

for example,

N < t

“2 hd

Theorem 4.4]

test homological dimension with Tor^(k, . ),

dimension must be we have hd

we have hd

~ Tor

left hand side of

left H -module H ® 3 3

[32;

(k,H^®„ N). 3

then the right hand side of the

a sub Hopf

Therefore we also obtain

1

H.,

is

the

^

follows by a well-known result

as

of rings

2

spectral sequence

(13).

associated a change

left H„-module)

«

E

is

and since

f.gldim F = gldim F =

an

iterated extension of

1

if F

is

a

free

algebra.

Now we shall prove Theorem prove its

a more

graded

1(b)

about coherence.

We

shall

general result about A-dimension that we shall need in

form.

Definition.

Let H be

a positively graded connected k-algebra.

214 We say that H has the

(left,

following holds.

\-dimension

graded)

< n

Each exact sequence of

(A-dim H

< n)

if

(positively graded)

left H-modules M

O where the F. one

are

step to the

(16)

O,

finitely generated free H-modules, left to an exact sequence

-> F . n+1

A-dimension has been studied in detail

proceedings,

we shall

li)ce

presented.

(16) ,

just recall here

It follows

o

< i

easily from

is

[36]

(graded) that

finite-dimensional as

Note also that

< n.

in

[36]

of these

that if we have an exact

then we say that M is

saying that Tor^(k,M) for

O

is a finitely generated free H-module. .

Since

sequence

M

O

n+1 where F

can be continued

A-dim H

< O

(
■ k

1 be an exact sequence of Hopf algebras with H^ free. A-dim H^

^

Proof.

We

sequence

(14)

A = A-dim H^

1

+ A-dim H^. consider again the change of rings

above,

(we

where T is

(k,M)

-j- F

-> N

is

that Tor^^^(kfM)

Present M as a quotient of a O

any

suppose that it is

H2“niodule such that Tor^

We want to prove

Then

left H^-module.

finite).

spectral Put

Now let M be any

finite-dimensional for

is

finite-dimensional

i < A

+

1.

too.

finitely generated free H^-module F,

M

O.

say

(17)

H, If we

apply the functor Tor

(H^, . ) ^

to the exact sequence

(17)

we

J

obtain the isomorphism

""2

Tor^

(H3,N)

^2

'^2

- Tor^_^^ (H3,M)

^2

for i >

But Tor^^^(H3,M)

-Tor^^^(k®^ H3,M)

since gldim H^

1.

-

1.

^1

(18)

^ Tor^j3(k,M)

Combining this with

(18)

= O for i

we obtain

>

1,

215 »2 Tor^ (H^,N)

= O for i >

for T = N degenerates

Tor

and therefore the

inta an

N) n

1,

Now the

»2 fact that Tor^ (k,M)

. . implies

(use

(17))

and therefore, But since

by

A-dim H

that Tor (19)

=

is

^2 s

A,

finite-dimensional

H

Now by

2

Tor^_^2

that is

3

and thereby also

'

(19)

is

Theorem 3

the remaining part

is best possible

associative

two variables.

algebras,

each in

of k with k

But H is

not

equality in

Remark only assume

for s

1

< A,

is

(A+1)-

this gives that

(b)

and Theorem

of Theorem 1.

+

1

=

and,

in the

following sense.

tensor product of two free Then H is

according to Theorem

an extension

3,

2.

(20)

coherent [35],

and therefore

A-dim H > 2.

Thus we have

(20). 2.

If we have

that

has

an extension as

3.

in Theorem 3,

finite global dimension,

a similar argument that A-dim

Remark

+

□

the

1

N also

again,

Let H = k®j^k be

A-dim H ^

X

Any extension of a finitely generated exterior

1.

1.


O.

the H -module H ® 3 3

3

(14)

isomorphism

~ Tor i:k,N) , n

3 H

spectral sequence

One would

^ gldim

like

to have

+ the

it also

where we

follows by

A-dim H^. following generalization

of our previous Theorem 3. Let -► H3 -. k

k -.

be an extension of Hopf algebras. A-dim

^

A-dim

+

Then

A-dim H^.

We do not even know if this the

right of

(21)

are O,

that is

is

(21) true when

the

A-dimensions on

We do not know whether the Hopf

algebra extension of two noetherian Hopf algebras is noetherian.

216 The following result will be used several

(graded, connected over

kj.

X-dim A < X-dim B;

(b)

f.gldim A < f.gldim B. Assume that

Assume

X-dim B = X

Then since B is A-free,

Tor^(k,M) 1

~ Tor^(k,B® M), 1 A

that M is

member of

(22)

§4 and

§5.

< “.

Let M be

we have

as before

a left an isomorphism

i > O. .

(22)

X-finitely presented over A,

is

B

Then

(a)

A-module.

in

Let A be a sub Hopf algebra of a Hopf algebra

THEOREM 4.

Proof.

times

finite-dimensional for i

that is

< X.

that the

Then

(22)

left

implies

g that Tor.(k,B® M) is finite-dimensional for i < X, so that 1 A B A Tor, ^(k.B® M) ~ Tor, ^(k,M) is also finite-dimensional. Therefore X+1 A X+1 X-dim A < X and

(a)

The proof of

is proved. (b)

is

similar.

Let f.gldim B = y

is any A-module of finite homological dimension,

< “.

If M

it follows

from

(22)

g that Tor^(k,B®^M) since

is

f.gldim B = y

zero for i < “.

Thus

>> O, (22)

and thus gives

is

zero for i

that hd^M < y

>

and

y,

(b)

is

□

proved.

COROLLARY

Golod ring.

2

(of Theorem 3

Then

*

Ext

dimension equal to 1.

and Theorem 2).

•

(k,k)

Let •

(R,m)

be a

•

is coherent and has finitistic global

□

Part of this was proved in

[33],

where we also had some more

precise results.

Does the preceding Corollary 2 have a converse?

PROBLEM.

3. In rings of

Golod maps [25]

Gerson Levin

§ 1.

Definition. of the

introduced a relative version of the Golod

A surjective

local

ring map

(R^,m^)

-»-

(R^,m^)

form O -^ a —^ R^ -^

2 with ^ c

is called a

Golod map

*

Ext

O

if the two natural maps

*

(k,k)

->■ Ext

(23)

*

(k,k)

and Ext

*

(m

,k)

——>■ Ext

(m

,k) '

are onto.

217 If R

is

regular,

then the ring map R

O Golod if and only the

if R^

characterizations

maps.

R^

O is

of Golod rings

We briefly state (a)

a Golod ring in the

some

in

§1

(Z)

sense of

is

§1.

have analogues

ring map R

All

for Golod

->■ R^ we have a

1

O

P

(23)

results.

For any surjective

(coefficientwise)

in

inequality of Poincare-Betti

series

«

(24)

*1 1

with equality in (b) of rings

Z. (P

(24)

R

(Z)

-

spectral

1)

o

local

Ri

d

R

R

^ (k,Tor ^(R^,k)) P q 1

=> Tor *^(k,k) n

R„ ->■ R^

0

1

=0,

P/q is

r>2,q>0,

and E

°°

= O,

p,q

If

R^ -^ R^

is

a Golod map,

,

the

change

q

> O,

if and only if

then the kernel of the

Hopf algebra map

★

Ext

(k,k)

->■ Ext

1 {Ext

i-1 R

Remark.

in

(Z) = P

case R

R^

o

(Z).[1 - z. (P

10 unlike

the

(k,k)

is

the

free

o

algebra on the graded vector space

but,

1

a Golod map.

•*

P

-> R

^

(c)

surjective

a Golod map.

ring map R

r satisfies

is

sequence

= Tor

P,q

■> R^

if and only if R^ -

For any surjective

2 E

-

{R.,jk)} 1

O

is Golod,

^ (Z)

- 1)]

-1

. 2

we have by

(a)

that

,

(25)

o

case when R

O

is

regular,

rationality of P

(Z)

R °

not necessarily imply rationality of P

(Z)

in general,

R

Levin

[25]

series

in

(25),

non-rational, series

(the Artin-Rees

THEOREM

(Levin

noetherian ring.

[25]).

is

given by the

lemma is

Let

(Z)

^O

A very interesting case of a Golod map, the relative

if R^ is non-regular.

where we

can

calculate

following theorem of

used in an essential way).

(R,m)

1

since P

^1 might be an infinite, perhaps

does

be any local commutative

Then there exists a number

such that

218 -)-

R

is a Golod map for all

R/m^

In this

special

*

and *

> O.

□

> v^.

case we can even choose

that all the maps Extj^(R/m n > V

n

n

,k)

In this

*

-Ext^(R/m

case,

the

n+1

so large

,k)

are

(cf.

zero

[25])

for

long exact sequences obtained

® * by applying Ext ( .,k) to R

*■'

O ->■

->■ R/m^ -O

decompose into short exact sequences

(Z) - 1 =

.P^(Z) .

^

j

if n > v^,

giving the

formula

\ dim^Cm^/m®’'’'’) . (-Z)^|,

n >

v^,

'■s^n

and this, after

together with

some P

formula

(25)

for R -^ R/m

now gives,

simplifications,

n

(Z)”"*

- P„(Z) R

= -(-Z)

O

Y dim

.

R/m for n > V

,

(m^/m^"^^) . (-Z) ^, k — —

s>n .

But the

last series

is

rational,

since

the Hilbert

series y dim, (m^/m^^"') . Z^ iio is

“

rational.

Therefore P^(Z)

is rational

if and only if P

(Z) R/m

is rational questions

(for some and then for all n > v^)

are reduced to the

and the

rationality

artinian cases.

Further examples of Golod maps

(cf.

[26]).

(a)

If

(R,m)

is

2 local and if x e m

is a non-zerodivisor,

then R ->■ R/(x)

is

a

Golod map. (b)

More generally,

if

(R,m)

is

local,

x

e m is

”

zerodivisor,

and I

that R ->- R/xI

is

is

an

ideal

in R such that x.I

~ 2

cm

,

a nonit follows

a Golod map.

More examples are given in

[26]

and

[?].

Note that it is not

true in general that the composites of Golod maps are again Golod maps.

The most illuminating counterexample

complete

intersection R,

~

R-sequence

(t,^,...,t^)

R —^ R/(t.j) It follows

from

(a)

is probably that of a

which is obtained by dividing out an

~2

in m

~

of a regular

-R/(t,^,t2)

->■

...

above that each map

local

ring R:

we have

-^ R/(t,^, . . . ,t^) =R. in

(26)

is a Golod map.

(26)

219 But if composites of Golod maps were Golod, were a Golod ring.

it would follow that R

But tjiis is impossible for v > 1.

This known

result could be deduced in an over-sophisticated way, using Extalgebras,

and the theory of §2...

A atass of local icings which might have rational Poincave-

4.

Betti series Let M be the class consisting of those local rings completions)

R which can be reached by a finite sequence of Golod

maps from a regular local ring R,

R.

R

(having

as in

R, “2

1

R

= R.

s

This class ^ contains the Golod rings and the complete intersections. It also contains all quotients of a regular ring R by monomials in an R-sequence [16,11,18] map).

(by iterated use of Example

It might also contain every quotient of

R

(b)

of a Golod

by a determinantal

ideal. It is known that if R can be reached by < 2 Golod maps from a regular local ring P

(Z)

which

(or even from a complete intersection),

So far, no local ring

is rational [27].

p (z) is not rational.

Part

R

(i)

R e

then

^ -is known for

of the following theorem

will be used in §5 to prove that the known counterexamples to rationality of P

R

THEOREM 5.

(Z)

do not belong to AG. —

Let (R,m) be a local ring whose completion can be

reached by a sequence of s Golod maps from a regular local ring (R,m),

say R

1

s

(27)

= R.

Then (i)

f.gldim Ext

(ii)

A-dira Ext

Proof.

R

(k,k)

(k,k)

< s;

< s.

We can assume that R = R.

induction on s.

(27)

Let us prove Theorem 5 by

The case in which s = 1 is Corollary 2

3 and Theorem 2)in §2. follows from

H

If the Theorem is proved for

and property

(c)

an exact sequence of Hopf algebras

(of Theorem (s^2),

it

of Golod maps in §3 that we have

220 *

k -^ F

*

-^ Ext

(k,k)

^ where F

is a free algebra.

-Ext (k,k) -^ k Vi

Now it follows from Theorems 2 and 3

s that *

*

f.gldim Ext

(k,k)

< f.gldim Ext

s

(k,k)

+ 1

s-n

and *

*

A-diin Ext

(k,k)

< A-dim Ext

^s respectively,

and Theorem 5 follows.

Remark 1. 5:

(k,k)

+ 1

s-1-I D

We even have a stronger result than

(i)

in Theorem

the Hopf algebra kernel of * * Ext

(k,k)

R

->- Ext~(k,k) R

S

has gldim < s. Let (R,m) be a local ring whose completion can be

COROLLARY.

reached by s Golod maps from a regular local ring.

Let E be a sub

•k

Hopf algebra of Ext (k,k) of finite global dimension. R

Then

gldim H < s.

Proof.

D

This follows from Theorem 5 and Theorem 4.

Remark 2.

in §5 we shall show that the examples of

(R,m)'s

with P

(Z) not rational that have been constructed up to now are all ^ * such that Ext (k,k) contains sub Hopf algebras of arbitrarily high R finite global dimension. In view of the Corollary above, these (R,m)'s cannot belong to AG.

Remark 3.

There are rings R that can be reached from a

regular ring by three Golod maps, properties.

Here is an example

studied in [34] R = k[x

1'

and that have rather strange 3 (R,m), where m =0, which we

(I thank Calle Jacobsson, who told me why R is in AGI). .,X^]/{X^,

.X^,X,(X2+.

■

■“5>'’‘2’=3'V5'hV4>

Since R' = k[[x ,...,X 1]/(X^,...,X^,X„X.,X X^) I

Z

-D

only has monomial relations,

3

it is,

Z

O

4

D

as we remarked above,

in AG.

Now observe that X^ is a non-zerodivisor in R' and that R = R'/X.j.I, where I is the ideal in r'

generated by X^,X2+..-+X^,X2X^.

therefore follows from Example

(b)

R' -^ R'/X^I = R ts a Golod map,

It

of Golod maps in §3 that so that R e AG.

In this case

221 ★

Ext

K

cannot be generated as an algebra by a finite number of

(k,k)

*

elements

(when char k ^ 2)

Furthermore

P^(Z) K

(cf.

X-dim Ext

R

*

(k,k)

= 3 and gldim Ext

R

(k,k) =3.

[34]),

= (1 - Z)/(1 - 6Z + 11Z^ - 8Z^),

and this expression is not of the form

(1 + Z)

embedding dim(R) , , . , . /polynomial in Z,

which we always obtain for rings that can be reached by < 2 Golod maps from a regular ring

(or from a complete intersection).

Thus

new phenomena occur when one studies rings in ^ with > 3 Golod maps, and perhaps one should be careful with conjectures here.

5.

Att the known counterexamptes to rationality of Pj^(z) are

outside the class ^ We do not describe in detail here the ingenious constructions found earlier by Anick [2], products of free algebras, of the Introduction,

using certain quotients of semi-tensor that give counterexamples to Question II

and thereby also counterexamples to Question I.

Instead, we shall briefly discuss a construction by Lofwall and myself [31] that is perhaps slightly easier to explain and to put into use to prove the assertion in the title of this section. We start with some general remarks.

It follows from the

Theorem of Levin and the discussion following it in §3 that the rationality questions for Poincare-Betti series of local rings can be reduced to the artinian case. The first non-trivial case is that 3 of (R,m) with m =0, and from now on we shall stick to that case. ~

Consider Ext

*

R

(k,k)

and its subalgebra A generated by Ext

1 R

(k,k).

Here A turns out to be the enveloping algebra of a graded Lie algebra, generated by elements in degree 1,

and having relations in degree 2,

so that A is a Hopf algebra of a very special type A is

(1,2)-presented).

Conversely,

come from a suitable local ring described.

all

(R,m)

(we shall say that

(1,2)-presented Hopf algebras

in the way we have just

Indeed, given such a Hopf algebra A, we obtain an R by

taking the "diagonal part" of the cohomology algebra of the Hopf algebra A,

and by dividing out the cube of the augmentation ideal of

the cohomology algebra.

More details can be found in [28,29], where

222 it is also proved that,

in the equicharacteristic case, R and A

determine each other.

Furthermore we have the following formula,

which was first discovered by Lofwall [28,29]

(see also [34] for an

alternative treatment of this formula): P

R

(Z)"^ =

where A(Z)

=

^

(1 + z"’’) .A(Z)"^ - z"\h

K

(-Z) ,

(28)

(dim A^).Z^ is the Hilbert series of the graded

i>0 (non-commutative)

algebra A,

and H

1

Now x^ for all

is i

into

first place

look

is

to

>

1.

Similarly,

(x^,...,x.)

[V-V],rs^]).

satisfying such equalities

But even for all

m

i+1 /

i

for all

j i+1 /, v and m / (x)

large

i.

form m^

certain local cohomology modules. look at a cohomological

modules

is

forced to

x^ , . . . ,x . =

if one i,

it

is

satisfy

look at modules

_ i+1 n m

Now modu] es of the

to

So one

in gr R

n

true that certain carefully chosen minimal reductions will such equalities

for

at a degree one

a non-zerodivisor

if and only if (cf.

return to this point

naturally come

= m/m^ © m^/m^ ffi...

system of parameters X'|,...,x^.

cannot find elements

shall

form m^''"^/xm^

in another way.

in grm

for all

We

to try to gain more

"^/xm^

arise when one

The purpose of this

looks

at

exposition is

interpretation of these numbers

and

insight into the problem of computing

Hilbert functions. The additional B = R ffi ^ ffi m^ ffi..,

notation will be as and B^

is

the

follows.

First,

ideal of B which is

the direct sum

233 of components =

1.

of positive

Secondly,

degree.

G = gr

Note

that height B_^ = grade

= R/m 0 m/m^ © m^/m^ 0...

the direct sum of components of positive Set

(X,(D ) X

= Proj

B,

interpretation of minimal sets

reduction

then

is

B.

For

d U D

X =

The geometrical

that the corresponding open

(x.)

i=1 geneous

degree.

the blow-up of R at m.

give a m.inimal cover of Proj

minimal reduction of ra,

if X =

,...,x^ is any

because the only homo

^

ideal of B containing x^,...,x^ is B^. is

local

can be expressed in

cohomology modules H’ B

(E)

the

If E

generated graded B-module and E

Then,

is any

finitely

corresponding sheaf,

the

terms of the

+

Grothendieck cohomology of the sheaves © H'(X,E(n))

and G_^ = gr m,

by [EGA;

E{n).

(2.1.5)],

Set H'(X,E(*))

=

there exist degree O

neZ canonical

isomorphisms,

H^(X,E(*))

and an

5

functorial in E, (E)

for p >

of graded B-modules

1

exact sequence of degree O homomorphisms, O ^ H° B Now,

(E)

^ E ^ H°{X,E(*))

^

B

+

for

i

> O,

the

local

functorial

in E,

O.

(E)

+

cohomology module H

(E)

can be

+ computed as

a direct limit of

on sequences the

d-th H

= x^,...,x^:

local

d B

>c

, , (B)

cf.

[Hz-K],

of Koszul

Consequently,

complexes

we have

for

cohomology. =

n

+

cohomology modules

,. dk+n , k {d-1)k+n lim m /x m ——> — — “ k

and H

d G

where (The

, , (G) +

= n

the maps

in the

dk+n, k (d-1)k+n dk+n+1 /x m + m ,

direct systems

idea will be to try

vanishing of H exploiting (the

lim m z ^ k

(B)

for i

the

fact that

result of

Serre,cf.

H^(X,E(n))

are

to assume as
N and all i

> O.

the modules

there

exists

N

234 For E = B or G we get the

following additional information after

first choosing a special minimal reduction ance i

there exists

for

By prime

avoid¬

a minimal reduction x.for m and an integer 1 d —

such that ^ (x.

,

) Dm

,x,

1 +1 =

,... ,x,

(x,

Let r be the reduction number for

dk+n/k

(d-1)k+n m

Fix

x.

then for

,r,-d-n +r+1),

m

for alli>i

1 max(0,r -d). X

The same notation has been used for

Proof.

i

+

k

in R d

in B.

kd-d+j Tx

/ N (k-l) d+j (x)

must have

j > 1 some

and each > k,

and

(d-1)k+r +j Thus

^dk+r^-d/^k^{d-1)k+r^-d ^ ^dk+r^-d+1

^ ^dk+r^-d/^k^(d-1) k+r^-d

235

^

and m

(d-1)k+r -d+j m 21

/x

^ . = O for

]i

^

1.

.^1 (ii)

O —>■ B(-1)

—1 B —>■ B/x^B —>■ O is exact and so we get

that

B_^

is

exact;

but H

(iii)

Since R is

B

(B) +

(B)

n-1

B_|_

(B/x B)

O

n

= O,

of dim d >

B_^

(B/x B) In

and so the desired conclusion

© H nel

C.-M.

= O.

O

(B)

(X,B(n))

—>■

(B)

follows.

—»- O is exact.

B 2,

H

0

(X,B)

k.

=

k

U (m :m ) k>0 -

= R,

and so

From the exact sequence x^

O —> B(-1)

—> B —> B/x^B

O,

we get that

H°

O

(B/X B)

d Hi

+ and,

as

~

(iv) n

(B)

B

+ and

B +

(B/x^B)^ = x^R,

Since depth G S d-1,

(B/x^B)

In

...

O

is exact,

it follows that

S B(R/x^R)

In

for all

> O where B(R/x^R)

and r

is

= R/x^R © m/x^R ©

the reduction number for m/x^R with respect

of X2/...,x^ in R/x^R.

^

We have

The x^

1

fact means

and n

H'"(B) n+1

X

that every element of that hMb)„

in

argument shows

that H

-

Since n > 0, (B)

O for all n

n

is

n

(B/x.B) . 1 n+1

. . .

—>

> max(0,r -d) .

(B)

= O

= O.

A similar

for all n S max(0,r -d). ^

sequence (G)

. n+1

annihilated by a power of

O —>■ mB —»- B —>■ G —O exact

-> H^(B)

.0,

H

exact sequence

to get the

images

X

n the

(B/x B) 1 n+1

> max(0,r -d).

to the

the exact sequence

(B) H

Set i =

(m/x^R)^ ffi. . .

H^(mB) — . T n H

1+1

(mB) — n

H^(B)

(G)

Now use

236 If i and

is

any integer such

(since

all n

i

>

e 2.

1)

that 1

H^(mB) = — n

But since


V.2,

J With the notation as above,

PROPOSITION 5. (i)

A

(ii)

A

(b).

{ => a^

(b)

(B)

(iii)

1) ,

-1

For any m-primary ideal q we have

A(q^/yq)

= e(q)

+ A(R/q)

A(mq^/yqm) A(yq/yqm) (i*)

-

(i*)

A(q/q2),

a m.inimal reduction of q.

= A(R/yR)+A(yR/yq)-A(R/q)

From


X^,

as

is

(Avramov,

the

is

straight

[5]

where

in the

the relation H B

on

local morphisms A ->■ B Very

instance

little

...).

y-series

is

minus

that a

form a(T + T^)

The

situation can

case.

But for

to have been

= H H AC

[1])

conceived;

studied as

see

a condition

(with fibre C) .

is known about

is probably the

see

example

but it might always have the Roos

a trivial

This polynomial

shown by the simple

Hilbert series nothing similar appears however

Andre;

fibre of A ->■ B,

coefficients).

conjectured by Andre,

said to be even more

inequality as

(3).

conclusion,

The

simplest lundecided

for A,B of Krull dimension

3,

that the multiplicity of A should not exceed the multiplicity of B. No

counter-example

be known,

to

the

stronger assertion that

but it would of course

could prove

that there

always

Let us

A

(T) (1 - T)”"^

< H

finally remark

and Poincare

series

B

already be a great thing if one

exists a natural number n,

depend on the morphism A -B, H

(19)

of

(T) (1 - T)

that there

of a

[8],

References 1.

M.

Andre,

locaux",

is a relation between the Hilbert

local ring which might make of Poincare

it possible series

(cf.

to the

which can be generalized).

(further "Le

that may

such that

attack Hilbert series problems by means formula

seems to

references

caractere

preprint,

can be

additif des

found in

[6])

deviations des

Ecole Polytechnique Federale

anneaux

de Lausanne,

1982. 2.

H.-B.

Foxby,

"A homological

Preprint Series Matematisk 3.

H.-B.

B.

Institut,

Foxby and A.

flat base

4.

1981,

Herzog,

change",

No.

Vol.

1,

Thorup, Proc.

"Minimal injective

Amer.

"Effect of certain

Teubner Texte

78-93.

K(z)benhavns Universitet,

1981.

local Hilbert functions".

pp.

theory of complexes of modules".

19,

Math. flat

67

resolutions under (1977),

local extensions on

Seminar D.

zur Math.

Soc.,

Bd.29

Eisenbud-B. (Teubner,

27-31. the

Singh-W.Vogel, Leipzig,

1980),

246 5.

B.

Herzog,

"On a relation between

to a local homomorphism",

J.

the Hilbert fxonctions belonging

London Math.

Soc.(2),

25

(1982),

458-466. 6.

T.

Larfeldt and C.

Lech,

couples of local rings", 7.

P.

Roberts,

rings,

"Analytic ramifications

and flat

Acta Math.,

201-208.

J.-E.

Roos,

and of local

Proceedings,

Mathematics pp.

over commutative

superieufes

Montreal,

(Les Presses de

1980).

"Relations between the Poincare-Betti

loop spaces Dubreil,

(1981),

Homological invariants of modules

Seminaire de mathematiques

I'Universite de Montreal, 8.

146

740

rings"*,

Paris

(Springer,

series of

Seminaire d'algebre Paul

1977-78, Berlin,

Lecture Notes

Heidelberg,

in

New Yorlc,

1979) ,

285-322.

Department of Mathematics, University of Stocltholm, Box 6701, S-113

2.

85

Stoclcholm,

Sweden.

Problems on asymptotic prime divisors (submitted by L.J.

Ratliff, Jr. and received on 8 September, 1981) Throughout, A

*

(I)

=

Ass R/I

I n

is

an ideal in a Noetherian ring R/

and A

*

(I)

= Ass R/(I

n

)

for all

large n,

where

I

3.

denotes

the integral

closure of I in R.

the sets Ass R/l^ are equal holds

for Ass R/(I^) PROBLEM 2.1

increasing;

,

by

for all

large n,

shov/n

in

[1]

that

and a similar result

[6].)

It is known

that is,

3

(It is

if P is

[6]

that the sets Ass R/(I

a prime divisor of

(I

k

)

are

^ for some k >

)^

1,

^

then P e i

>

A

1.

(I)

The

and in

fact P is

first part of this

a prime divisor of is not true

(I

for the

)

for all

^ k sets Ass R/l ,

★

by [1]. l'^

Is

it true that if P e

for some k

>

1,

then P

PROBLEM 2.2.

(I)

Given a finite

It is

set S

is

a prime divisor of

shown in

for all i

of positive

an ideal I

a prime divisor of I

PROBLEM 2.3.

and P

is a prime divisor of

there exist a Noetherian ring R, such that P is

A

in R,

integers,

and P e

if and only if k e [1]

that the

>

1? do

Spec R

S?

sets Ass

(I^/I*^^^)

*

are equal shown in B

*

(I)

u

for all L2;

r

iP:

(This was

large n.

Proposition I

c p £

asked in

Let B

Ass r}. p.

denote

10 and Corollary 1

[2;

(I)

13]

Characterize B

75].)

*

this

set.

that A (I)

n

Then

(I)

{P:

I

it is

= c p

£

ass r}.

247 PROBLEM 2.4.

(R,M)

It is shown in [4;

of altitude n+1

(16)] that a local domain

> 1 ‘is a Cohen-Macaulay ring if and only if R

is quasi-unmixed and there exists an integrally closed ideal of the principal class of height n in R(X^,...,X ) =r[x^,...,X ]

r

Inin MRLX^,...,X

1 n (An ideal I is of the principal class in case I can be generated by h = height I elements.) essential?

Is the quasi-unmixedness assumption

(This was asked in [4;

PROBLEM 2.5.

J

In [3;

(18)].)

Theorem 4],

Noetherian domain such that A

(I)

was essentially characterized.

the

set jj/ =

{R: R is a

★

/s *

= A

(I)

for all ideals I in r}

However,

it was left open in [3]

if there exists an integrally closed local domain R of altitude three in sJ, but it is shown there that any such R cannot satisfy the alti¬ Does there exist such an integral domain xn

tude formula.

7

(The only known example of such a ring is the recent example of T. Ogoma that shows that the strongest of the catenary chain conjectures, the Chain Conjecture catenary),

(the integral closure of a local domain is

is false.)

It is shown in [6]

that if there exists a

local UFD of altitude three which does not satisfy the altitude formula, so,

then it must be in j"-/.

Does there exist such a ring?

then the weakest of the catenary chain conjectures,

Chain Conjecture

(If

the Normal

(if the integral closure of a local domain R

satisfies the first chain condition for prime ideals, integral extension domain of R does), PROBLEM 2.6.

In

[5],

then every

is false.)

it is shown that asymptotic sequences

in R are an excellent analogue of R-sequences in general Noetherian rings.

B

(Elements b

R and g n, where B [5]

(B

^

n

,...,b 1

)

in R are an

P ^ : b.R = (B,

asymptotic sequence in case

n

. ) for i = 1,...,g and for all large a 1 1-1 a = (b^,...,b.)R (i = O,1,...,g).) It is also shown in 1 1 1 that the asymptotic grade of an ideal I in R satisfies most of 1-1

the basic properties of the usual grade of an ideal.

(The asymptotic

grade of I is defined as the maximum of the lengths of asymptotic sequences contained in I.) interesting,

The literature is crowded with useful,

and important results concerning R-sequences and grade(I).

To what extent are the asymptotic versions of these results valid?

248 References 1.

M.

Brodmann,

Math. 2.

S.

Soc.,

74

(1979),

McAdam and P.

(1979), 3.

"Asymptotic stability of Ass(M/IM)",

Proc.

Amer.

16-18.

Eakin,

"The asymptotic Ass", J.

Algebra,

61

71-81.

L.J. Ratliff,

Jr.,

prime divisors".

"Integrally closed ideals and asymptotic

Pacific J.

Math.,

91

(1980),

445-456.

4.

L.J. Ratliff, Jr., "New characterizations of quasi-unmixed, unmixed, and Macaulay local domains", J. Algebra, to appear.

5.

L.J.

Ratliff,

Jr.,

"Asymptotic sequences", preprint. University

of California at Riverside, 6.

L.J.

Ratliff,

Jr.,

1981.

"Asymptotic prime divisors",

in preparation.

Department of Mathematics, University of California, Riverside, California 92521,

3.

Miscellaneous problems

PROBLEM 3.1

April,

U.S.A.

1982).

(submitted by Winfried Bruns and received on 19

Let R be a noetherian commutative ring and let n be

a positive integer.

What can be said about the set

(n) V ^ ' {a)

) 1 }

r {p e Spec R: ^ c p

for an ideal ^ of R? PROBLEM 3.2

1981).

Let

(submitted by E.G. Evans and received on 24 July,

(R,ra,k)

be a regular local ring for which Vi is infinite,

and let M be a finitely generated R-module.

For x e M,

let

O

(x) = {f(x) If e Horn (M,R)}. M R Does there exist an x e M for which height O

(x)

PROBLEM 3.3

1981).

< rank M?

(submitted by E.G. Evans and received on 24 July,

In the situation of Problem 3.2,

can one put reasonable

assumptions on M and then predict the behavior of some O

(x) (for M their depth,

example,

can one predict their primeness or othe2rwise,

etc.)?

The hope would be to generalize to height greater than 2 the

results obtained in [eg]. [eg]

E.G. Evans and P. Griffith, "Local cohomology modules for normal domains", J. London Math. Soc. (2), 19 (1979), 277-284.

249

(submitted by E.G. Evans and received on 24 July,

PROBLEM 3.4

1981).

In the situation

Problems 3.2 and 3.3,

and one knows all the Oj^(x) all have height > r), about its depth,

1982).

Is X-dim R g I(k)

(for example,

then what can one say about M

its syzygyness,

PROBLEM 3.5

February,

for x e M - mM

if M is reflexive that they

(for example,

etc.)?

(submitted by Jan-Erik Roos and received on 5 Let < “?

(R,m)

be a commutative noetherian local ring.

(The notation is the same as that on p.191.)

This problem is equivalent to Problem 3.6 below. PROBLEM 3.6

February,

1982).

(submitted by Jan-Erik Roos and received on 5 Let

(R,m)

be a commutative noetherian local ring.

Does there exist an integer a(R) generated R-module with Ext^(M,R) then we have that Ext PROBLEM 3.7

February,

1982).

(M,R)

such that,

if M is a finitely

of finite length for 1 < i < a(R),

is of finite length for all i > 1?

(submitted by Jan-Erik Roos and received on 5 Let

(R,m)

be a commutative noetherian local ring.

Does there exist an integer 6(R)

such that,

if,

for a finitely

generated R-module M, we have Ext^(M,R) = O for 1 < i < 6(R), it . R follows that Ext^(M,R) = O for all i > 1? R PROBLEM 3.8 (submitted by Jan-Erik Roos and received on 5

February,

1982).

Let

(R,m)

be a commutative noetherian local ring.

Does there exist an integer Y(R) R-module is a Y(R)“th syzygy, PROBLEM 3.9

June,

1982).

(R,m) — r+1

m —

such that,

then it is a

if a finitely generated (y(R)

+ 1)-st syzygy?

(submitted by Judith D. Sally and received on 2

is there a d-dimensional Cohen-Macaulay local ring

having minimal reductions x.,...,x^ and y.,...,y^ such that 1 d 1 d =

r

(x.,...,x )m T a — PROBLEM 3.10

r+1

but m —

?=

r

(Y.,» • • • - Y .) m ? I a ~

(submitted by R.Y.

Sharp).

if a commutative

Noetherian ring A possesses a dualizing complex, must A be a homo¬ morphic image of a finite-dimensional Gorenstein ring? PROBLEM 3.11

(submitted by R.Y. Sharp).

Let A be a commutative

Noetherian local ring and let M be a balanced big Cohen-Macaulay A-module

(this terminology is explained on p.

74).

Let p be an

associated prime ideal of M/(a^,...,a^)M for some M-sequence

250 a^,...,a

1

r

.

Must it be the case

that M

p

is

a balanced big Cohen-

Macaulay A -module?

P This question has an affirmative answer when A is a catenary local domain:

[s]

see [S;

Theorem 4.3].

R.y.

Sharp, "A Cousin complex characterization of balanced big Cohen-Macaulay modules", Quart. J. Math. Oxford (2), to appear.

S'2-b°l‘