Free Resolutions in Commutative Algebra and Algebraic Geometry. Sundance 90 9780867202854, 9781138454293, 0867202858, 9781003420187

Table of contents :
Cover
Half Title
Series
Title
Copyright
Contents
Contributors
Introduction
Structure and Size of Free Resolutions
Problems on infinite free resolutions
Problems on Betti numbers of finite length modules
Multiplicative structures on finite free resolutions
Wonderful rings and awesome modules
Green's Conjecture
Green's conjecture: an orientation for algebraists
Some matrices related to Green's conjecture
Other Topics
Problems on local cohomology
Recent work on Cremona transformations
The homological conjectures
Remarks on residual intersections
Some open problems in invariant theory

Citation preview

Free Resolutions in Commutative Algebra and Algebraic Geometry

Research Notes in Mathematics Volume 2

Free Resolutions in Commutative Algebra and Algebraic Geometry Sundance 90

Edited by

David Eisenbud

Department o f Mathematics Brandeis University Waltham, Massachusetts

Craig Huneke

Department o f Mathematics Purdue University West Lafayette, Indiana

CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

First issued in hardback 2019 © 1992 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13: 978-0-86720-285-4 (pbk) ISBN-13: 978-1-138-45429-3 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any fixture reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access w w w .copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http ://www.taylorandfrancis.com and the CRC Press Web site at http ://www.crcpress.com Library of Congress Cataloging-in-Publication Data Free resolutions in commutative algebra and algebraic geometry : Sundance 90 / edited by David Eisenbud and Craig Huneke. p. cm. - (Research notes in mathematics ; 2) ISBN 0-86720-285-8 1. Free resolutions (Algebra)--Congresses. 2. Commutative algebra—Congresses. 3. Geometry, Algebraic- Congresses. I. Eisenbud, David. II. Huneke, C. (Craig) III. Series : Research notes in mathematics (Boston, M ass.); 2. QA169.F72 1992 512\24~dc20 91-45056 CIP

DOI: 10.1201/9781003420187

Contents Contributors Introduction Structure and Size of Free Resolutions

1

Problems on infinite free resolutions L. Avramov

3

Problems on Betti numbers of finite length modules H. Charalambous and E.G. Evans, Jr.

25

Multiplicative structures on finite free resolutions M. Miller

35

Wonderful rings and awesome modules G. Kempf

47

Green's Conjecture

49

Green's conjecture: an orientation for algebraists D. Eisenbud

51

Some matrices related to Green's conjecture D. Bayer and M. Stillman

79

v

vi

Contents

Other Topics

91

Problems on local cohomology C. Huneke

93

Recent work on Cremona transformations S. Katz

109

The homological conjectures P. Roberts

121

Remarks on residual intersections B. Ulrich

133

Some open problems in invariant theory J. Weyman

139

Contributors Numbers in parentheses refer to the pages on which the authors' contributions begin. Luchezar L. Avramov (3), Department o f Mathematics, Purdue University, West Lafayette, Indiana 47907 Dave Bayer (79), Department of Mathematics, Barnard College, New York, New York 10027 Hara Charalambous (25), Department of Mathematics, State University of New York at Albany, Albany, New York 12222 David Eisenbud (51), Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254 E. G. Evans, Jr. (25), Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 Craig Huneke (93), Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 Sheldon Katz (109), Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078 George R. Kempf (47), Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218 Matthew Miller (35), Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 Paul C. Roberts (121), Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 Mike Stillman (79), Department of Mathematics, Cornell University, Ithaca, New York 14853 Bemd Ulrich (133), Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 Jerzy Weyman (139), Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

vii

Introduction Free resolutions arise from systems of linear equations over rings other than fields. A system of linear equations in finitely many unknowns over a field has a basic set of solutions in terms of which all others can be expressed as linear combinations, and these basic solutions can be chosen to be linearly independent. Over a polynomial ring, or more generally any Noetherian ring R, a system of linear equations in finitely many unknowns still has a finite system of solutions in terms of which all others may be expressed, but now these solutions cannot in general be taken to be linearly independent. To find the dependence relations on a given system of solutions requires solving a new system of linear equations. Iterating this process, one gets a whole series of systems of equations, which makes up a free resolution of the original problem. If one considers the cokernel M of the matrix expressing the original system of equations as a module over ii, one speaks of a free resolution of M. The free resolution expresses certain properties which are implicit in, but not at all obvious from, the original system of equations. If the ring R is a polynomial ring or power series ring, then free resolutions can be taken to be of finite length (that is, the systems of equations produced eventually have linearly independent sets of basic solutions); this is essentially the celebrated ” Hilbert Syzygy Theorem” , which started the subject off. But over more general rings, the free resolution may involve infinitely many sets of equations. Free resolutions and questions related to them occur in many areas of commutative algebra and algebraic geometry. In May of 1990 there was a small and informal conference in Sundance, Utah, organized by David Eisenbud, Craig Huneke, and Robert Speiser, on the topic of free resolutions and their uses in commutative algebra and algebraic geometry. A good deal of the conference was devoted to discussions of the current state of work on some of the central problems in the area. These discussions seemed worth transmitting to a broader audience, and we were able to convince a number of the participants to write up accounts of areas in which they are expert. Some of these writeups develop groups of current prob-

ix

x

Introduction

lems which seem likely to influence future development of the field. Others are basic expositions of areas of current interest; and some contain new research, not otherwise published. A notion of the diversity and richness of the subject (and of the conference) can be obtained from a description of the topics treated: The first group of papers treats basic questions about the relations between the systems of linear equations in a resolution. Much current interest relates to the size and ranks of these intermediate systems of equations (the “ranks of syzygies” ), a subject related to questions about possible ranks of vector bundles on projective spaces. A central test case is that of a resolution of a module of finite length over a polynomial ring, and the state of our current knowledge and conjectures about this case are surveyed by Charalambous and Evans. Avramov treats the case where the free resolution is infinite, concentrating on what regularity may eventually appear in the resolution. Miller surveys the state of our knowledge about multiplicative structures on resolutions. In the most basic case, if the module M is of the form R/I, then the multiplicative structure of R/I extends (nonuniquely) to a commutative but only homotopy-associative algebra structure on the resolution. In some cases this multiplicative structure allows one to compare different parts of the resolution. The results one gets are closely related to the (originally geometric) notion of linkage, which plays an important role. Kempf briefly surveys some cases that correspond to the best possible behavior of an infinite free resolution of a graded module over a graded ring. These are the cases in which the systems of linear equations only involve linear polynomials. Surprisingly many resolutions of geometric interest have this property; noone yet really understands why. The next pair of papers concerns Green’s conjecture and some variants of it. Green’s conjecture connects rather subtle geometric properties of an algebraic curve with the shape of the free resolution of the homogeneous coordinate ring of the curve. Eisenbud presents a new algebraic version of the conjecture, and explains, from an algebraic point of view, some of the motivation. He then surveys some of the approaches that have been tried and the partial results that they have yielded. In particular, there are several approaches which lead to absolutely explicit (but large) numerical

Introduction

xi

matrices; determination of the ranks of these matrices would settle a leading case of the conjecture. Although several people had worked out such approaches, till now none of them has been written up. In the second paper, Bayer and Stillman exhibit two such matrices, from their own research, and explain the derivation of them. All those who are good at linear algebra, take note! Here is an opportunity to solve a central problem in the theory of Riemann surfaces by elementary methods. The last five papers concern another range of algebraic and geometric topics: Craig Huneke explains what is known and conjectured about the mysterious and interesting local cohomology modules Hj(Af) := lim Ext3R(R/In ,M ) . >n with respect to an arbitrary ideal I. Cremona transformations are by definition the birational automorphisms of projective spaces. Sheldon Katz explains how free resolutions can be used for constructing and testing examples in this very classical geometric subject. Paul Roberts gives a survey of work in the area of the “homological conjectures” , where much has been done since Hochster’s famous CBMS notes of 1975. Bernd Ulrich treats "residual intersections” : The central problem here is to say something about some of the (primary) components of an ideal /, given information about the ideal itself and information about the other components. This problem generalizes the problem of linkage, which is essentially the case in which / is a complete intersection. Finally, Jerzy Weyman explains the current problems on invariants connected with the free resolutions of ideals associated to strata in representations, and some of the ideas of Kempf in this direction.

The conference was made possible by the generous support of Brigham Young University, in Provo, Utah. We personally found the conference highly productive, and the place and atmosphere unusually pleasant and congenial. Robert Speiser, our coorganizer, took charge of all the local

xii

Introduction

arrangements with great spirit and success; the conference ran smoothly through the lubrication of his hard work. We, and the other participants, are grateful to the University for underwriting the conference, and also to the NSF for providing travel funds for some of the participants. David Eisenbud and Craig Huneke

August 1991

Dept, of Mathematics Brandeis University Waltham MA 02254 [email protected] Dept, of Mathematics Purdue University W. Lafayette IN 47907 clh® macaulay.math .purdue .edu

Structure and Size of Free Resolution

Problems

on

infinite free

resolutions

LUCHEZAR L. AVRAMOV1' 2 Institute of Mathematics, ul. "Akad. G. Boncev" blok 8, 1113 Sofia, Bulgaria Throughout this survey R w ill denote either a noetherlan local ring with maximal Ideal m and residue field £ , or a graded algebra R = ©n^o Rn with ( irrelevant) max­ imal ideal m = ©n^i Rn and R0 =

k

The minimal number of generators of m is called

the embedding dimension of R and denoted edim R . A local (resp., graded) ring 0 with maximal ideal n is said to be a deformat ion of R of codimension c if R = Q /(xi, ... , xc) for a 0-regular sequence x = xi , ... , xc , which is assumed to be homogeneous when R is a graded ring; in case x is contained in n2 , the deformation is said to be embedded. If R is a local ring whose m-adic completion $ can be deformed to a regular local ring 0 , then R is said to be a complete intersection ; in the graded case, this notion refers to a graded ring R which can de deformed to a graded polynomial ring 0 . Only fin ite ly generated R-modules w ill be considered; in the graded case homomorphisms of modules w ill be assumed to be compatible with the gradings and homogeneous of degree zero. A free resolution (F, d) : ... ----- > Fn

Fn-1 ----- >

> Fi

F0 ( ----- > M

»0)

is said to be minimal if dn( Fn) ci mFn-i for n ^ 1 . I t is wel l known that any two min­ imal resolutions of M are isomorphic as complexes of R-modules. Thus, the module dn(Fn) is defined uniquely up to isomorphism: it is called the n'th syzygy of M and denoted Syzn(fc)- Sim ilarly, the rank of the free R-module Fn does not depend on the choice of F : it is called the n'th Betti number of R and denoted bf? (M) . Betti numbers are usually computed (and frequently defined) via the equalities

b|?(M) =

1 P a rticip a tio n at the Sundance Conference on Free Resolutions in A lg eb ra ic Geometry* and Commutative Algebra was supported by the NSF under the U.S.-Bulgarian Program in Algebra and Geometry. 2 Present address: Department of Mathematics, Purdue University, West Lafayette, IN 47907.

DOI: 10.1201/9781003420187-2

3

4

L. Avramov

dim^Tor^(M , £ ) = dim^ExtR(M, £ ) . It isoften oftenadvantageousto studyBetti numbers wholesale rather than individually, andthe standard vehicle for the relevantinformation is the formal power series P n(t) = ¡E n>obn(M )tn , called the Poincare series of M ; by traditional abuse of language the series P ^(t) is also called the Poincare series of R .

1. Poincare series The conjecture that p\(t) always represents a rational function in t has been a powerful motivation for the study of in fin ite free resolutions for almost 25 years. However, in 1979 Anick [1 ] constructed an artinian (graded) ring with m 3 = 0 , em­ bedding dimension 5 , and length 13 , such that P ^(t) is transcendental. A related con­ struction was then proposed by Lofwall and Roos [41 ]. Building up on this, B0 gvad [21 ] produced an artinian (graded) Gorenstein ring with transcendental Poincare series: it has m 4 = 0 , embedding dimension 12 , and length 2 6 . Instructive surveys on the (non-)rationality of P%(t) have been given by Roos [47] and by Babenko [16]. The negative results notwithstanding, quite general statements on rationality are known for several classes of local rings. Namely, the formal power series pR(t) is ra­ tional for every fin ite ly generated

R-module M when R satisfies one of the following

conditions: (a) R is a complete intersection [31 ]; (b) R is a Golod ring [2 9 ]; (*)

(c) edim R - depth R ^ 3 [15]; (d) edim R - depth R = 4 , and R is Gorenstein [3 6 ]; (e) R is directly linked to a complete intersection [15]; (f) R is linked in two steps to a complete intersection, and R is Gorenstein [15].

Problems on infinite free resolutions

5

Our firs t problem (which can be refined when R is Gorenstein) puts the focus on the gaps which s till remain between the cases when rationality or irrationality is known : Problem 1. Find "natural" sufficient conditions on R for P ri(t) to be a rational function for each R-module M . In particular test the following properties: edim R depth R = 4 ; length R $ 12 ; R is in the linkage class of a complete intersection. As a firs t step, do they imply the rationality of the Poincare series of R ? Partial results are available: Backelin and Froberg [18] have obtained rational expressions for P^(t) when R is graded of length $ 7, and Palmer [4 2 ] has proved that Pm (t)

is rational for all M over certain almost complete intersections with edim R -

depth R = 4 , cf. also the note at the end of the paper. However, Jacobsson [3 5 ] has constructed a ring R with rational P ^(t) , but which has a module M with transcendental. If pR (t)

Pm (t)

is rational for all M , then the next problem asks whether

there can be an infinite number of essentially different Poincare series. Problem 2.

Assume that

R has the property that the Poincare series of any

fin ite ly generated R-module is rational. Does a polynomial DenR(t) with integer coef­ ficients then exist, such that for each R-module M the series DenR( t ) p f i( t )

is a

polynomial ? In all cases when the hypothesis of Problem 2 is known to hold, the answer is pos­ itive, and furthermore one can take DenR(t) to be the denominator of P%(\) written as an irreducible rational function. This extra information has been used in an essential way in [9 ] for the study of a conjecture of Eisenbud on modules with bounded Betti numbers, cf. Section 2 below. It is very interesting to study quantitatively the gap between rational and non-ra­ tional series. At least for equicharacteristic rings, this can be put in concrete terms as follows. Let R = t ® m be a fin ite dimensional ^-algebra with nilpotent ideal m , and let V denote an n-dimensional ^-vector space. A bilinear pairing h: R x V —> V w ill

6

L. Avramov

be Identified with an element of R* ® V* ® V . Those n which define on V a structure of (le ft)

R-module form an algebraic subset M odn(£) of R* ® V* 0 V . The linear

group GL(n, £ ) acts in an obvious way on Modn( £) and the orbits of this action are in one-to-one correspondence with the isomorphism classes of R-modules M of dimension n over £ . There is a single closed orbit, given by the module £n (for details cf. e:g. [2 7 ]). Problem 3. Study the geography of the set {M € Modn(£) I Pm U) is rational} : When is it non-empty? If it is, then does it (or its complement) contain a non-empty open subset? Does there exist an irreducible component consisting of modules with ra­ tional (or non-rational) series? What is the maximal dimension of an orbit which con­ tains such a module? There are versions of these questions for the set LocAlgn>m( O c V* V* ® V which parametrizes the commutative fc-algebra structures on £© V such that Vm = 0 , 2 $ m ^ n+1 (cf. e.g. [3 4 ] for a description of the corresponding constructions, and the recent a rticle [4 8 ] for information on the components of LocAlgn,3( O ) • For example, what can be said about LocAlgi2,3( fc) ? (By Anick's example this is the firs t case where it is known that a non-trivial situation occurs.) Note that in these questions the re ­ striction to artinian rings and fin ite length modules is natural in view of results of Levin [4 0 ] which reduce rationality problems for Poincare series to the artinian case.

2. Bounded Betti numbers The projective dimension of a non-zero R-module M may be introduced by the formula: pdR M = sup { n e IN | bp ( M) * 0 }. From this point of view, the simplest mod­ ules of infinite projective dimension are those whose Betti numbers are bounded. They have received some attention in literature, but very natural and important questions are s till open, even the existential one:

Problems on infinite free resolutions Problem 4.

7

Which rings have modules of in fin ite projective dimension with

bounded Betti numbers? That the answer has to be non-trivial is shown for instance by the existence of rings R such that the Betti numbers s tric tly increase for each non-free R-module M and for n £ 1 ( in [4 4 ] such R are introduced by the name BNSI rings). For an elemen­ tary example consider a ring

R with m * m2 = 0 . The firs t syzygy of a non-free

module M is then a direct sum of copies of £ . As it is easily seen that bp ( £ ) = (edim R)n for n >, 0 , it follows that R is a BNSI ring when edim R > y 2 . More general constructions of BNSI rings are given by Ramras [4 4 ], Gover and Ramras [3 0 ], and Lescot [37]. As a potential source of bounded Betti sequences, consider periodic modules: if m is a positive integer, then a non-zero R-module M is said to be periodic of period m if it is isomorphic to its m’th syzygy (and then SyzR(M) = SyZf?+m(M) for n € IN). The Betti numbers of a periodic module are obviously bounded, and the converse is true when R is artinian with £ is algebraic over a fin ite field, cf. [2 8 ]. Thus, a step in an­ swering the question raised in the preceding problem w ill be a solution of the following one. Problem 5. Characterize the rings which have a periodic module. The only general construction for both problems seems to be for rings which have an embedded (codimension one) deformation R a* Q/(x) (cf. [2 6 ] for regular 0 , and [3 3 ] or [13] otherwise). There are then homomorphisms of free O-modules G such that: rankR F = rankR G ; (M*, R ) = 0 for n > 0 : the R-dual of a minimal resolution of M* then shows M = M **

is an in fin ite syzygy. Recall now that in Ausländer and Bridger's [3 ] theory of

G-dimension, a module M is said to be of G-dimension zero if it is reflexive and satis­ fies Ext£( M*, R ) = 0 = Extß( M, R ) for n > 0 . Thus, if G-dirriR M = 0 , then M is an in fin ite syzygy provided pdR M* = °o . Furthermore, it is proved in [3 ] that for any module N over a Gorenstein ring R there is an equality G-dimR N + depthR N = dim R . It follows that if M is a maximal Cohen-Macaulay module, then G-dimR M = 0 . As M* also is then maximal Cohen-Macaulay, pdR M* is fin ite only if M* is free, and then M = M ** is free. Thus a non-free maximal Cohen-Macaulay R-module M is reflexive and has pdR M* = oo , which establishes our assertion. Since in fin ite syzygies have been constructed over some artinian (graded) rings which are not Gorenstein and have no deformations, cf. [12], Problem 8 is open even for artinian rings.

3. Asymptotic behavior of Betti sequences If M is an R-module of in fin ite projective dimension, then for its (depth R)'th syzygy N there exists a maximal R-regular sequence y which is also N-regular. It follows that for n e IN there are equalities

b n + d e p th R

(

M) =&n(N) = b^/ i y ) ( N /(y) N ).

Thus, when dealing with asymptotic properties of Betti sequences one may assume that the ring R has depth zero. The firs t question on unbounded Betti sequences seems to be

Problems on infinite free resolutions

11

whether they also satisfy lim n-*oo bfj(M) = oo , cf. [4 5 ]. The following one, proposed in [6 ], asks for more and includes Problem 6: Problem 9. Is the Betti sequence of an arb itra ry

R-module eventually non-de­

creasing? One might try to start with modules of fin ite length, where length considerations have often proven effective, cf. [46, 28, 25]. Before describing cases where the answer to Problem 9 is known, we shortly discuss the asymptotic behavior of sequence of Betti numbers. The simplest estimate on such a sequence is given as follows: complexity

d , denoted

cx r

M is said to have

M = d , if d is the least integer such that b§(M) $ and“ 1

holds for some positive a € R and for n >> 0 , cf. [8 ]. A module M is said to have strong polynomial growth of degree

d if there are polynomials p(X) and q(X) e R [ x ]

which are both of degree d , have the same leading term, and satisfy p(n) ^ bft(M) $ q(n) for n >> 0 . Sim ilarly, M is said to have strong exponential growth if there is a real numbers o< > 1 such that cxn $ b^ ( M) for n >> 0 . We refer to [9 ] for a discussion of related conditions, as well as for a proof of the well-known fact that for any module M there is a p e R such that b^ ( M) $ pn for n >> 0 . The following questions are taken from [9 ]; note that in the firs t unknown case - that of complexity 1 - the last question coincides with that raised in Problem 6. Problem 10. Does every R-module have one of the two types of growth described above? If the complexity of M is fin ite , does the inequality

cx r

M $ edim R - depth R

then hold? Does a module of fin ite complexity have strong polynomial growth? Answers to both problems are known in roughly the same situations: when M = £ [4 9 , 32, 7 ]; when R is artinian of length s< 7 or artinian Gorenstein of length $ 11 [2 8 ]; when m3 = 0 [37]; when M has fin ite virtual projective dimension, in particular when R is a complete intersection [8, 13]; when R is aGolod ring [39, 9]. Although

12

L. Avramov

some of the proofs exploit the rationality of Pm U ), it seems that the growth and ratio­ nality properties of Betti sequences are in general unrelated: witness the result of Anick andGulliksen [2 ], which for any ring R produces a ring R' with ( m ')3 = 0 , such that the Poincare series of either one is rationally expressible in terms of the other one's. Growth problems in some cases in which rationality is not known have been treated by Choi [24, 25]. As a ring with edim R - depth R ^ 2 is either a complete intersection or a Golod ring, Problems 9 and 10 then have positive solutions. The same conclusion holds for Gorenstein rings with edim R - depth R = 3 , since then either R is a complete inter­ section or there is an odd integer

r ^ 5 such that every

R-module has a rational

Poincare series with denominator 1 - r t 2 - r t 3 + t 5 . These observations however do not suffice to prove the growth of Betti numbers for all the rings described by conditions (*) in Section 1. As for the asymptotics of Betti sequences, in this case they are ''almost'' known, in the sense that the asymptotics of the fir s t sum-transform

Pn(M ) =

Z i=o b?( M) are determined in [9]. The scarcity of results and the considerable effort spent in proving those described above illustrate both the apparent difficulty of the problems and the extent of our igno­ rance concerning infinite resolutions. While the rings described in (* ) might yield to al­ ready available techniques, there seems to be a need for a whole new arsenal of tools to tackle such questions in general: we shall come back to this in the next section. Another useful measure of the asymptotic growth of the Betti sequence of M is given by the radius of convergence of its Poincare series, that is, the real number pr( M) = 1/lim sup ( bn ( M )) 1/n . We call it the radius of convergence of M ; note that pr(M) = if and only if pdR M < °o , and that otherwise there are inequalities 0 < pr( M) $ 1 . In general, the modules whose radius of convergence is extremal seem particularly interesting. It is known that the inequality

pr(M)

> ,p r(£ )

holds for all

M

. This fol­

lows from the more precise result, that for each M there is a positive real number y such that the inequality b n ( M ) ^ y b n ( £ )

holds for all n g IN . When M is artinian,

Problems on infinite free resolutions

13

the assertion is immediate from the long homology exact sequence, which shows that one can then take y equal to the length of M . The general case is reduced to the artinian one by a result of [4 ] and [4 0 ] which proves that for each M there is a positive integer r such that the equality b{?(M/mr M) = bfi(M ) + bf?_i (m r M)

holds for all

n . At the

other extreme, it is clear that a module of fin ite non-zero complexity has radius of con­ vergence 1 , and it has been proved that p r ( £ ) >, 1 if and only if R is a complete intersection, cf. [6, 7]. Finally, note that the radius of convergence of a module is equal to that of any of its syzygies, and that for each positive real number p the R-modules M which satisfy the inequality

p r

(M)

{ p form, in the category of all fin ite ly generated R-modules, a

subcategory which is closed under extensions. Thus the radius of convergence may be useful in studying the Grothendieck group G( R) of a local ring R . The preceding remarks provide the context for the next group of questions. Problem 11. Does p r (M )^ 1 imply that M has fin ite complexity? Is there a constant pr < 1 , such that p r ( M) < 1 implies pr ( M) < pr ? Is there a ring for which, the set { pr ( M)} , where M ranges over all finitely generated R-modules, is infinite? We finish this section by taking a quick look at the in itia l portions of Betti se­ quences. While for modules of fin ite projective dimension the whole problem is here, in case of infinite dimension this piece of the resolution seems to be almost beyond control. This adds interest to Huneke’s remark (at the Sundance Conference), that the following question, which for modules of fin ite projective dimension has recently focused much at­ tention and is discussed elsewhere in these Proceedings, cf. [23], is also open in general: Problem 12. Does the inequality b^(M) > ^d1mR^dimMj h o ld fo ra lln ?

14

L. Avramov 4. Structures on resolutions By general nonsense, Yoneda pairings provide E x t? !(t, £ )

with a structure of

graded ^-algebra, and Exti>(M, £ ) with a structure of a graded left module over it. The key to some of the results described in the preceding sections has been provided by an understanding these pairings at a non-trivial level. The information on the Yoneda alge­ bra is concentrated in a much smaller object functorially attached to R , the homotopy L ie algebra

tt

* ( R ) . This is a graded Lie (super-)algebra whose universal enveloping

algebra is ExtR( £, £) , cf. [6 ]. Thus, the cohomology module of M is in a natural way a graded representation of the homotopy Lie algebra. Very simple structural questions are s till unanswered, for example the following one: Problem 13. For what M is the Extp>( £ , £) -module ExtR(M, O

fin ite ly gen­

erated?

Even when the cohomology module is known to be fin ite ly generated, it might s till be difficult to gather from this fact numerical information, due to the complexity of the homotopy Lie algebra tt*(R ) , which is conjectured to contain a non-abelian free subal­ gebra unless R is a complete intersection, cf. [5 ]. An original and sometimes efficient way to circumvent this inconvenient has been proposed by Backelin and Roos [2 0 ], who investigate the cohomology of ExtR(M, £ ) over the cohomology algebra of ExtR(fc. t ) . which is a (skew-)commutative graded ^-algebra. Qn the other hand, there are indica­ tions of correlations between fin ite complexity and restrictions on the structure of the cohomology module of M . For instance, the theory of modules of fin ite virtual projective dimension [8 ] is entirely based on the portrait of M provided by its cohomology module, which in this case is fin ite ly generated over a central polynomial subalgebra of Ext]i>( k., kS). Thus, in some cases the action of a large part of the Yoneda algebra may be irrelevant. The next problem takes this observation into account.

Problems on infinite free resolutions

15

Problem 14. When is ExtR(M, £) finitely generated over a noetherian subalge­ bra or factor algebra of Extp>( k £) ? Note that B0 gvad and Halperin [2 2 ] have proved that if

ExtR(£, t )

itself is

noether ian, then R is a complete intersection (this had been conjectured by Roos; the converse is also true and had been known for some tim e ). If M is periodic of period m , then the in itia l portion of its minimal resolution provides a length m exact sequence: o

► M -------► Fn M

Fm-2

...... ........ * F,

F0

>

M ►0 .

Right multiplication by the congruence class h e ExtR(M . M ) of this exact sequence provides isomorphisms Extj}( M, N ) —» ExtR+m( M, N )

for each R-module N and for

n e N . If one trie s to find an explanation of periodicity phenomena in the structure of ExtR(M, £ ) , the next problem arises (its solution is positive for modules of fin ite v ir ­ tual projective dimension, hence for all modules over the rings described by the condi­ tions (* ) in Section 1). Problem 15.

If M is periodic of period m , does there exist X e ExtR(£, £)

such that left multiplication by X provides isomorphisms ExtR( M, £ ) —> ExtR+m( ti, £) for n e IN ? The ExtR( t,-fc)-module structure on Ext|( M, O reflects the "linear part" of the differential of (F, 3) : for M = t

an explanation of this admittedly cryptic statement

may be found in [61 The examination of the Yoneda Ext-algebra from the point of view of its homotopy Lie algebra nucleus is strongly connected with ideas and techniques from ra­ tional homotopy theory, which are discussed at length in [6 ] and [1 4 ]. This approach also suggests that the differential of a minimal free resolution might be "reconstructed" from some higher order structures. One such structure was associated by Eisenbud [2 6 ] with a deformation of R . When R = Q/(x) for a non-zero divisor x , let ( F, 5 )

be a sequence of homomorphisms of

16

L. Avramov

free O-modules such that ( F R , d 5, but not a complete intersection. There are very few results of this sort in the literature. A trivial argument shows that I is a complete intersection if Tf 0. In grade four there is the simple result [13] that dim Tj2 > ( 1/ 2) dim T2 implies J is a complete intersection; a quick glance at the classification theorem in [15] shows that dim T 2 = dim T iT2 < 3 unless there is a distinguished element e in T\ so that T 2 = eT\ (and in any case dim T f = 0). In grade g > 5 the Tor-algebra of the Herzog ideal has dim T " = (5~*) for n < g — 1 and T ®-1 = 0 (see [5, Proposition 6.3 and

Multiplicative structures on finite free resolutions

43

Example 5.11]). It is possible that the Huneke-Ulrich ideals exhibit the maximum possible non-trivial multiplication in T for ideals that are not complete intersections: if the grade is five, then dim Ti = dim T4 = 7, dim T2 = dimTs = 22, dim T 2 = = 15, dim T f = 0, dim T iT 2 = 7, dimTjTs = 6, and dim T| is either 6 or 7 according as 2 is a unit in R or not. If g = 2n — 1 (and I is therefore generated by 2n + 1 elements), then the multiplication on T (see [19] or [8, Theorem 6.2]) satisfies dim Tj = ( 2” ) = (ff"^1) for i < n and T ” = 0. After the Gorenstein ideals the next best class (certainly from a linkage-theoretic point of view) axe the perfect almost complete intersections. In grade four the ideals of Palmer’s thesis have T = E' ix W , where E' = E/E± is a truncation of the exterior algebra E = [\kA and W is a trivial ¿^'-module. In particular, d im T[ = (1) for i < 3. The Tor-algebras of the Northcott ideals given in [5] reveal the same structure, with dim T[ = (f) for i < g — 1 and T f — 0. The HunekeUlrich almost complete intersections exhibit the phenomenon of having a large number of higher order Koszul relations in their minimal resolutions in an exceptionally strong way: in grade g = 2n the Tor-algebrasatisfies dimTi = (2” +1) = (9"|"1) for i < n and d im T " +1 = 0 (see [8]), that is, the Koszul relations on all the minimal generators appear in the resolution as far as half-way back! In this case, however, we don’t know the full structure of Tor^(5, k). Huneke has pointed out that by taking the product Si ®jt S2 of two such (as a quotient of the product of disjoint polynomial rings), one obtains a perfect ideal of deviation two and grade 4n with dim 7\2 = (4"2+2) • This process can be iterated so that one can produce perfect ideals of arbitrarily high deviation (but also with correspondingly high grade) such that dim T 2 = (,i^ ) . In Kustin’s recent preprint [10] one finds grade four almost complete intersections with 6 < dim T 2 < 10 = Q ), dim Tj3 = 10 — dim T 2, and various possibilities for dim T\ T j.

44

M. Miller

Problem 13. For which ideals is dimTj2 = ( M^ ) ? If we consider the classification problem from the perspective of Weyman’s analysis, then the same computational evidence points to the following problem. Here the emphasis is less on the size of T *, but rather on how “decomposable” the product structure must be.

Problem 14. The image of the map b*: T2* —►f \ 2 T* lies in the space of n x n alternating matrices, where n = dim Ti. Excluding the examples Si S2 mentioned above, determine whether each image b*() is a matrix of rank at most g + 1. (In fact, in all known cases, except for the Huneke-Ulrich Gorenstein ideals of deviation two, one can replace g + 1 by g\ and since the rank of an alternating matrix is even, one can lower an odd upper bound by one automatically.) Finally, Weyman has suggested that one consider how T is generated as a algebra (intending application to resolutions of coordinate rings of Grassmannians.) At one extreme we have the exterior algebra structure T = /\Ti for a complete intersection. On the other hand there are resolutions with long linear strands, so that generators of T must appear in high homological degrees. Suppose that R /I is a quotient of a polynomial ring R = k[Xj , . . . , X n] by a homogeneous ideal. Then there is a graded minimal resolution and this induces a bigraded structure Ttj = T ot^(R /I, k)j. If I is generated by quadrics and the resolution of R/I has a linear strand, say Titi+i 7^ 0 for 1 < i < p, then T is generated over To by this linear strand in many, but not all, examples (and, indeed, in all cases for which p > 2). Problem 15. Find necessary and sufficient conditions for the linear strand in a resolution of an ideal generated by quadrics to generate T. A C K N O W L E D G M E N T S . Thanks to the organizers and participants of the conference at Sundance for stimulating a fresh

Multiplicative structures on finite free resolutions

45

look at this part of the theory of resolutions, and especially for posing new problems! Thanks also to Andy Kustin for very helpful discussions during the preparation of this article. Ref erences 1. L. A vram ov, On the Hopf algebra of a local ring, A p p en d ix by V. H inic, Izv. A kad. N auk. S S S R Ser. M at. 3 8 (1 9 7 4 ), 2 5 3 -2 7 7 , E n glish tra n sla tio n , M ath. U SS R -Izv. 8 (1 9 7 4 ), 2 5 9 -2 8 4 . 2. L. A vram ov, Obstructions to the existence of multiplicative structures on minimal free resolutions , A m er. J. M ath. 1 0 3 (1 9 8 1 ), 1 -3 1 . 3. L. A vram ov , Homological asymptotics of modules over local rings, C o m m u ta tiv e A lgeb ra, M ath. Sci. R esearch In stitu te P u b lica tio n s 1 5 , Springer V erlag, 1989, pp. 3 3 -6 2 . 4. L. A vram ov and H .-B . Foxby, Gorenstein local homomorphisms , B u lletin A m er. M ath . S oc. (1 9 9 0 ), 1 4 5 -1 5 0 . 5. L. A vram ov, A . K u stin , and M. M iller, Poincare series of modules over local rings o f small embedding codepth or small linking number,

J. A lgeb ra 1 1 8 (1 9 8 8 ), 1 6 2 -2 0 4 . 6. D. B u ch sb au m and D . E isen b u d , Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3,

A m er. J. M ath. 9 9 (1 9 7 7 ), 4 4 7 -4 8 5 . 7. H. C h aralam b ou s and E. G. E van s, Problems on betti numbers of finite length modules.

8. A . K u stin , The minimal free resolutions o f the Huneke-Ulrich deviation two Gorenstein ideals, J. A lgeb ra 1 0 0 (1 9 8 6 ), 2 6 5 -3 0 4 . 9. A . K u stin , Gorenstein algebras of codimension four and characteristic two , C om m . A lg. 1 5 (1 9 8 7 ), 2 4 1 7 -2 4 2 9 . 10. A . K u stin , Classification of the Tor-algebras of codimension four almost complete intersections , p reprint, 1991. 11. A . K u stin , The minimal resolution of a codimension four almost complete intersection is a DG-algebra, p reprint, 1991. 12. A . K u stin and M. M iller, Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four , M ath. Z. 1 7 3 (1 9 8 0 ), 1 7 1 -1 8 4 . 13. A . K u stin and M. M iller, Algebra structures on minimal resolutions of Gorenstein rings, C o m m u ta tiv e Algebra: A n a ly tic M ethod s, Lecture N o tes in P ure and A p p lied M ath. 6 8 , ed. R. D raper, M arcel Dekker, 1982. 14. A . K u stin and M. M iller, Multiplicative structure on resolutions of algebras defined by Herzog ideals, J. London M ath. Soc. (2) 2 8 (1 9 8 3 ), 2 4 7 -2 6 0 .

46

M. Miller

15. A . K u stin and M. M iller, Classification of the Tor-algebras of codimension four Gorenstein local rings, M ath. Z. 1 9 0 (1 9 8 5 ), 3 4 1 -3 5 5 . 16. S. P alm er, Multiplicative structure on resolutions of grade four almost complete intersectionsy P h .D . d isserta tio n , U n iversity o f S ou th C arolina, 1990. 17. C. P esk in e and L. Szpiro, Liaison des variétés algébriques I, In ven t. M ath. 2 6 (1 9 7 4 ), 2 7 1 -3 0 2 . 18. H. S rin ivasan , Algebra structures on some canonical resolutions, J. A lgeb ra 1 2 2 (1 9 8 9 ), 1 5 0 -1 8 7 . 19. H. S rinivasan, Minimal algebra resolutions for cyclic modules defined by Huneke-Ulrich ideals, J. A lgeb ra 1 3 7 (1 9 9 1 ), 4 3 3 -4 7 2 . 20. H. Srin ivasan , The non-existence o f minimal algebra resolution despite the vanishing of Avramov obstructions, preprint, 1989. 21. B. U lrich, Sums of linked ideals, T rans. A m er. M ath. Soc. 3 1 8 (1 9 9 0 ), 1 -4 2 . 22. J. W eym an , On the structure o f free resolutions of length 3 , J. A lgeb ra 1 2 6 (1 9 8 9 ), 1 -3 3 .

Wonderful rings and awesome modules George R. Kempf / The Johns Hopkins University Let A = k © A\ © . .. be a connected graded commutative ring over a field k. Let M be a finitely generated graded A-module. The M is awesome if T

o r k) is purely of

degree i for all i if M has a free resolution ...

A0ni

A0n° — ►M — ►0

where all the > 0. Therefore high enough Veronese embedding of projective varieties have wonderful coordinate rings. (Kempf) if L is an ample invertible sheaf on an abelian variety X , the ring n®or ( X ,L 0n) is wonderful if L is an at least 4 power of some N and if M is (alge-

DOI: 10.1201/9781003420187-5

47

48

G. Kempf

braically equivalent to) an at least third power of iV, the module

n>0

T (X , M ® N®n) is

awesome [3]. (Butler unpublished) let C be a complete smooth curve of genus g of characteristic zero. If W is a invertible on C of slope > 2g + 2, then n ® o r(X ,S y m " HO is wonderful. Also if U is another invertible sheaf on d of slope >

2< 7,

then

® r(X ,C /® S y m n W )

n> 0

is awesome. (Kempf [2 ]) Let

k

. . . ,p r be points in IPn where r < 2 n which are in general linear

position. Then the projective coordinate ring of these points is wonderful. If C as above has no g^g's or g5 , then the canonical ring n>or( X , ii®n). (This is to appear), It seems natural to study projective coordinate rings of surfaces. Motivated by the examples of awesome modules over wonderful ring. I asked the following problem. If A is wonderful then for fixed M then define the graded module M[r] by M r 1 • = | ^ r+*

lrl,‘

1 0

if * > 0 otherwise.

Then the problem is to show th a t M[r] is awesome if r » k

0.

This problem was solved by

Avramov and Eisenbud at the conference. R eferen ces 1. G. Kempf, Some wonderful rings in algebraic geometry, Journal of Algebra, to appear. 2.

_______ , Syzygies of points in projective space, to appear in Journal of Algebra.

3. _______ , Projective coordinate rings of abelian varieties, Algebraic Analysis, geometry and Number Theory, Edited J. I. Igusa, The Johns Hopkins Press 1989, pp. 225-236.

Green's Conjecture

Green's Conjecture: An Orientation for Algebraists by David Eisenbud 1 Department of Mathematics, Brandeis University Waltham MA 02254 [email protected] These notes parallel the introduction to Mark Green's conjecture on the free resolutions of canonical curves (Green [1984] and Green-Lazarsfeld [1985b]) that I presented at the Sundance 90 conference. They have a two-fold purpose: to introduce commutative algebraists with a modest background in algebraic geometry to a formulation of Green’s conjecture that is more algebraic than the usual one; and to survey some of the approaches to Green's conjecture that have been tried. The first section leads up to an algebraic conjecture (somewhat wild) generalizing Green's conjecture. The second section tries to explain the attractiveness of canonical rings of curves (for algebraists; geometers already know this!) and explains the connection between the algebraic conjecture of section 1 and the usual version of Green's conjecture. The third section surveys some promising approaches to Green's conjecture. 1.

I d e a ls g e n e r a t e d b y q u a d r ic s a n d 2 - l i n e a r r e s o l u t i o n s

N o t a t io n Let S = k[xo, ... ,x r ], and let R = S /I be a h o m o g e n e o u s fa cto r ring of S. If w e a ssu m e th a t I c o n ta in s no lin ea r fo rm s, and th e p r o je c tiv e d im en sio n of S /I is m , th e n w e m a y re p r esen t th e m in im a l free reso lu tio n IF of S /I b y a b e t t i d ia g r a m (as in th e "betti" c o m m a n d of th e p rogram M a ca u la y of B a y er and S tillm a n ) of 1. Thanks to Brigham Young University, which very generously supported the conference at which these notes originated. I am also grateful to the NSF for partial support during the preparation of this document.

DOI: 10.1201/9781003420187-7

51

52

D. Eisenbud

th e form

m (degree of syzygy) (step of resolution)

0

1

^ 2

-

3

-

a3 ^1

^2

^3

m e a n in g th a t 7 ca n be w r itte n as

0 «— S /I «— S in d u ce s a split m o n o m o rp h ism on th e 2 lin ea r p a rt of *5. At th e 0 th step f o = 9o = S, and th e re su lt is clea r. S in ce J c o n ta in s no lin ea r fo rm s, th e q u a d ra tic m in im a l g en er a to rs in th e ideal I a re a m o n g th e m in im a l g en er a to rs of J; th is is e x a c tly e q u iv a le n t to th e d esired s ta te m e n t. An e a s y in d u ctio n c o m p le te s th e proof. / / Thus one kind of a cc ep ta b le "explanation" for th e len g th (or size...) of th e 2 -lin e a r p art of th e resolu tion of S /I w ou ld be t h a t I c o n ta in s an id eal of so m e sta n d a rd form , w h ic h is k n o w n to h a v e a long (or large...) 2 -lin e a r p art. This is th e form w h ic h G reen’s c o n je c tu r e ta k es. B efore sta tin g th e c o n je c tu r e , w e w ill d escrib e th e id eals of "standard form" t h a t arise: E x a m p le s If I is th e id eal g en er a ted b y th e 2 x 2 m in o r s of a g en eric pxq m a tr ix , th e n th e 2 -lin e a r stra n d is k n ow n (L ascoux [1978]) to h a v e len g th > p +q -3, w ith e q u a lity in c h a r a c te r istic 0 (and m o st lik ely all th e tim e .) This tu r n s out to be tr u e of id eals of 2 x 2 m in o r s of co n sid er a b ly m o r e gen era l m a tr ice s: follow ing E isenbud [1988] w e d efin e a m a tr ix L = (Ajj) of lin ea r fo rm s to be 1 - g e n e r i c if no e n t r y Xy is 0, an d n o n e ca n be m a d e 0 b y row and co lu m n tr a n sfo r m a tio n s. W e h a v e T h e o r e m 1 . 2 : If L is a 1 -g e n e r ic m a tr ix of lin ea r fo rm s o v er a p o ly n o m ia l ring S, and I is th e ideal g en er a ted b y th e 2 x 2 m in o r s of L, th e n th e m in im a l free resolu tion of S /I h a s a 2 -lin e a r stra n d of len g th > p+q-3.

54

D. Eisenbud

This r e su lt w a s p roved u n d er v a r io u s e x tr a n e o u s h y p o th e se s b y Green an d L azarsfeld (Green [1984;A ppendix]); in a m o re alg eb ra ic se ttin g b y m y s e lf (un pu blish ed ); and th e n , in a sim p ler w a y , b y Koh an d S tillm a n [1989]. The m o st fa m ilia r ca se of T h eorem 1 . 2 is th a t in w h ic h on e of p, q, s a y p, is 2: A ssu m in g th a t if th e 2 x q m a tr ix of lin ea r fo rm s L is 1 -g e n e r ic , it is not h ard to sh o w t h a t th e ideal of m in o r s of L is of gen eric co d im en sion (= q -1 ), so th e m in im a l free reso lu tio n T is th e th e E ag o n -N o rth co tt co m p lex , w h ic h co n sists e n tir e ly of a lin ea r stra n d of len g th e x a c tly p+q-3 = q -1 . For a r b itr a r y p,q an d a g en eric m a tr ix L, th e c o m p u ta tio n of L ascoux sh o w s t h a t th e 2 -lin e a r stra n d h a s len g th e x a c tly p+q-3; th e con d ition of 1 -g e n e r ic ity is su ffic ien t (b u t is n ot n ece ssa r y ; it w ould be n ice to k n ow so m e n e c e ssa r y con d ition s) to p r e se r v e a t le a st one of th e req u ired sy z y g ie s. A n a lg e b r a ic c o n j e c t u r e : The boldest possible c o n je c tu r e w ou ld n o w be to sa y th a t a sort of c o n v e r se to T h eorem 1 sh ould be tru e: th a t is, a n y ideal I su ch th a t 2L P(S/I) = n sh ould co n ta in an ideal of 2 x 2 m in o r s of a 1 g en er ic pxq m a tr ix w ith p+q-3 = n. A las, th is is false. F irst of all, it is possible to sta r t w ith a 1 -g e n e r ic m a tr ix and rep la ce so m e of its e n tr ie s w ith 0 ‘s w ith o u t spoiling th e le n g th of th e 2 -lin e a r p a rt of th e resolu tion of its ideal of 2 x 2 m in o r s, so th a t so m e m a tr ic e s w h ic h a re n ot 1 -g e n e r ic m ig h t p erfo rm th e sa m e fu n ctio n . H ow ever, if w e a ssu m e t h a t I is a p rim e ideal n ot c o n ta in in g a n y lin ea r fo rm s, th e n of co u rse I ca n n o t co n ta in a n o n zero d e te r m in a n t of a 2 x 2 m a tr ix of lin ea r fo rm s w ith on e e n tr y = 0. T hus I could n ot co n ta in a n o n tr iv ia l ideal of 2 x 2 m in o r s of a n y m a tr ix of lin ea r fo rm s o th e r th a n a 1 -g e n e r ic m a tr ix (or a 1 -g e n e r ic m a tr ix exp an d ed w ith ro w s and c o lu m n s of zeros.) So if w e a ssu m e th a t I is p rim e, th is o b jection b eco m es void. S econ d , th e r e a re id eals w h o se 2 -lin e a r p a rt is n o n tr iv ia l b ut w h ic h co n ta in no d e te r m in a n ts at all!. For n o te th a t a n y d e te r m in a n t of a 2 x 2 m a tr ix of lin ea r fo rm s is a q u a d ric of ra n k < 4 (th e ra n k m u s t be 3 or 4 if th e m a tr ix is 1 -g en er ic.) Thus th e ideal I g e n e r a te d b y a sin gle ran k s q u ad ric, w ith s>5, h a s a 2 -lin e a r p art

Green’s conjecture: an orientation for algebraists

55

of len g th 1 b u t c e r ta in ly c o n ta in s no 2 x 2 m in o r s of in te r e s t. The sa m e h olds for so m e q u ad rics of ran k < 4 if th e field is n ot a lg e b r a ica lly closed: for e x a m p le x 2+ y2+z2+ w 2 is n ot a d e te r m in a n t of a m a tr ix of lin ea r fo rm s o v er th e real n u m b e rs. B u t w h a t a b ou t p rim e id eals g en er a ted b y q u a d rics of ra n k < 4 (or m o re g e n e r a lly , p rim es w h o se d egree 2 p a rts a re sp a n n ed b y q u a d rics of ra n k < 4) o v er an a lg e b r a ica lly closed field? T here a re still c o u n te r e x a m p le s , d ue to S c h r e y e r [1986, 199?], in c h a r a c te r is tic 2 (an alogou s ca ses, in c h a r a c te r istic s * 2, a re k n o w n n ot to be c o u n te r e x a m p le s.) H ow ever, one m a y be o p tim istic an d feel th a t th e th e o r y of id eals w ith lots of q u a d ra tic g en er a to rs m ig h t w ell be a little d iffer en t in c h a r a c te r istic 2. (Of co u rse th e p e ssim istic w ill feel in stea d t h a t w e a r e being w a rn ed .) If w e su ppose t h a t c h a r k = 0, or e v e n th a t it is * 2, I k n ow no fu r th e r c o u n te r e x a m p le s , so I r a s h ly m a k e th e C o n j e c t u r e 1 . 3: Let k be an a lg e b r a ica lly closed field of c h a r a c te r is tic * 2, and let I c S = k[xo, ... ,x r ] be a p rim e ideal, co n ta in in g no lin ea r form , w h o se q u a d ra tic p a rt is sp a n n ed b y q u a d rics of ran k < 4. If 2LP(S/I) = n, th e n I c o n ta in s an ideal of 2 x 2 m in o r s of a 1 -g e n e r ic pxq m a tr ix w ith p+q-3 ■ n. G reen’s c o n je c tu r e , from th e algeb raic point of v ie w , is ju s t th e sp ecial ca se of th is w h e r e w e a ssu m e in addition: a) b) c) d)

S /I is n o r m a l (= in te g r a lly closed) d im S /I = 2 S /I is G orenstein d egree S /I = 2r

(con d itio n s c,d could be replaced b y th e e q u iv a le n t co n d tio n s th a t S /I is a C o h e n -M a c a u la y d om ain su ch th a t m od u lo a m a x im a l reg u la r se q u e n c e of lin ea r fo rm s it h a s Hilbert fu n ctio n 1, y, y, 1 for so m e in te g e r y, w h ic h is r -1 in th e n o ta tio n above.) T here is r e a lly no a r g u m e n t co n n ectin g a n y of th e se four con d itio n s to th e co n clu sion of th e co n jec tu re ; b u t th e r e a re so m e g e o m e tr ic te c h n iq u e s (described below ) w h ic h h a v e lead to th e v e r ific a tio n of Green's c o n je c tu r e u n d er th e se e x tr a h y p o th e se s in

56

D. Eisenbud

m a n y sp ecial ca ses (for ex a m p le in all ca ses w ith r < 7; see S c h r e y e r [1986].)

2.

C a n o n ic a l r i n g s of c u r v e s

In th e p rev io u s sectio n I cla im ed t h a t th e sp ecial h y p o th e se s a) - d) m ig h t w ell be ir r e le v a n t to th e c o n je c tu r e . In th is se ctio n I w a n t to exp lain w h y g e o m e te r s n e v e r th e le s s find a )-d ) so e n tr a n c in g , and d escrib e a g e o m e tr ic re fo r m u la tio n u n d er th e se con d ition s. W h at is th e sim p lest in te r e stin g kind of v a r ie ty ? A c u r v e , of cou rse! (W ell, a lm o st of course: ra tio n a l v a r ie tie s also h a v e ex ercised so m e c la im s, esp e cia lly in ce r ta in periods, b eca u se t h e y a re so e a sy to sp ecify . For e x a m p le a ra tio n a l su r fa c e ca n be sp ecified as a p la n e w ith so m e m a rk ed p oin ts (to blow up) and c u r v e s (to blow dow n.) B ut for ou r p u rp oses h ere , th e a n w e r is c e r ta in ly "a curve." ) It's n a tu r a l to con sid er first o n ly n o n sin g u la r p r o je c tiv e cu rv es; and one fin d s th a t th e se t of iso m o rp h ism cla sses of th e s e c u r v e s b rea k s up b y g en u s in to w e ll-b e h a v e d algeb raic v a r ie tie s , th e "moduli spaces." Of co u rse a n y a lgeb raist w ould r a th e r h a v e a ring th a n a v a r ie t y . Do th e s e c u r v e s g iv e rings? Not im m e d ia te ly . The sim p lest an d m o st a t tr a c t iv e w a y for a ring to co m e from a v a r ie t y is as th e h o m o g e n e o u s co o rd in a te ring of t h a t v a r ie t y in s o m e p r o j e c t i v e e m b e d d in g . T hus to get a ring, one w a n ts n ot o n ly a c u r v e b u t an em b ed d ed c u r v e . The sp ace of em b ed d in gs of a c u r v e is n ot too bad, b u t th e r e is a sim p ler w a y ou t of th is d ilem m a th a n stu d y in g all em b ed din gs: L eavin g asid e th e so -ca lled h y p e r e l l i p t i c c u r v e s, w h ic h a re in a n y ca se q u ite w ell u n d erstood , e v e r y c u r v e co m e s w ith a u n iq u e ly d istin g u ish ed em b ed d in g in p r o je c tiv e sp a ce, called , for ob v io u s re a so n s, th e c a n o n i c a l e m b e d d in g — th a t is, th e r e r e a lly is a d istin g u ish ed c a n o n i c a l r in g , th e h o m o g e n e o u s co o rd in a te ring of th e th e c a n o n ic a lly em b ed d ed c u r v e , co rresp on d in g to ea c h a b str a c t (n o n -h y p e r e llip tic ) n o n sin g u la r c u r v e 2. W ith m o r e so p h istica tio n th e h y p e rellip tic c u r v e s ca n be 2. Other embeddings are interesting too. See for example Eisenbud [transcanonical]. and Martens-Schreyer [1986] for some interesting cases, and Green-Lazarsfeld [1985b] for some general conjectures.

Green’s conjecture: an orientation for algebraists

57

in clu d ed h e r e too, b u t w e w ill n ot w o r r y ab o u t th is point. M oreover, th e ca n o n ica l rings of c u r v e s tu r n o u t to be q u ite sim ple: t h e y a re (p recisely ) th e graded d o m a in s w h ic h a re alg eb ra s o v er a field k and sa tisfy p rop erties a )-d ) from th e la st sectio n . The n u m b e r y in tro d u ced th e r e is g -2 , w h e r e g is th e g en u s of th e c u r v e . The fa ct t h a t th e ca n o n ica l rin gs of c u r v e s a re C o h e n -M a c a u la y is called N oeth er's th e o r e m (see A rbarello et al [1985]). The G orenstein p ro p erty follow s e a sily from th is u sing d u a lity th e o r y . That th e q u a d ra tic p art of th e ideals is sp an n ed b y q u a d rics of ra n k 4 w a s c o n je c tu r e d b y A n d reo tti and M ayer and p roved in g en era l b y Green [1984] (see also S m ith and V a rle y [1989].) Let u s n ow ta k e a look at th e free reso lu tio n of th e ca n o n ica l ring of a c u r v e of g en u s g (as a m od u le o v er th e h o m o g en o u s c o o rd in a te ring of th e p r o je c tiv e sp ace in w h ic h th e c u r v e is c a n o n ic a lly em b ed ded: if th e c u r v e h a s g en u s g, th e n th is is S = k[xo, ... ,x r ], w ith r = g-1.) From th e G orenstein p ro p erty and p ro p erty d) it follow s th a t th e resolu tion h a s b etti d ia g ra m of th e form :

0 0 1 2 3

1

1 al ag-3 — “ -

2

...

g-4

a2

#. . . ••

-

ag-4 -

••• ...

g-3

g-2 -

ag-4 ag-3 a2 al ~ “

1

N ote th e s y m m e t r y b e tw e e n th e ith row and th e ( 3 -i) th ro w , and th a t bj (in our p rev io u s d iagram ) is n ow g iv en as a g_2 -i- Also from th e H ilbert fu n ctio n in d) one com p u tes: ai - a g_ i - i = C Ì 2) - ( g - l - i ) ( f ; |)

for i = 1 ... g -2 ,

so th a t in fa ct all th e b etti n u m b e r s a re k n o w n if w e k now

k

aLg/2j

a g-3-

58

D. Eisenbud

As w e h a v e a lr e a d y m e n tio n e d , th e v a n ish in g of a 4 im p lies t h a t of a i+i. S im ila r ly if a* is 1 a i+i v a n ish e s (probab ly a i+i * 0 a c tu a lly im p lies t h a t aj is r a th e r large.) For sm a ll g and in c e r ta in o th e r c a se s w e k n ow a lot ab ou t th e se q u en ce (*). H ow ever, in g en er a l, w e k n ow little ab o u t w h ic h se q u e n c e s of n u m b e r s (* ) ca n occu r. The sa m e r e m a r k s a re valid for th e free resolu tio n of a n y h o m o g e n e o u s R = S /I s a tisfy in g h y p o th e se s c) an d d) from se ctio n 1 — th e a ssu m p tio n s t h a t R is 2 -d im e n sio n a l, n o rm a l or e v e n a d o m a in , a re ir r e le v a n t. Of co u rse th e p ossib ilities for th e se q u e n c e (* ) m a y w ell ex p an d if w e w e a k e n th e h y p o th e se s in th is w a y . B u t also in th e g en er a l ca se, w e do n ot k n ow w h ic h se q u e n c e s (* ) occu r. It is n ot e v e n k n o w n w h e th e r th e se q u en ce can be 0, ... , 0! In fa ct a leadin g sp ecial ca se of G reen’s c o n je c tu r e (th e so -ca lled “g en eric case") is ju s t this: G e n e r ic G r e e n ’s C o n j e c t u r e 2 . 1: T here e x is ts a h o m o g e n e o u s G orenstein ring (r e sp e c tiv e ly a h o m o g en eo u s G orenstein n o rm a l d o m a in ) as a b o v e w ith

aLg/2j= 0 (and t h u s ai = 0 for i > [g/2].) W e w ill call th e tw o v er sio n s of th is c o n je c tu r e , w ith and w ith o u t th e "norm al dom ain" con d ition , th e stro n g an d w e a k fo rm s, r e sp e c tiv e ly . The stro n g form is k n ow n to hold for g < 17, b y v ir tu e of c o m p u te r w ork (a t le a st for so m e c h a r a c te r istic , and a fortiori for a lm o st all c h a r a c te r is tic s , in clu d in g c h a r a c te r istic 0) u sin g th e a p p roach of W e y m a n d etailed below . It is also k n o w n th a t ea c h of th e c a se s {g odd) an d {g e v e n ) im p lies th e o th e r (w ork of Ein, B a y e r S tillm a n an d W eym an ...). To m a ssa g e Green's c o n je c tu r e in to its u su a l s t a te m e n t , and to r e la te it to C o n jectu re 1 . 3, w e q u o te a (r a th e r e a s y ) re su lt d eriv ed from ou r [19 8 8 , pp. 5 4 9 -5 5 2 ]: T h e o r e m 2 . 2: Let X c IPr be a red u ced irred u cib le lin ea r ly n o r m a l n o n d e g e n e r a te c u r v e . There is a 1 -g e n e r ic p xq m a tr ix of lin ea r fo rm s L w h o se 2 x 2 m in o r s v a n ish on X iff th e h y p e r p la n e

Green’ s conjecture: an orientation for algebraists b u n d le //

k

k ca n be factored as £ i ®

£ 2

w ith h ° £ i > p an d h ° £

59 2

* q*

Here th e co n d itio n s "linearly n o rm a l n on d egen erate" m e a n t h a t th e n a tu r a l m a p H ° ( $ p r (l)) -* H ° (0 x (D ) Is an iso m o rp h ism . R ecall t h a t th e Clifford in d ex of a lin e b u n d le £ on a c u r v e C is b y d efin itio n th e n u m b e r Cliff £ := g+1 - ( h ° ( £ ) + h ° (u > ® £ - 1 )) - deg £ - 2( h ° ( i : ) - l ) and th a t th e Clifford in d ex of C (in ca se th e g en u s of C is > 3) is th e m in im u m of all Clifford in d ices of b u n d les £ w ith h ° £ > 2 and h ° c o ® £ “1 > 2. P u ttin g th is to g eth e r, w e get C o r o lla r y 2 . 3: Let C be a c u r v e of Clifford in d ex c c a n o n ic a lly em b ed d ed in P 8“1. If c' is th e m a x im u m n u m b e r su ch t h a t th e r e is a 1 -g e n e r ic pxq m a tr ix of lin ea r fo rm s L w h o se 2 x 2 m in o r s v a n is h on X an d p+q-3 = c , th e n c = g - 2 -c . / / T hus C o n jectu re 1 . 3 b eco m es, in th e ca se of ca n o n ica l cu rv es: C o n j e c t u r e 2 . 4 (G r e e n ): The len g th of th e 2 -lin e a r p a rt of th e reso lu tio n 7 of th e ca n o n ica l ring S /I of a c u r v e of g en u s g and Clifford in d ex c is 2LP (S /I) = g - 2 -c . E q u iv a le n tly , w ith n o ta tio n as in section 1 a b o v e, w e h a v e b i = .... = bc_ i = 0, b u t bc * 0. In G reen’s te r m in o lo g y , th is sa y s th a t a c u r v e of Clifford in d ex c sa tisfies con d itio n Nc_ i b ut n ot Nc. N ote th a t th e fa ct t h a t th e len g th of th e lin ea r p a rt is a t lea st g - l - c (e q u iv a le n tly bc* 0 ) c o m e s “for free" from T h eo rem s 1 . 2 and = = Cited Item Has B een D eleted s = ; th is "easy h a lf” w a s first p roved b y Green an d L azarsfeld in th e ap p en d ix to G reen’s [1984].

60

D. Eisenbud

The gen eric c u r v e of g en u s g is k n ow n to h a v e Clifford in d ex — in th e se n se th a t th is is th e v a lu e of th e Clifford in d ex ta k e n on b y th e c u r v e s in an open d en se set of th e m o d u li sp ace. Thus th e g en eric form of G reen’s c o n je c tu r e becom es:

L(g-l)/2j

G e n e r ic G r e e n 's C o n j e c t u r e ( g e o m e t r i c v e r s i o n ) 2 . 5: The free reso lu tio n of th e ca n o n ica l ring of a g en eric c u r v e of g en u s g h a s

aLg/2j> -

* a g -3 = °- 0 ........ °-

S in ce th e con d ition in th e con clu sion of th e c o n je c tu r e is a Zariski open con d ition in fa m ilies of c u r v e s, it w ould be th e sa m e to a sse rt t h a t th e r e e x ists a sm o o th c u r v e w h o se ca n o n ica l ring sa tisfie s th is co n clu sio n — or e v e n a locally G orenstein, sm o o th a b le, o n ed im e n sio n a l, c a n o n ic a lly em b ed d ed sc h e m e w ith th is p ro p erty .

3. S p e c ia l c a s e s a n d g e n e r a l a p p r o a c h e s In th is se ctio n w e w ill list th e k n ow n a p p ro a ch es to G reen’s c o n je c tu r e . In ord er to g iv e so m eth in g a b so lu te ly ex p licit, w e s u m m a r iz e an ap p roach d ue to W e y m a n givin g g e n e r a to r s and r e la tio n s for c e r ta in graded m o d u les of fin ite len g th o v er p o ly n o m ia l rin gs w h o se h ilb ert fu n c tio n s d e te r m in e th e d esired b etti n u m b e r s. W e apologize in a d v a n c e to th e ex p er t w h o se fa v o r ite bit w e skipped.

I.

D e g e n e r a t i o n s a n d t h e s t r o n g g e n e r ic c o n j e c t u r e

This se ctio n d escrib es ap p ro a ch es to th e ’’g e n e r ic ” G reen’s c o n je c tu r e (1.2 or 1.5 ab o v e) u sing th e te c h n iq u e of d eg en era tio n s. These a p p ro a ch es to th e (stron g form of th e ) g en eric Green's c o n je c tu r e lead q u ite q u ick ly to exp licit id eals of ca n o n ica l c u r v e s, on e for ea c h g en u s, w h ic h e x p e r im e n ta lly h a v e th e kind of reso lu tio n p red icted b y G reen’s c o n je c tu r e for th e g en eric c u r v e . If on e could p ro v e th a t on e of th e se id eals re a lly h a s th e desired r e so lu tio n , th e n on e w ou ld h a v e a proof of th e g en eric G reen’s c o n je c tu r e , sin ce ea c h of th e se e x a m p le s r e p r e se n ts a " d egen eration ”

Green’s conjecture: an orientation for algebraists

61

of a sm o o th ca n o n ica l c u r v e of g en u s g. T here is e v e n on e v er sio n of th is, co m in g from th e Ribbon ap p roach of B a y e r-E isen b u d , th a t w ould lead to a proof th a t G reen’s c o n je c tu r e holds for a c u r v e of ea ch Clifford index. T here is an in te r e stin g o p p o r tu n ity for an a lg eb ra ist h ere. All th a t is in v o lv e d is to find th e free reso lu tio n of a p a r tic u la r ideal. So far a t le a st, g eo m e tr ic id eas h a v e not b een of m u c h u se. T hough th e p rob lem s in v o lv ed are of th e m o st c o n c r e te and sp ecific kind (specific id eals w h o se m in im a l re so lu tio n s m u s t be ca lcu la ted — s o m e tim e s w ith ex p lic itly k n o w n n o n -m in im a l re so lu tio n s, so th a t th e co m p u ta tio n c o m e s d ow n to d ecidin g th e ran k of an ex p lic itly k n ow n m a tr ix w ith in te g er en tries!), people h a v e n ot w r itte n m u c h up. A. cuspi C u s p id a l r a t i o n a l c u r v e s a n d t h e t a n g e n t d e v e l o p a b le s u r f a c e . (B u ch w e itz, S c h r e y e r , B a y e r -S tillm a n , W e y m a n , Green, Kempf...) The follow in g co n str u c tio n w a s n oted b y B u c h w e itz and S c h r e y e r so m e tim e arou n d 1983: One w a y to get a ca n o n ica l c u r v e of g en u s g is to ta k e a ra tio n a l c u r v e w ith g cu sp s in IP8"1. S u ch c u r v e s tu r n ou t to be th e h y p e r p la n e se ctio n s of an a r ith m e tic a lly C ohenM a c a u la y su r fa c e (a c tu a lly a d eg en er a te K3 su rfa ce , if th e r e a re a n y g e o m e te r s listen in g ) ob tain ed as th e ta n g e n t d ev elo p a b le su r fa c e ( s th e u n ion of th e ta n g e n t lin es) to th e ra tio n a l n o rm a l c u r v e in IPg. This is th e ra tio n a l su rfa ce w ith a ffin e p a r a m e tr iz a tio n A 2 -» A g g iv en b y (x ,y ) »-> (x ,x 2 ,...,xg) + y ( l,2 x ,3 x 2,...,gxg_1) = (x+1, x 2+ 2 x y , ... ). S in ce th e se c u r v e s a re sm o o th a b le, a proof of th e g en eric v er sio n of G reen’s c o n je c tu r e w ould follow if w e could ch eck th e c o n je c tu r e on th e re so lu tio n of a n y on e of th e se c u r v e s, or, e q u iv a le n tly , on th e reso lu tio n of th e ta n g e n t d evelop ab le su rfa ce itself.

62

D. Eisenbud

W e w ill sh o w b elow t h a t if w e ta k e th e ra tio n a l n o rm a l c u r v e to h a v e eq u a tio n s g iv en b y th e m in o r s A a>b of th e m a tr ix Let A a b (0 < a , b < g -1 ) be th e m in o r in v o lv in g th e co lu m n s a an d b of th e m a tr ix f x 0 X! ... x g_ i ] I X! x 2 ... x g J w ith th e u su a l c o n v e n tio n th a t A a b - “A b >a. If w e ta k e th e ra tio n a l n o rm a l c u r v e to be th e c u r v e w ith e q u a tio n s A a>b = 0

(0 < a , b < g -1 ),

th e n w e w ill sh o w b elow t h a t th e q u a d ra tic eq u a tio n s of th e ta n g e n t d evelo p a b le su r fa c e are r a ,b :■ A a+2,b " 2 A a+i fb+i + A a b+2 ■ 0

(0 < a < b < g -3 ).

(For g > 6 th e q u a d ra tic eq u a tio n s g e n e r a te th e ideal of th e ta n g e n t d ev elo p a b le su r fa c e , b u t in a n y ca se o n ly th e q u a d ra tic e q u a tio n s a re in v o lv e d in ch eck in g Green's c o n je c tu r e .) Thus:

To prove the (strong) generic version of Green's co n je ctu re , and indeed to prove the co n je ctu re for generic sm ooth cu rv es, it is enough to show that the 2 -lin ea r part of the m inim al resolution of the ideal J generated by the linear com binations A B,b +2 • 2 A a +i (b+i + A a+2 ,b

0 s a < b s g-3

of the m inors of the m atrix M has length s (or, eq u ivalen tly, - ) Lg/2 j - 1 . W e w ill n o w d eriv e th e se eq u a tio n s for th e ta n g e n t d ev elo p a b le su rfa ce . W e w ill th e n ex h ib it an exp licit n o n -m in im a l free reso lu tio n for th e ideal J in a form d iscovered b y J e r z y W ey m a n . All th is w ill be d one in te r m s of m u ltilin e a r algeb ra and th e r e p r e se n ta tio n th e o r y of SL 2 ; th e rea d er w ith o u t a ta s te for su ch c o n str u c tio n s m a y w ish to skip to th e d escrip tio n of "W eym an's Modules", below :

Green’s conjecture: an orientation for algebraists

63

W e begin w ith an in v a r ia n t d escrip tion of th e id eal of th e r a tio n a l n o r m a l c u r v e . W e th in k of P 1 as P(V ), th e sp a ce of 1 q u o tie n ts of a 2 -d im e n sio n a l v e c to r sp ace V, and w r ite A ut P 1 = S U V ) = SL2 . The a m b ie n t sp ace P g of th e ra tio n a l n o rm a l c u r v e is th e n P(SgV ), w h e r e Sg d en o te s th e gth s y m m e tr ic p o w er fu n c to r , and w e th in k of SgV as th e fo rm s of degree g on IP1. The sp a ce of all q u a d ra tic fo rm s on th is IPg is S 2 (SgV). The r e str ic tio n m a p of th e s e to t h e r a tio n a l n o r m a l c u r v e ( = IP1) is th e n th e n a tu r a l m a p S2SgV -> S2gV, an d th e q u a d r a tic p a rt of th e ideal of th e ra tio n a l n o r m a l c u r v e is th e k ern el of th is m a p . It is c o n v e n ie n t to d en o te a b asis of V b y { l,x }, an d m o r e g e n e r a lly a basis for SgV b y { l,x , ... ,x g), in w h ic h ca se th e m a p S 2 SgV -* S2gV a b ove m a y be w r itte n

xa.xb ^ xa+b In c h a r a c te r is tic 0 th e k ern el is e a s y to d escrib e an d a n a ly z e , as follows: W e h a v e S2SgV = S2gV © S2g_4V © S2g_8V © ... . The q u a d r a tic p a rt of th e ideal of th e ra tio n a l n o rm a l c u r v e itself co n sists of all b u t th e first of th e se su m m a n d s. It is th u s S2Sg- 2V = S2g_4V © S2g_8V © ... , and (as w e sh all sh o w ) th e q u a d ra tic p art of th e ideal of th e ta n g e n t d ev elo p a b le su r fa c e co n sists of all b ut th e first tw o . It is th u s

s 2s g-4v ■ S2g- 8V © S2g_12V © ... . In b oth c a se s th e in clu sion m a p s a re g iv e n b y "inner m u ltip lication " by a := x 2 *l - x*x € S2S2V,

64

D. Eisenbud

w h e r e b y in n e r m u ltip lic a tio n w e m e a n th e e q u iv a r ia n t p airin g S2SaV S2SbV -

S2Sa+bV

g iv e n b y x u ,x v ® x s ,x t -> x u+s-x v4t + x u+t-xv + s. U n fo r tu n a te ly , th is d escrip tion does not w o rk in a r b itr a r y c h a r a c te r istic , so w e adopt a n o th e r , o n ly a little less sim p le w h ic h does (I a m g ra te fu l to J e r z y W e y m a n for d iscu ssio n s of th is m a tte r ): W e ca n id e n tify th e sp ace Q of q u a d ra tic fo rm s in th e ideal of th e r a tio n a l n o r m a l c u r v e w ith th e r e p r e se n ta tio n A 2Sg_ iV (w h ic h is th e sa m e as S2Sg_2V in c h a r a c te r istic 0). To get th e in clu sio n m a p , n o te th a t th e k ern el of S2SgV -* S2gV is sp a n n ed by th e e le m e n ts A a b = x ax b+1 - x a+1x b € S2SgV

for 0 < a < b < g -1 ,

w h ic h a r e th e im a g e s of th e ob viou s b asis v e c to r s x a ^ x b € A 2Sg_1V (th is is h a lf th e im a g e of th e p rod u ct of ( x ® l - l © x ) an d x a ® xb- x b® xa in S gV ® SgV, u n d er th e m a p to S2SgV; th e m a p is t h u s e q u iv a r ia n t.) W e cla im n e x t th a t th e q u a d ra tic p art of th e ideal of th e ta n g e n t d ev elo p a b le su rfa ce is giv en b y th e re p r e se n ta tio n A 2Sg. 3V c A 2Sg_iV . As in th e c h a r a c te r istic 0 d escrip tion , th is m a p ca n be d escrib ed as a n "inner m u ltip lic a tio n w ith th e e le m e n t oc", w h e r e th is tim e in n er m u ltip lic a tio n r e fers to th e pairing A 2SaV ® S2SbV -» A 2Sa+bV

Green’ s conjecture: an orientation for algebraists

65

g iv en b y x u - x v x^ x* •-* x u + s- x v+t + x u + t- x v + s. The im a g e of x a ^ x b in S 2 SgV is n ow ea sily co m p u ted to be r a>b - A a+2>b - 2 A a+i ib+i + A a|b+2

for 0 < a < b < g -3

W e n ow cla im th a t th e q u a d ra tic p art of th e ideal of th e ta n g e n t d evelo p a b le su r fa c e is g en er a ted b y th e se. It is u sefu l to fall back on th e c h a r a c te r istic 0 co m p u ta tio n : sin ce oc*S 2 Sg_ 2 V g e n e r a te s a m a x im a l in v a r ia n t su b sp ace of S 2 SgV, th e sa m e w ill be tr u e of S 2 S g _ 4 in S 2 Sg_ 2 . A d irect c o m p u ta tio n sh o w s th a t th e im a g e of S 2 S g _ 4 in S 2 Sg is th e n sp an n ed b y th e e le m e n ts Fa,b+1 ■ r a+ l,b

for 0 < a < b < g-4

— th is is th e im a g e of x a*xb. B ut o v er Q th e se sp an th e sa m e sp a ce as th e e le m e n ts r a b th e m se lv e s. Thus o v er b sp an a m a x im a l in v a r ia n t su b sp ace of th e ideal of th e ra tio n a l n o rm a l c u r v e , and it su ffic es to sh ow th a t t h e y v a n ish on th e ta n g e n t d evelo p a b le su rfa ce. By in v a r ia n c e it is en ou gh to sh o w th a t th e se fo rm s v a n ish on a single ta n g e n t line to th e ra tio n a l n o rm a l c u r v e . For e x a m p le , th e ta n g e n t lin e at th e point "x^O", th a t is, (1 ,0 , ... ,0), is th e lin e co n sistin g of all e le m e n ts (s, t, 0, ... ,0). This v a n ish in g o ccu rs b eca u se th e o n ly m in o r A u v not v a n ish in g on th is lin e is A o ,i, an d th is on e n e v e r o ccu rs in th e exp ressio n for r a>bW e h a v e n ow sh o w n th a t th e r e p r e se n ta tio n A 2Sg_ 3 V c S 2 SgV d efin es th e q u a d ra tic p art of th e ideal of th e ta n g e n t d ev elo p a b le su r fa c e in c h a r a c te r istic 0. To co m p lete th e proof in all c h a r a c te r istic s it su ffic es to sh ow th a t, if w e ta k e V for a m o m e n t to be Z2, th e q u o tien t S2SgV /{ r a>b | 0 < a < b < g -3 } is torsion free, w h ic h a m o u n ts to finding a m in o r s of th e a p p ro p ria te size in th e in clu sion m a tr ix w h ic h a re r e la tiv e ly p rim e. This is a str a ig h tfo r w a r d c o m p u ta tio n . B eca u se th e ideal J of th e ta n g e n t d evelo p a b le su r fa c e is SL 2 -

66

D. Eisenbud

in v a r ia n t, it is n a tu r a l to t r y to w r ite d ow n a reso lu tio n in te r m s of r e p r e se n ta tio n th eo ry ; for ex a m p le, th e Koszul co h o m o lo g y groups m ig h t be w r itte n in te r m s of r e p r e se n ta tio n s of SL 2 . S e v e r a l people se em to h a v e tried th is w ith o u t su ccess; an in v a r ia n t b ut n o n m in im a l re so lu tio n s a re k n o w n , b u t n ot h o w to m a k e a n y of t h e m m in im a l! One su ch c o n stru ctio n is d etailed in th e p ap er of B a y e r an d S tillm a n e lse w h e r e in th e se proceed in gs. Here is su ch a c o n str u c tio n of a n o n m in im a l re so lu tio n , from W e y m a n 's p r iv a te n o te s (ab ou t 1986). It tu r n s o u t th a t th is is a sp ecial ca se of th e c o n str u c tio n p u rsu ed in W ey m a n 's p aper [1989], (w h e r e th e id eal J of th e ta n g e n t d evelop ab le is called J g- i ) to w h ic h th e re a d e r m a y go for m o re details. Let R be th e p o ly n o m ia l ring k[SgV], and let J be th e id eal of th e ta n g e n t d evelop ab le su rfa ce , as ab ove. W ey m a n 's idea is based on a co m p u ta tio n of th e free resolu tion of th e n o r m a liz a tio n A of R /J , realized b y p u sh in g forw ard th e s tr u c tu r e sh e a f from th e d esin g u la riz a tio n of th e ta n g e n t d evelop ab le su rfa ce. He finds: 1) A d eco m p o ses as S L (V )-m od u le as A = ®d>0 S (g_1)dV ® S dV. 2) The cok ern el C := A /(R /J ) d ecom p oses as C = © d 2l Sdg. 2V ® A 2V. 3) The m in im a l free resolu tion of A o v er R is 0 “ d for t = 0, — 1, 0| : = ( s - Z{ t ) 2 . If A C {0 ..< / — 1}, define polynomials 9a := UaeA(s ~ 2«iz«• ^he matrix II is obtained in a straightforward manner by using $

•

= ( “ I)**

cB\m#U{0}i

for B € A(ft + 1 , l..g - 1 ), mf = bj € 5 , as a basis for the column space, and = ^

Res

2 j4u{0},m

for A € A (ft - 1, 1 ..g - 1 ), m £ A y1 < m < g —1, as a generating set for the row space. Finally, Theorem 3.5 follows immediately: dim Tor/,(5/7, K % + 1 = dim kerd ~

^ jj

■ (’¡ 1) +(*+0 0, the number of associated primes of H j ( M ) is finite.

In this form, the conjecture is completely open. A more plausible conjecture might be, C o n je c t u r e 5.2. Let (i£,m ) be a regular local ring, and let I be an ideal of R.

Then for all j >

finitely many associated primes.

0

H {(R ) has only

104

C. Huneke

The second conjecture has recently been proved in characteristic p in [HuS]. Unfortunately, there does not seem to be any hope of using reduction to characteristic p when it comes to the vanishing of elements in local cohomology. Nonetheless, this result in characteristic p gives hope in all other cases. Another suggestive result was given in the proof of Faltings of Proposition 3.1. One of the key steps in the proof in showing the modules were finitely generated was to show that the number of associated primes was finite by an induction. Of course, after the fact this is true in Faltings’ case since he was showing the modules were actually finitely generated. On the negative side the result of Huneke and Koh (3.5) shows that even Hornr ( R /I , H " ( R ) ) will not be finitely generated for n > bight(J) in a regular local ring. While this does not mean there are infinitely many associated primes, it certainly gives one pause. An interesting problem in this connection, first raised as far as I know by Markus Brodmann, is the following question. Question 5.3. Let R be a noetherian ring and let I and

J be ideals of R.

Fix an integer j > 0. Suppose for each maximal ideal m of R, Jm is contained in the nilradical of the annihilator of H j (Rm). Then is J contained in the nilradical of Hj(R)? m This question would have an immediate positive solution if the local cohomology module had only finitely many associated primes. There has been some progress recently on this question by Raghavan. Another somewhat whimsical question I raised some years ago is the following.

Problems on local cohomology

105

Question 5.4. Let ( R , m) be a noetherian local ring and let x i , . . . x n be any collection of elements of R. Consider the set of all associated primes of all ideals generated by monomials in the Xi. Is this a finite set?

Of course, the well-known result of M. Brodmann saying that the number of associated primes of powers of ideals is a finite set would be an immediate corollary of this question. The only classes of elements for which I know a positive solution axe when the i t- form a regular sequence, or when n = 1, both being rather trivial cases. Very likely it is true if the x,- form a d-sequence, and it might be a good place to start to see if it is true in this case. We close this paper with a discussion of the local cohomology modules of a prime ideal in a complete regular local ring (R, m) of dimension d containing a algebraically closed residue field of characteristic 0 for d < 5.

In case p = m, everything is very well known. All the local cohomology vanishes except at d. If the ht(p) = 1, then p is principal and cd(p,R) = 1. The other extreme is when dim(R/p) = 1. In this case, Hartshome showed that cd(p,R) = d — 1 = ht(p) (recall in general that Hj(R) = 0 for j < grade(J) = ht(I ) and is nonzero if j = ht(I)). If dim(i2/p) = 2, then (2.1) shows that cd(p,R) = d — 2. Putting these remarks together, we see that if dim (ii) < 4, then cd(p,R) = ht(p) for all primes in R. The first interesting case is if ht(p) = 2 and d = 5. In this case (2.3) shows that cd(p, R) < 3. Let H be the third local cohomology with

106

C. Huneke

support in p. Then H q = 0 for all q with ht(q) < 3 by using the Hartshome-Lichtenbaum result mentioned above. W hat about Hq if ht(q) — 4? Since Rq is not complete, we may lose the primeness of p after completion and not be able to apply (2.1)—it is even possible that the punctured spectrum mod p in the completion may not be connected. Thus it is possible that H q ^ 0 for infinitely many primes q of height 4. If there were infinitely many, this would provide a counterexample to the finiteness of associated primes of local cohomology. Essentially it comes down to this question: how many primes q of height 4 have the property that the punctured spectrum of (Rqy/p(RqY is disconnected? Besides the problem with the completion there is also another problem. The residue field k(q) is no longer separably closed. If it were true that H q = 0 for all height 4 ideals (for example in case that R /p is normal, or that R /p has an isolated singularity) then the support of H would be {m}. Since cd(p, R) < 3, (4.1) applies to show that H is at least Artinian. However Raghavan has pointed out that the last nonvanishing local cohomology module is never finitely generated, so H could not be of finite length without being 0. We close this paper with the observation that many of the theorems listed in this paper are known only in the case the ring contains a field. I feel sure all of them remain true in mixed characteristic, and might not even be hard to show given a concerted effort.

Problems on local cohomology

107

Bibliography [B]

M. BRODMANN, Einige Ergebnisse aus der lokalen Kohomologietheorie und Ihre Anwendung, Osnabrucker Schriften

zur Mathematik, 5 (1983). [BH]

M. BRODMANN a n d C. H uneke, A quick proof of the Hartshome-Lichtenbaum vanishing theorem, preprint (1990).

[BR]

M . B rod m a n n AND J. R u n g, Local cohomology and the connectedness dimension in algebraic varieties, Comment.

Math. Helvetici, 61 (1986), pp. 481-490. [CS]

F. W . C a l l AND R . S h arp, A short proof of the local Lichtenbaum-Hartshome theorem on the vanishing of loca1 cohomology, Bull. London Math. Soc., 18 (1986), pp.

[Fl]

261-264. G. FALTINGS, Uber lokale Kohomologiegruppen hoher Ordnung, J. fur d. reine u. angewandte Math., 313 (1980), pp. 43-51.

[F2] ------------------, Uber die Annulatoren lokaler Kohomologiegruppen, Arch. Math., 30 (1978), pp. 473-476. [F3] ------------------, Some theorems about Formal Functions, Publ. RIMS Kyoto Univ., 16 (1980), pp. 721-737. [Gl] A. GROTHENDIECK, Local Cohomology, notes by R. Hartshome, Lecture Notes in Math., 862, Springer-Verlag, 1966. [G2] ----------------------, Cohomologie locale des faisceaux et theoremes de Lefshetz locaux et globaux (SGA 2), Amsterdam: North Holland Publ. Co. (1969). [H l]

R . HARTSHORNE, Cohomological dimension of algebraic

varieties, Annals of Math., 88 (1968), pp. 403-450.

[H2] ----------------------, Aßine duality and coßniteness, Inventiones Math., 9 (1970), pp. 145-164.

108

C. Huneke

[HS

R. H a r t s h o r n e a n d R. S p eiser, Local cohomological dimension in characteristic p, Annals of Math., 105

(1977), pp. 45-79. [HuK

J . K oh, CoSniteness and vanishing of local cohomology, preprint (1990).

[HuL

C. H u n ek e a n d G. L yubeznik, On the vanishing of local cohomology, Inventiones Math., 102 (1990), pp. 73-93.

[HuS

C. H u n ek e a n d R. S h a rp , in preparation.

[Ly

C.

G.

HUNEKE AND

LYUBEZNIK, Some Algebraic Sets of High Local Co-

homological Dimension in Projective Space, Proc. Amer.

Math. Soc., 95 (1985), pp. 9-10. [O

[PS

A.

OGUS, Local cohomological dimension of algebraic va-

rieties, Annals of Math., 98 (1973), pp. 327-365. C . PESKINE AND L . S z p i r o , Dimension Projective Finie

et Cohomologie Locale, I.H.E.S., 42 (1973), pp. 323-365.

[R

K. N. R agh ava n , developing thesis, Purdue Univ. (1991).

[SI

R . SPEISER, Cohomological dimension of non-complete

hypersurfaces, Invent. Math., 21 (1973), pp. 143-150.

[S2

--------------- , Cohomological dimension and abelian varieties, Amer. J. Math., 95 (1973), pp. 1-34.

[S3

--------------- , Projective varieties of low codimension in characteristic p > 0, Trans. Amer. Math. Soc., 240 (1978), pp. 329-343.

Recent W ork on Cremona Transformations Sheldon Katz Oklahoma State University Stillwater, OK 74078 This survey article contains a very biased treatment of work done in the area of Cremona transformations during the past decade or so. As in classical times, this field is still rich in special cases with pretty descriptions, and divides into very different topics— no general theory has yet emerged. I have merely chosen the subjects that are most interesting to me. Most of these topics will be treated very briefly. A little more time will be taken up by a discussion of special Cremona transformations, i.e. those whose base scheme Y is smooth and irreducible. The connection to the theme of this conference is that certain rational mappings can be demonstrated to be Cremona transformations by using the syzygy matrix of the forms defining the mapping. This idea is due to Schreyer. I have refrained from considering the more general subject of birational mappings between arbitrary varieties, especially in dimension 3. One instance of this is recent work extending the Iskovskikh-Manin approach to non-rationality of varieties (i.e. showing that there are few birational automorphisms). I will settle for merely listing two references here, the original Iskovskikh-Manin paper [IM] and its extension by Pukhlikov [P]. Another interesting topic not considered here is the work on Hanamura [H1,H2] which constructs the scheme Bir(X) of birational automorphisms of X , and shows that it behaves nicely if X is not uniruled. This uses some techniques related to the minimal model program (MMP). To my knowledge, the MMP has never been applied to the study of Cremona transformations. I. Generalities Definition. A Cremona transformation is a birational mapping from P r to P r. The set of all Cremona transformations of P r forms a group, denoted

C rr.

Let $ = (/o, •••, f r) ' P r ►P r denote a Cremona transformation, with fi forms of degree non P r. The base locus X of $ is the scheme of zeros of the /, . We can blow up X (or more conveniently, perform a sequence of blow ups with smooth centers) to get a variety P r and a morphism

DOI: 10.1201/9781003420187-11

109

110

S. Katz

$ : P r —►P r such that the following diagram commutes. E

C

| ir

| X

Pr

C

Pr

\ $

-

Pr

Here, E = UEi denotes the exceptional divisor, with irreducible components E i. Let H denote the hyperplane class on P r. Then H = m t *H — Em,-Ei for some m,- > 0 . Lemma

1

[CK1] dim\mr*H —

= r, i.e. $ is defined by a complete

linear series.

Proof: If the conclusion of the lemma were false, then $ would factor through a projection P r —►P r, with r' > r. The image of the factorization map must have degree 1, hence be linear. But mappings defined by complete linear systems are nondegenerate, a contradiction. Q.E.D. The moral is that alternatively, one can describe Cremona transformations by specifying a base locus X , degree n, and multiplicities mt-. The geometry of X then translates to information about the structure of For instance, if C C P r is a curve whose proper transform C C P r satisfies C (m r*H —ErriiEi) = 0, then $ (C ) is a point. The nicest case is when $ is a blow down mapping (it is not true in dimension > 3 that every birational mapping is the composition of blow ups and blow downs; counterexamples appear in [H], [O], and [C]). Two elementary examples come to mind. 1. The quadratic transformation $ = (^ i« 2 > # 2 ^0 ) ^o^i) *• P 2 P 2. Here X = {p x, p2, P3 } = {(1,0,0), (0,1,0), (0,0,1)}. and

► =

2w*H — E i — E 2 — E$ if Lij is the line joining pi and pj, then Lij =

7r*H —Ei —E j . Since °

of degree m over P 2.

/

Let L fm n = Spec I ^ ^ O p l x p 1(—km i —kn) 1 bethe total space of the

.

W °

line bundle of bidegree (m, n) over P 1 x P 1. Let F ^ n = Spec [ S y m { O p 1 (—n)

O p 1(—n)^ be the total space of

the vector bundle O pi(m ) 0 O p 1(n) over P 1. E'Jn denotes a particular A 1 bundle over for details. T heorem

6

/

depending on /. See [U4]

(I) Let G be a connected algebraic group in C r 3 . Then G is

contained in the conjugacy class of one of the following algebraic operations:

(PI) (PGL4, P 3),

(P 2) (PSO5, quadric C P4). (El) (PGL2, PGL2/T), T is (E 2) (PGL2, PGL2/r), T is

(Jl) (J2) (J3) (J4) (J 5) (J 6 )

an octahedral subgroup of PGL2. an icosahedral subgroup 0/PGL2.

(PGL3 x PGL2, P 2 x P 1). (PGL 2 x PGL 2 x PGL2, P 1 x P 1 x P 1). (PGL2 x Aut°FJn, P 1 x F'm) where m isan integer > 2. (PGL 3 , PGL3 /B ) where B is a Borel subgroup of PGL3 . (PGL2, PGL2/D2n) where n is an integer > 4 . (G ,G /# m>„ ) where G = G m x SL2 x SL2

»■n, = {(« ■ ( o' ^ ) •( o’

))

6

and m,n are integers with m > 2 , — 2 > n. (J7) (Aut°JJn, J^) where m is an integer m > 2. (J8 ) (Aut°L^ n L^ n) where vn^n are integers with m > n > 1. (J9) (Aut°FJn n, n) where m,n are integers with m > n > 2. (J10) (Aut°FJn m, FJ„ m) where m is an integer > 2.

118

S. Katz (J 1 1 ) (Aut°E^, E^) where l,m are integers with m > 2, / > 2 or m = 1, 1 > 3. (J 12) Generically intransitive operation (P G L i, X r ) with general orbits isomorphic to (PGL2, PGL2/Gm), where ir : C\ —► C 2 is an étale 2-covering of a rational curve C 2 with genus

(Ci) > 1- (These operations are effectively parametrized by the

moduli space o f nonsingular elliptic or hyperelliptic curves of genus > 1).

(II) The rThe (conjugacy classes of) algebraic subgroups of O det 3 ermined by the above operations (PI), (P 2), (El), (E 2), (Jl), ••• ,(J12) are maximal (conjugacy classes o f ) algebraic subgroups o fC r 3.

References [C] B. Crauder. Birational morphisms of smooth threefolds collapsing three surfaces to a point. Duke Math. J. 48 (1981) 589-632. [CK1] B. Crauder and S. Katz. Cremona transformations with smooth irreducible fundamental locus. Amer. J. Math. I l l (1989) 289-309. [CK2] B. Crauder and S. Katz. Cremona transformations and Hartshorne’s conjecture. Amer. J. Math. 113 (1991) 269-285. [ESB] L. Ein and N. Shepherd-Barron. Some special Cremona transformations. Amer. J. Math. I l l (1989) 783-800. [G] M.H. Gizatullin. Defining relations for the Cremona group of the plane. Math. USSR-Izv. 21 (1983) 211-268. [H] H. Hironaka. An example of a non-Kahlerian complex-analytic deformation of Kahlerian complex structures. Ann. Math. 75 (1962) 190-208. [HI] M. Hanamura. On the birational automorphism groups of algebraic varieties. Comp. Math. 63 (1987) 123-142. [H2] M. Hanamura. Structure of birational automorphism groups, I: nonuniruled varieties. Inv. Math. 93 (1988) 383-403. [HKS] K. Hulek, S. Katz, and F.-O. Schreyer. Cremona transformations and syzygies. Math. Z., to appear. [I] V.A. Iskovskikh. Proof of a theorem on relations in the two-dimensional Cremona group. Russian Math Surveys 40 (1985) 231-232.

Recent work on Cremona transformations

119

[IM] V.A. Iskovskikh and Yu.I. Manin. Three-dimensional quartics and counterexamples to the Liiroth problem. Math. USSR Sbornic 15 (1971) 141-165. [K] S. Katz. The cubo-cubic transformation of P 3 is very special. Math. Z. 195 (1987) 255-257. [O] T. Oda. Convex Bodies and Algebraic Geometry. Introduction to the Theory of Toric Varieties. Springer-Verlag. Berlin-Heidelberg-New York 1988.

[p] A.V. Pukhlikov. Birational isomorphisms of four-dimensional quintics. Inv. Math. 87 (1987) 303-329.

[ST1] J.G. Semple and J.A. Tyrrell. The Cremona transformation of Se by quadrics through a normal elliptic septimic scroll XR 7. Mathematika 16 (1969) 88-97. [ST2] J.G. Semple and J.A. Tyrrell. The T 2,4 of Se defined by a rational surface 3F S. Proc. London Math. Soc. (3) 2 0 (1970) 205-221. [U l] H. Umemura. Sur les sous-groupes algebriques primitifs du groupe de Cremona a trois variables. Nagoya Math. J. 79 (1980) 47-67. [U 2 ] H. Umemura. Maximal algebraic subgroups of the Cremona group. Nagoya Math. J. 87 (1982) 59-78. [U3] H. Umemura. On the maximal connected algebraic subgroups of the Cremona group I. Nagoya Math. J. 8 8 (1982) 213-246. [U4] H. Umemura. On the maximal connected algebraic subgroups of the Cremona group II. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), 349-436. Adv. Stud. Pure Math. 6 North-Holland, Amsterdam-New York 1985.

The Homological Conjectures Paul C. Roberts In 1975 Hochster [12] published a monograph in which he discussed a group of conjectures, often referred to as the “Homological Conjectures” , since they are all related in one way or another to homological algebra, and, in particular, to properties of modules of finite projective dimension. Since th at time there has been progress in several directions; some of the conjectures have been proven, some shown to be false, and many new conjectures have been added to the list. We do not intend to give an exhaustive list here of all the developments in the field, but rather to summarize the present state of research on these conjectures, discussing which of these questions are still open and the simplest cases for which they are not known. We begin by reproducing Hochster’s diagram: The Hom ological Conjectures (Hochster 1975)

For complete statem ents of all of these conjectures we refer to Hochster [12]. We will discuss most of them here, and will refer to their number in the above diagram by a number in parentheses. We collect these conjectures into groups of similar problems. The outline is as follows:

DOI: 10.1201/9781003420187-12

121

122

P. Roberts

1. Rigidity (1). 2. Intersection (2), (3), (4), (5). 3. Cohen-Macaulay modules ( 6 ), (7). 4. Monomial (10), (11). 5. Multiplicities (8 ), (9). 6.

Codimension (12), (13).

1. R ig id ity . We first discuss the Rigidity Conjecture, partly because it is at the top of the diagram, and partly because so little progress has been made th at it will not take long to deal with it. This conjecture states: C o n je c tu re 1 (R ig id ity (1 )). Let M be a module of finite projective dimension and let N be any module. If T or^{M ,N ) = 0, then T o rf(M ,N ) = 0 for all i > 0. This conjecture was one of a number of conjectures coming from the work of Serre [28]. The idea behind all of these conjectures comes from the method of reduction to the diagonal for complete equicharacteristic regular local rings, by which arbitrary modules of the form The rThe Nr ) = 0 are computed using Koszul complexes. If K mis a Koszul complex, then, for any finitely generated module M , if H i ( K 0 M ) = 0, we have H i ( K mM ) = 0 for a li i > 0, and hence (as shown by Serre) the rigidity property holds for equicharact eristic regular local rings. It was proven by Lichtenbaum [17] for general regular local rings. However, very little progress has been made on this conjecture for non-regular rings.

2

. I n te rs e c tio n C o n je c tu re s.

These conjectures ((2) through (5)) derive from the Intersection Theorem of Peskine and Szpiro [18],[19] and its consequences, and are now known in general (Peskine-Szpiro [19] for rings of positive characteristic or essentially of finite type over a field, Hochster [1 1 ] for the general equicharacteristic case, Roberts [25],[26] for the mixed characteristic case). We state a newer version of the basic theorem:

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Theorem (N ew Intersection Theorem : A newer version o f (3 )). Let F . =

0 —►.Ffc —►. . . —►F0 —►0 be a non-exact complex of free A-modules with homology of finite length. Then k > dim(A).

There are various refinements of this theorem which are still conjectural. In one direcrfree module tion, this theorem can be interpreted as a lower bound on the rank of th The e in a complex Fmsatisfying the hypotheses of the thoerem and with Ho(F*) ^ 0 (th at is, Fd ^ 0, so the lower bound on the rank is 1). There are other conjectural lower bounds for the ranks of the other free modules in such a complex which are open even for regular local rings; we refer to the article on Betti numbers by Charamboulos and Evans [3] in these proceedings for a discussion of these questions. Another stronger version of the Intersection Theorem is: Conjecture 2 (Im proved N ew Intersection Conjecture). Let F , =

0 —> Fk —►. . . —►F q —►0 be a non-exact complex of free A-modules with homology of finite length except possibly for H o ( F 9 ).

Assume there is a minimal generator of Ho(Fm) annihilated by a power of the

maximal ideal of A. Then k > dim(A).

This conjecture was introduced by Evans and Griffith [6 ] as a lemma in the proof of their Syzygy Theorem. It is known for equicharacteristic rings, where it can be proven by reduction to positive characteristic and use of the Frobenius map. However, the technique of using the local Chem characters of Baum, Fulton and MacPherson [2], which was used in the proof of the New Intersection Theorem in mixed characteristic, does not appear to extend to the improved version, and this question is still open in mixed characteristic. We will return to this conjecture in the section on the Monomial Conjecture, since results of Hochster [13] and D u tta [4] have shown th at it is equivalent to the Monomial Conjecture.

124 3.

P. Roberts C ohen-M acaulay M odules.

Conjecture 3 (Sm all Cohen-M acaulay m odules (6 )). Let A b e a complete domain of dimension d. Then there exists a finitely generated A-module of depth d.

Such a module can be constructed easily in dimension 2 by talcing the integral closure of A. This conjecture is open in dimension three and higher in every characteristic. There does not appear to be a natural way of constructing such modules, but, on the other hand, there does not seem to be a way of showing th at no such modules exist over a given ring. We remark th at since A is assumed to be a complete domain, we can find a regular local subring over which it is a finite module, and, since a small Cohen-Macaulay module would be free over R , the existence of a small Cohen-Macaulay module is equivalent to the existence of an .R-algebra homomorphism from A to a ring of matrices over R. It is easy to construct examples where no such homomorphism exists in certain cases when A is a finite extension of a ring R which is not regular. Conjecture 4 (B ig C ohen-M acaulay modules (7 )).

Let A be a local ring of

dimension d, and let x \ . X d be a system of parameters for A. Then there exists an Amodule M (not necessarily finitely generated) such that ( x i , . . . Xd)M ^ M and such that x \ , . . . Xd is a regular sequence on M .

A module M satisfying the conclusion of this conjecture is called a big Cohen-Macaulay module. Big Cohen-Macaulay modules were introduced by Hochster [1 2 ], who showed that their existence implies many other conjectures, as shown in the diagram reproduced in the introduction. He proved th at big Cohen-Macaulay modules do exist for equicharacteristic local rings. Griffith [10] has extended this and shown th at (still for equicharacteristic rings) if A is a finite extension of a regular local ring i?, then a non-zero A-module exists which is free over R. This conjecture is open in mixed characteristic for dimension 3 and higher, and it appears to be of roughly the same level of difficulty as the Monomial Conjecture, although it is somewhat stronger. In a slightly different direction, one can define the notion of “Cohen-Macaulay complexes” in such a way th a t their existence has most of the same implications as the existence of big Cohen-Macaulay modules. The existence of Cohen-Macaulay complexes can be shown over the complex numbers by analytic methods (Roberts [22]). Recently Hochster and Huneke [15] have shown th at in positive characteristic, the integral closure of a domain in the algebraic closure of its quotient field is Cohen-Macaulay in this sense. This shows the existence of a big Cohen-Macaulay algebra (in the equicharacteristic case), and it also provides a natural construction of one for rings of positive

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characteristic. Using this construction many of the consequences of the existence of big Cohen-Macaulay modules can be strengthened or their proofs simplified. 4.

T he M onom ial Conjecture.

The conjectures discussed here are all known in the equicharacteristic case and unknown in mixed characteristic. Conjecture 5 (M onom ial Conjecture (1 1 )). Let x

\

b

eters for A. Then x\x\ . . . x fd is not in the ideal generated byThe

ebe a system r . . . x^+1.

of param-

The im portance of this conjecture was shown by Hochster [13]. The crucial situation for this conjecture, as shown by Hochster, is when A is a finite extension of a complete regular local ring R and the system of param eters is a regular system of param eters for R. Furthermore, (and this is not so obvious) R can be assumed to be unramified and thus a power series ring over Zp. The (equivalent) Direct Summand Conjecture (10) states th at R is a direct summand of A in this situation. This conjecture is trivially true when the degree of the extension is prime to p by using the trace map; apart from this, the strongest general result, due to Koh [16], states th at the result is true for Galois extensions of degree p (and some extensions of degree p2). It has been shown (Hochster[13], Dutta[4]) that the Monomial Conjecture is equivalent to the “Canonical Element” conjecture and to the Improved New Intersection Conjecture mentioned above. The Canonical Element conjecture can be stated in several ways (see Hochster [13] ). We give one version here. Conjecture 6 (Canonical Element Conjecture).

Let x \ b ebe a system of

;parameters for A. Let K m be the Koszul complex on x\ ,...X d , let Fm be a minimal free

resolution of A /( x \ , . .. Xd), and let m: K • —►Fmbe a map of complexes lifting the identity map in degrees 0 and 1. Then d{Kd) 2 m^d (where m is the maximal ideal of A .)

We describe one recent approach to the Monomial Conjecture in dimension 3, an approach which also shows the relationship with some of the other conjectures. Denote the system of param eters X, Y, Z (when the ring has mixed characteristic p, we will let Z denote p.) Let R be a regular subring of A such th at X ,Y ,Z is a regular sequence of param eters for R. Let I be the ideal generated by X t+1, y i+1, Z t+1, X tY tZ t, and let be denote the free resolution of R / I over the regular subring, tensored with A. The Monomial Conjecture states th at there is no element of the form (a, 6 , c, 1 ) in the kernel of jFi —►F q .

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C o n je c tu re 7. If A is a local ring of dimension d and 0 — ►Fd —►. . . —►F 0 —►0

is a complex of free modules with homology of finite length, then the module of cycles is integral over the module of boundaries in every degree > 0 .

In the case under consideration, the element ( a , 6 ,c, 1 ) could not possibly be integral over the boundaries, since the boundaries have entries in the maximal ideal. Evidence for this conjecture is th a t it holds in the equicharacteristic case via tight closure (see Hochster and Huneke [14]); it has also been proven by Rees [21] in the case of the Koszul complex on a system of parameters. Examination of this question leads to the study of the algebra generated by the boundaries in the symmetric algebra on F\. Let this symmetric algebra be A[5, T, U, V], where the indeterm inates correspond to the generators of the ideal I. Let D denote the derivation m + x 'Y 'z '-& If there exists an extension of R over which an element of the kernel of D exists of the form aS + bT + cU -f V (this is what a counterexample to the Monomial conjecture looks like in this form), its norm would be in the kernel of the derivation D and have coefficients in R with V n coefficient 1 . Conversely, such an element of R[S, T, U, V] will give rise to a counterexample if it has a linear factor over some extension of the quotient field of R. Since the existence of a linear factor in an extension field can be expressed by the vanishing of equations on the coefficients, this makes the whole problem one of the existence of solutions to equations (albeit complicated ones) in the regular ring i2[S, T, U, V] without reference to a specific extension. Finally, we note th at if there is an element (a, 6 , c, 1) of F\ which goes to zero in Fq , then the truncated complex 0 —* 2*3 —►F 2 —> F\ —t►0 is a counterexample to the Improved New Intersection Conjecture, since this element defines a minimal generator of the homology at JFi which is annihilated by a power of the maximal ideal. Thus, since the Monomial Conjecture implies this conjecture, if there is a counterexample to the Improved New Intersection Conjecture (in dimension 3), then this particular complex must be a counterexample for some extension of R.

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5. M u ltip lic ity C o n je c tu re s. These conjectures include the conjectures of Serre on intersection multiplicities defined by Euler characteristics over regular local rings and various generalizations of them. Like the Rigidity Conjecture, the multiplicity conjectures generalize properties of Koszul complexes. We first state them in a general form: C o n je c tu re

8

(M u ltip lic itie s (( 8 ) fo r re g u la r rin g s, (9 ) fo r a r b i tr a r y rin g s)).

Let M be a module of finite projective dimension, and let N be a module such that M is a module of finite length. Let x(M , iV) = J3 (—l)*length(Torf (M , N )). Then we have

(Mo) dim(M)+dim(iVr) < dim(A). (M i) (V an ish in g ) If dim (M )+dim (iV) < dim(A), then x(M , JV) = 0. (M 2 ) (P o s itiv ity ) If dim(M)+dim(iV’) = dim(A), then

be

N ) > 0.

The first of these three statem ents was proven by Serre for regular rings. It is also true if M has projective dimension 1 by using the determinant and Krull’s Principal ideal theorem. It is not known in general for non-regular rings in any characteristic. The vanishing conjecture was proven by Serre [28] in the equicharacteristic regular case, and by Roberts [23] and Gillet-Soulé [8 ], [9] in the general regular case. In addition, whenever both modules have finite projective dimension and the Euler characteristic is closely related to local Chem characters (see Roberts [24]) vanishing is known; this includes complete intersections and any ring with singularity locus of dimension at most 1. We note th at since Mi is now known for regular rings, the diagonal arrow from small CM modules in the diagram in the introduction can end simply at M 2 , so th a t the existence of small Cohen-Macaulay modules implies the Serre Positivity Conjecture. The entire Multiplicity Conjecture has been proven for graded modules over graded rings by Peskine and Szpiro [20]. Conjectures Mi and M 2 are also true for general local rings if N has dimension 1 (this was shown by Foxby [7]). In the case in which only one module has finite projective dimension, Mi and M 2 are false, as shown by the example of D utta, Hochster, and McLaughlin [5]. They construct an example where A = k[[X , F, Z, W ^ /{ X Y — Z W ), where M is a module of finite length (its length is 15) and finite projective dimension, and where x(A/,7V) = —1 for N = A / ( X , Z). Thus vanishing, and hence also positivity, is false in this generality.

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P. Roberts

The positivity conjecture was proven for equicharacteristic regular rings by Serre. For mixed characteristic regular rings it is still open. We remark on one partial result in this case. Let A be a regular local ring, and let P and Q be prime ideals such th at dim ( A / P ) + dim {A IQ) = dim(A) and such th at P + Q is primary to the maximal ideal. The usual reduction, using the fact th at all modules have finite projective dimension, shows that this case implies the general case when the ring is regular. If the subvarieties defined by P and Q are not tangent, then the intersection multiplicity is simply the product of the multiplicities of A / P and A / Q, which is positive (Tennison [29]). In general there is another term, which should itself be positive if the two subvarieties are tangent at the point. In the non-regular case, it is seen from the example of D utta, Höchster, and McLaughlin mentioned above th at positivity is false if only one of the modules is required to have finite projective dimension. If we require both modules to have finite projective dimension, the question is open. On the other hand, an example (see Roberts [27]) shows th a t positivity is not so natural in this case. There exists a (non-regular) ring A of dimension 4 and two complexes Fmand G* of free A modules with the following properties: first, the supports of the complexes have dimension two and intersect only at the closed point. Second, for every ideal P of height two, the Euler characteristic x ((*F«)p ) is non-negative, and for one such P it is positive, and the same holds for G *. On the other hand, x(-^» ® &•) < 0. We remark th at for regular local rings, if positivity as stated in M 2 above holds, then no example with these properties can exist. However, there is no known example of the failure of the Positivity Conjecture in which Fmand G mare resolutions of modules. 6.

G rade (C odim ension) Conjectures.

I would like to thank Hans-Bjom Foxby for many useful suggestions in preparing this section. The main conjecture, due to M. Ausländer, states the following: C o n je c tu re 9 (C o d im e n sio n (1 3 )). Let M be a module of finite projective dimension. Then grade(M) = dim(A)— dim(M).

We recall th at the grade (also sometimes called codimension, which gave the name to this conjecture) of a module M is the maximal length of a regular sequence contained in the annihilator of M . Equivalently, the grade of M is the smallest integer i such that E x t A) ^ 0. Thus if I is the annihilator of M , this conjecture states th a t there is a

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regular sequence of length dim (A)—dim (M ) in I. If M has finite length, this result states th at A is Cohen-Macaulay, which is a special case of the New Intersection Theorem. In fact, more can be deduced from the Intersection Theorem: suppose there is a maximal regular sequence for A contained in the annihilator of M (so th at after dividing by this sequence there is an element annihilated by the maximal ideal m of A). Let k be the dimension of M , let y i , . . . y* be a sequence of elements of m such th a t M / ( y i , . . . y*) has finite length, and let B = A / ( y i , . . . y*). Let r be grade of M ; under our assumption, r is also the depth of A. By the Auslander-Buchsbaum equality the projective dimension of M is less than or equal to r. If F . is a projective resolution of M , then tensoring Fmwith B gives a complex satisfying the hypotheses of the New Intersection Theorem, so we have r > dim (B) = dim(A) — k. Thus we have grade(M ) + dim(Af) > dim(A), so the Grade Conjecture holds in this case. Now let M be any module of finite projective dimension and let r denote the grade of M . Let k = dim(A) —dim (M ). Suppose th at for each minimal prime ideal in the support

of M we have height(P) > k. If be . . . x r is a maximal regular sequence in the annihilator of M , we can find an associated prime ideal P of A / ( x i , . . . x r ) containing the annihilator of M; localizing at P , we are in the situation where x \ , . . . x r is a maximal regular sequence for A p, so, by the above discussion, we have r = dim (Ap) —dim (M p) > k. Thus the Grade Conjecture holds. It follows from this discussion th at the Grade Conjecture is really a question on the height of minimal prime ideals in the support of a module of finite projective dimension. By completing, we can assume th at the ring A is catenary. If A is also equidimensional, then the Grade Conjecture holds for A-modules of finite projective dimension. Peskine and Szpiro [20] showed th at if vanishing holds (Mi above in the section on multiplicity conjectures), then the codimension conjecture holds; using this result, they proved th a t the Grade Conjecture holds for graded modules over graded rings. While Mi does not hold in general, it does in codimensions 0 (this case is quite easy) and 1 , and thus the codimension conjecture holds in these cases; it is sometimes possible to show th at the multiplicity conjectures hold in a specific case, so th at the Grade Conjecture also holds in th at case. Putting these results together, one can deduce th at the lowest unknown case is when A has dimension 4 and depth 3, and M has grade 2, dimension 1 , and depth 0. The conjecture is unknown for cyclic modules in this case. In case M is not only a module but and A-algebra (still of finite homological dimension), there is another version of the Grade Conjecture due to Avramov and Foxby [1 ]. For a local ring 5 , we define the Cohen-Macaulay defect (abbreviated cmd) of 5 as follows: cmd(S) = dim(S') —depth(5).

130

P. Roberts

If we define the amplitude of a complex to be the difference between the degrees of the highest and lowest non-vanishing homology of the complex, then the Cohen-Macaulay defect of A is the amplitude of a dualizing complex for A. If : A —►5 is a ring homomorphism such th a t S is a finite A-module, one can define the dualizing complex of the homomorphism to be RHom(S, A); we then define the Cohen-Macaulay defect of to be the amplitude of this complex. P ro p o s itio n .

The Grade Conjecture holds for all module finite algebras S over A of

finite projective dimension if and only if for all such extensions we have the equality cmd ht I. a) An s-residual intersection of I is an il-ideal J such that ht J > s and J = a : I for some 5-generated H-i leal a properly contained in I. b) A geometric s-residual intersection of I is an 5-residual intersection J of I such that ht(I + J) > s + 1. Notice that if I is an unmixed ideal of height g in a Gorenstein ring R , then ^-residual intersection of I simply corresponds to linkage and geometric ^-residual intersection corresponds to geometric linkage ([15]). Another situation where residual intersection occurs naturally can be described as follows: Let a be an ideal in a Noetherian ring R with s = bight a = max{dim R? | p minimal prime of a} and assume that a can be generated by s elements; now consider a primary decomposition a = qi fl • • • n qr fl Q\ n • • • Qt with qi yjQ~j and ht Qj = 5 ; then J = Qi fl • • • fl Qt is a geometric 5-residual intersection of I = q\ fl • • • D qT. It goes without saying however, that many of the technical difficulties in dealing with residual intersections arise in the “embedded” case where the ideal I is contained in some associated prime of the residual intersection J. For further examples we refer to [6] and [9]. *Supported in part by the NSF

DOI: 10.1201/9781003420187-13

133

134

B. Ulrich

Cohen-Macaulayness In their paper [1], Artin and Nagata dealt with the question of when a residual intersection is Cohen-Macaulay. One of their main results however was not quite correct. This was noticed by C. Huneke who was also able to give a correct answer by adding the assumption that the ideal I is strongly Cohen-Macaulay ([6]). Recall that I is said to be strongly Cohen-Macaulay if all Koszul homology modules of some (and hence every) generating set of I are Cohen-Macaulay modules ([6]). According to Huneke’s result in [6], J is a Cohen-Macaulay ideal of height s , provided that R is a local Cohen-Macaulay ring, J is a geometric 5-residual intersection of / , and I is a strongly Cohen-Macaulay #-ideal satisfying Gs. Here, following [1], one says that I satisfies G a if the number of generators fi(Ip) is at most dim Rp for all prime ideals p with I C p and dim Rp < 5 — 1 (and I satisfies G if I is Ga for all 5 ). Later, Herzog, Vasconcelos, and Villarreal replaced the assumption of strong Cohen-Macaulayness by the weaker “sliding depth” condition, but they also showed that this assumption cannot be weakened any further ([5]). On the other hand, the condition Gs can be weakened; this is done in the following result, which also includes the case of non-geometric residual intersections: T h eo r em 2 ([8]). Let R be a local Gorenstein ring, and let I be an R- ideal that is linked in an even number of steps to a strongly Cohen-Macaulay R- ideal K satisfying Goo (e-g-, let I be an R-ideal in the linkage class of a complete intersection). Consider an s-residual intersection J = a : I of I (where as before fi(a) < s and s > g = ht I). Then J is a Cohen-Macaulay ideal of height 5 , depth R/a = dim R —5 , and the canonical module of R /J is the symmetric power $ 8 - 9 + l(J/ ht 7. There are some results concerning the generation of J if s < ht 7 + 1, and these yield satisfactory answers in case 7 is Gorenstein ([11]). The generators of J are also known if s is arbitrary, but 7 is a special type of ideal, namely a perfect ideal of grade 2 ([6]), a perfect Gorenstein ideal of grade 3 ([12]), or a complete intersection ([8]). As far as resolutions are concerned, it is not even known in the setting of Theorem 2, if a residual intersection of a perfect ideal is again perfect. However if R is regular, then the last Betti number of R /J can be read from the description of the canonical module in Theorem 2 (this is also true for the last graded Betti numbers in the graded case, cf. [13]). Furthermore, the resolution of any residual intersection has been constructed in case 7 is perfect of grade 2 ([4]), perfect Gorenstein of grade 3 ([12]), or a complete intersection ([2]). If the residual intersection is sufficiently general, then R /J is normal and its divisor class group has been computed ([9]). For each of the above three types of ideals 7 and for every sufficiently general residual intersection J of 7, there is an explicitly known family of complexes that resolve about “half” the modules comprising the divisor class group of R /J ([3], [12], [10]). . It would be desirable to add more special types of ideals 7 to this list, because even for relatively simple ideals 7, the residual intersections and their resolutions tend to provide interesting new classes of examples. Apart from that however, more general information about the resolutions of arbitrary residual intersections is needed, and although an analogue of Ferrand’s mapping cone construction may be too much to hope for, it should still be possible to estimate the Betti numbers and predict the degrees in the resolution of a residual intersection.

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Codimension It is interesting to find good upper bounds for the height of colon ideals, since this - among other things - allows one to show by induction on the dimension that two given ideals are equal. In fact it is known that an 5-residual intersection is unmixed of height exactly s if I satisfies the assumptions of Theorem 2 (by Theorem 2), if R is Cohen-Macaulay and s = ht I (trivial), or if R is Gorenstein, I is Cohen-Macaulay and generically a complete intersection, and s < ht I + 1 ([11]). Furthermore one has the following observation (which was independently noticed by C. Huneke): P r o p o sitio n 3. Let R be a Noetherian local ring which is quasi-unmixed (e.g., let R be a local Cohen-Macaulay ring), assume that a C I