Threading Homology through Algebra: Selected Patterns

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OX FO R D M AT H E M AT I C A L M O N O G R A P H S Series Editors

J. M. BALL W. T. GOWERS N. J. HITCHIN L. NIRENBERG R. PENROSE A. WILES

OX FO R D M AT H E M AT I C A L M O N O G R A P H S

Hirschfeld: Finite projective spaces of three dimensions Edmunds and Evans: Spectral theory and diﬀerential operators Pressley and Segal: Loop groups, paperback Evens: Cohomology of groups Hoﬀman and Humphreys: Projective representations of the symmetric groups: Q-Functions and Shifted Tableaux Amberg, Franciosi, and Giovanni: Products of groups Gurtin: Thermomechanics of evolving phase boundaries in the plane Faraut and Koranyi: Analysis on symmetric cones Shawyer and Watson: Borel’s methods of summability Lancaster and Rodman: Algebraic Riccati equations Th´ evenaz: G-algebras and modular representation theory Baues: Homotopy type and homology D’Eath: Black holes: gravitational interactions Lowen: Approach spaces: the missing link in the topology–uniformity–metric triad Cong: Topological dynamics of random dynamical systems Donaldson and Kronheimer: The geometry of four-manifolds, paperback Woodhouse: Geometric quantization, second edition, paperback Hirschfeld: Projective geometries over ﬁnite ﬁelds, second edition Evans and Kawahigashi: Quantum symmetries of operator algebras Klingen: Arithmetical similarities: Prime decomposition and ﬁnite group theory Matsuzaki and Taniguchi: Hyperbolic manifolds and Kleinian groups Macdonald: Symmetric functions and Hall polynomials, second edition, paperback Catto, Le Bris, and Lions: Mathematical theory of thermodynamic limits: Thomas-Fermi type models McDuﬀ and Salamon: Introduction to symplectic topology, paperback Holschneider: Wavelets: An analysis tool, paperback Goldman: Complex hyperbolic geometry Colbourn and Rosa: Triple systems Kozlov, Maz’ya and Movchan: Asymptotic analysis of ﬁelds in multi-structures Maugin: Nonlinear waves in elastic crystals Dassios and Kleinman: Low frequency scattering Ambrosio, Fusco and Pallara: Functions of bounded variation and free discontinuity problems Slavyanov and Lay: Special functions: A uniﬁed theory based on singularities Joyce: Compact manifolds with special holonomy Carbone and Semmes: A graphic apology for symmetry and implicitness Boos: Classical and modern methods in summability Higson and Roe: Analytic K-homology Semmes: Some novel types of fractal geometry Iwaniec and Martin: Geometric function theory and nonlinear analysis Johnson and Lapidus: The Feynman integral and Feynman ’s operational calculus, paperback Lyons and Qian: System control and rough paths Ranicki: Algebraic and geometric surgery Ehrenpreis: The radon transform Lennox and Robinson: The theory of inﬁnite soluble groups Ivanov: The Fourth Janko Group Huybrechts: Fourier-Mukai transforms in algebraic geometry Hida: Hilbert modular forms and Iwasawa theory Boﬃ and Buchsbaum: Threading homology through algebra

Threading Homology Through Algebra: Selected Patterns GIANDOMENICO BOFFI Universit` a G. d’Annunzio

DAVID A. BUCHSBAUM Department of Mathematics, Brandeis University

CLARENDON PRESS · OXFORD 2006

3

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press, 2006 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 0–19–852499–4

978–0–19–852499–1

1 3 5 7 9 10 8 6 4 2

A coloro che amo To Betty, wife and lifelong friend. Though she can’t identify each tree, she shares with me the delight of walking through the forest.

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PREFACE From a little before the middle of the twentieth century, homological methods have been applied to various parts of algebra (e.g. Lie Algebras, Associative Algebras, Groups [ﬁnite and inﬁnite]). In 1956, the book by H. Cartan and S. Eilenberg, Homological Algebra [33], achieved a number of very important results: it gave rise to the new discipline, Homological Algebra, it uniﬁed the existing applications, and it indicated several directions for future study. Since then, the number of developments and applications has grown beyond counting, and there has, in some instances, even been enough time to see various methods threading their way through apparently disparate, unrelated branches of algebra. What we aim for in this book is to take a few homological themes (Koszul complexes and their variations, resolutions in general) and show how these aﬀect the perception of certain problems in selected parts of algebra, as well as their success in solving a number of them. The expectation is that an educated reader will see connections between areas that he had not seen before, and will learn techniques that will help in further research in these areas. What we include will be discussed shortly in some detail; what we leave out deserves some mention here. This is not a compendium of homological algebra, nor is it a text on commutative algebra, combinatorics, or representation theory; although, it makes signiﬁcant contact with all of these ﬁelds. We are not attempting to provide an encyclopedic work. As a result, we leave out vast areas of these subjects and only select those parts that oﬀer a coherence from the point of view we are presenting. Even on that score we can make no claim to completeness. Our Chapter I, called “Recollections and Perspectives,” reviews parts of Polynomial Ring and Power Series Ring Theory, Linear Algebra, and Multilinear Algebra, and ties these with ideas that the reader should be very familiar with. As the title of the chapter suggests, this is not a compendium of “assumed known” items, but a presentation from a certain perspective—mainly homological. For example, almost everyone knows about divisibility and factoriality; we give a criterion for factoriality that ties it immediately to a homological interpretation (and one which found signiﬁcant application in solving a long-open question in regular local ring theory). The next three chapters of this book pull together a group of classical results, all coming from and generalizing the techniques associated with the Koszul complex. Perhaps the major result in Chapter II, on local rings, is the homological characterization of a regular local ring by means of its global dimension. Section II.6 includes a proof of the factoriality of regular local rings which is much closer to the original one, rather than the Kaplansky proof that is frequently quoted.

viii

Preface

We have also included a section on multiplicity theory, mainly to carry through the theme of the Koszul complex, and a section on the Homological Conjectures, as they provide a good roadmap for still open problems as well as a historical guide through much of what has been going on in the area this book is sketching. Chapter III deals with a class of complexes developed with the following aim in view: to associate a complex to an arbitrary ﬁnite presentation matrix of a module (the Koszul complex does this for a cyclic module), and to have that complex play the same role in the proof of the generalized Cohen–Macaulay Theorem that the Koszul complex plays in the classical case. We have made an explicit connection, in terms of a chain homotopy, between an older, “fatter” class of complexes, and a slimmer, more “svelte” class. We have also included a last section in which we deﬁne a generalized multiplicity which has found interesting applications, of late. Chapter IV applies some of the properties of these complexes to a systematic study of ﬁnite free resolutions, ending in a “syzygy-theoretic” proof of the unique factorization theorem (or “factoriality”) in regular local rings. The last three chapters and the Appendix not only focus on determinantal ideals and characteristic-free representation theory, but also involve a good deal of combinatorics. Chapter V employs the homological techniques developed in the previous part in the study of a number of types of determinantal ideals, namely Pfaﬃans and powers of Pfaﬃans. In Chapter VI we develop the basics of a characteristic-free representation theory of the general linear group (which has already made its appearance in earlier chapters). Because of the generality aspired to, heavy use is made of letter-place methods, an idea used more by combinatorialists than by commutative algebraists. As some of the proofs require more detail than is probably helpful for those encountering this material for the ﬁrst time, we decided to place these details in a separate Appendix: Appendix A. Much of the development of this chapter rests heavily on the notion of straight tableaux introduced by B. Taylor. In Chapter VII we ﬁrst present a number of results that immediately follow from this more general theory. Then examples are given to indicate what further use has been made of it, and in most cases references are given to detailed proofs. It is in this part of the chapter that we see the important inﬂuence of the work of A. Lascoux in characteristic zero. We give some of the background to the Hashimoto example of the dependence of the Betti numbers of determinantal ideals on characteristic. We deal with resolutions of Weyl modules in general, and skew-hooks in particular, and we make connections with intertwining numbers, Z-forms, and several other open problems. The intended readership of this book ranges from third-year and above graduate students in mathematics, to the accomplished mathematician who may or may not be in any of the ﬁelds touched on, but who would like to see what developments have taken place in these areas and perhaps launch himself into some of the open problems suggested. Because of this assumption, we are allowing ourselves to depend heavily on material that can be found in what we regard as comprehensive and accessible texts, such as the textbook by D. Eisenbud. We may at times, though, include a proof of a result here even if it does appear in such a text, if we think that the method of proof is typical of many of that kind.

CONTENTS

I

Recollections and Perspectives I.1 Factorization I.1.1 Factorization domains I.1.2 Polynomial and power series rings I.2 Linear algebra I.2.1 Free modules I.2.2 Projective modules I.2.3 Projective resolutions I.3 Multilinear algebra I.3.1 R[X1 , . . . , Xt ] as a symmetric algebra I.3.2 The divided power algebra I.3.3 The exterior algebra

1 1 1 6 8 8 13 17 21 22 28 30

II Local Ring Theory II.1 Koszul complexes II.2 Local rings II.3 Hilbert–Samuel polynomials II.4 Codimension and ﬁnitistic global dimension II.5 Regular local rings II.6 Unique factorization II.7 Multiplicity II.8 Intersection multiplicity and the homological conjectures

37 38 43 46 50 54 56 59 64

III Generalized Koszul Complexes III.1 A few standard complexes III.1.1 The graded Koszul complex and its “derivatives” III.1.2 Deﬁnitions of the hooks and their explicit bases III.2 General setup III.2.1 The fat complexes III.2.2 Slimming down III.3 Families of complexes III.3.1 The “homothety homotopy” III.3.2 Comparison of the fat and slim complexes III.4 Depth-sensitivity of T(q; f ) III.5 Another kind of multiplicity

69 69 70 72 80 82 83 85 88 91 94 99

IV Structure Theorems for Finite Free Resolutions IV.1 Some criteria for exactness IV.2 The ﬁrst structure theorem

103 104 110

x

V

Contents

IV.3 Proof of the ﬁrst structure theorem IV.3.1 Part (a) IV.3.2 Part (b) IV.4 The second structure theorem

115 115 118 119

Exactness Criteria at Work V.1 Pfaﬃan ideals V.1.1 Pfaﬃans V.1.2 Resolution of a certain pfaﬃan ideal V.1.3 Algebra structures on resolutions V.1.4 Proof of Part 2 of Theorem V.1.8 V.2 Powers of pfaﬃan ideals V.2.1 Intrinsic description of the matrix X V.2.2 Hooks again V.2.3 Some representation theory V.2.4 A counting argument V.2.5 Description of the resolutions V.2.6 Proof of Theorem V.2.4

127 128 128 131 132 134 136 137 138 139 140 143 145

VI Weyl and Schur Modules VI.1 Shape matrices and tableaux VI.1.1 Shape matrices VI.1.2 Tableaux VI.2 Weyl and Schur modules associated to shape matrices VI.3 Letter-place algebra VI.3.1 Positive places and the divided power algebra VI.3.2 Negative places and the exterior algebra VI.3.3 The symmetric algebra (or negative letters and places) VI.3.4 Putting it all together VI.4 Place polarization maps and Capelli identities VI.5 Weyl and Schur maps revisited VI.6 Some kernel elements of Weyl and Schur maps VI.7 Tableaux, straightening, and the straight basis theorem VI.7.1 Tableaux for Weyl and Schur modules VI.7.2 Straightening tableaux VI.7.3 Taylor-made tableaux, or a straight-ﬁlling algorithm VI.7.4 Proof of linear independence of straight tableaux VI.7.5 Modiﬁcations for Schur modules VI.7.6 Duality VI.8 Weyl–Schur complexes

149 149 149 153 154 156 156 159 164 164 165 167 169 174 174 176 181 183 186 187 187

VII Some Applications of Weyl and Schur Modules VII.1 The fundamental exact sequence VII.2 Direct sums and ﬁltrations for skew-shapes VII.3 Resolution of determinantal ideals

193 193 197 199

Contents

VII.3.1 The Lascoux resolutions VII.3.2 The submaximal minors VII.3.3 Z-forms VII.4 Arithmetic considerations VII.4.1 Intertwining numbers VII.4.2 Z-forms again VII.5 Resolutions revisited; the Hashimoto counterexample VII.6 Resolutions of Weyl modules VII.6.1 The bar complex VII.6.2 The two-rowed case VII.6.3 A three-rowed example VII.6.4 Resolutions of skew-hooks VII.6.5 Comparison with the Lascoux resolutions

xi

200 202 203 206 206 208 209 211 212 215 217 225 227

A Appendix for Letter-Place Methods A.1 Theorem VI.3.2, Part 1: the double standard tableaux generate A.2 Theorem VI.3.2 Part 2: linear independence of double standard tableaux A.3 Modiﬁcations required for Theorems VI.3.3 and VI.3.4 A.4 Modiﬁcations required for Theorem VI.8.4

237 237

References

249

Index

253

241 244 246

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I RECOLLECTIONS AND PERSPECTIVES

This chapter is neither a collection of results which we assume to be known nor the place to prove some results probably unknown to the reader, but needed in the following. Although it resembles a little of both things, it is essentially a selection of topics, some elementary, some more advanced, which we feel are adequate, or even necessary, to prepare the ground for the material of the chapters to come. Since it is almost impossible to tell which “basic” material is truly universally known, and which is not, we can only assure the reader that those terms in this chapter which are unfamiliar can be easily found in the book by D. Eisenbud, [41]. I.1 Factorization In this section, we deal with the basic topic of divisibility. In doing so, we review a few properties of some rings, which are of importance to us. For more details, we refer the reader to Reference [87]. I.1.1 Factorization domains Let R be an integral domain, that is, a commutative ring (with 1) having no zero divisors. Given a and b in R, we say that a is a divisor of b (written a | b) if b = ac for some c in R. If a | b and b | a, then b = ua for some unit u, and a and b are called associates. Being associate is an equivalence relation. a is a proper divisor of b if a divides b, but is neither a unit, nor an associate of b. In terms of ideals, a | b means (b) ⊆ (a), u being a unit is equivalent to (u) = R, a and b associates says (a) = (b), and a properly divides b if and only if (b) ⊂ (a). Deﬁnition I.1.1 An element c ∈ R is called a greatest common divisor (gcd) of a and b in R if c | a, c | b, and c is divisible by every d such that d | a and d | b. An element c ∈ R is called a least common multiple (lcm) of a and b in R if a | c, b | c, and c divides every d such that a | d and b | d. Given a and b in R, gcd(a, b) may or may not exist. If it does, it is unique up to associates. Similarly for lcm(a, b). Remark I.1.2 If a, b ∈ R −{0}, and lcm(a, b) exists, then also gcd(a, b) exists, and lcm(a, b) · gcd(a, b) = ab, up to units. If a, b ∈ R − {0}, and gcd(a, b) exists, lcm(a, b) may not exist (cf. Example I.1.11 later on). However, if gcd(a, b)

2

Recollections and perspectives

exists for all choices of a and b in R − {0}, then lcm(a, b) exists for all choices of a and b in R − {0} . If 1 is a greatest common divisor for a and b, we will say that a and b are coprime. In terms of ideals, c being a common divisor of a and b means (a, b) ⊆ (c), while c being a common multiple of a and b means (a) ∩ (b) ⊇ (c). The equality (a, b) = (c) implies gcd(a, b) = c (but not conversely), while (a) ∩ (b) = (c) is equivalent to lcm(a, b) = c. Deﬁnition I.1.3 A non-zero, non-invertible element a ∈ R is called irreducible if it does not have any proper divisors. A non-zero, non-invertible element a ∈ R is called prime if, whenever a | bc, then either a | b or a | c. In terms of ideals, a is irreducible if and only if (a) is maximal among the proper principal ideals of R; a is prime if and only if (a) is a prime ideal. If a is prime, then it is irreducible. But the converse does not hold. Deﬁnition I.1.4 An integral domain R is called a factorization domain if every non-zero, non-invertible a ∈ R can be expressed as a product of irreducible elements. A factorization domain is called a unique factorization domain (UFD) if every factorization into irreducibles is unique up to permutation of the factors and multiplication of the factors by units. In terms of ideals, an integral domain R is a factorization domain if and only if there is no strictly ascending inﬁnite chain of principal ideals in R. In particular, every principal ideal domain (PID) is a factorization domain: given any ascending chain of ideals, the union of these ideals is an ideal of the sequence. In fact, a PID is always a UFD, by part (ii) of the following proposition. Proposition I.1.5 equivalent.

Let R be a factorization domain. The following are

(i) R is a UFD. (ii) lcm(a, b) exists for every choice of a, b in R. (iii) Every irreducible element is prime. Proof (i) ⇒ (ii) As in Z, one expresses a and b as products of powers (with non-negative exponents) of suitable irreducibles (the same ones for a and b): max{si ,ti } a = fisi and b = fiti , say. Then lcm(a, b) = fi . (ii) ⇒ (iii) The gcd exists in R since the lcm does. If c = gcd(a, b), then cd = gcd(ad, bd) for every d. For, given any two ideals a and b, d(a ∩ b) = da ∩ db; hence dlcm(a, b) = lcm(ad, bd); using that adbd = gcd(ad, bd)lcm(ad, bd) and ab = clcm(a, b), we are through. Let an irreducible element c divide ab, and assume that c b: we claim that c | a. Since c is irreducible and c b, b and c are coprime, that is, 1 = gcd(b, c).

Factorization

3

It follows that a = gcd(ab, ac); since c divides ab by assumption, c must divide gcd(ab, ac), as claimed. (iii) ⇒ (i) As in the ring of integers. 2 Corollary I.1.6 If R is a UFD, then hdR (R/(a, b)) ≤ 2 for all a and b in R. (Here hd stands for homological dimension, sometimes called projective dimension and denoted by pd.) Proof If a = 0 = b, the quotient ring is R and the homological dimension is 0. If a = 0 and b = 0, the quotient ring is R/(b) and we consider the exact complex of R-modules: b

0 → K → R → R → R/(b) → 0, where K stands for the kernel of the map given by multiplication by b. Since R is a domain, cb = 0 implies c = 0, and K = (0). Hence hdR (R/(b)) ≤ 1. If both a and b are diﬀerent from 0, we consider the following exact complex: (a,b)

0 → K → R2 → R → R/(a, b) → 0, where K stands for the kernel of the map given by the matrix (a, b). We want to show that K is free over R, as this will give us our result on the homological dimension of R/(a, b). If (a) : b denotes the ideal {r ∈ R | rb ∈ (a)}, clearly K = (a) : b, because rb ∈ (a) if and only if rb = sa for some (unique) s ∈ R, that is, (−s)a + rb = 0. As R-modules, (a) : b ∼ = b((a) : b), and obviously b((a) : b) = (a) ∩ (b). By the previous proposition, (a) ∩ (b) is a principal ideal; hence it is a rank 1 free R-module, and so is K. 2 Because we have not made signiﬁcant use of homological dimension, we will put oﬀ giving a formal deﬁnition of that term here; the reader will ﬁnd it in the next section (Deﬁnition I.2.25). The crucial fact that we needed in the proof of the above corollary was just that K is free. We will see in Chapter II that if R is a noetherian local ring, then R is a UFD if and only if hdR (R/(a, b)) ≤ 2 for all a and b in R. This will lead to proving that regular local rings are UFD. Remark I.1.7 We have noticed in the proof of Proposition I.1.5 that if R is a UFD, then c = gcd(a, b) implies cd = gcd(ad, bd) for every d. It follows that gcd(a, b) = 1 and a | bc together imply a | c. In terms of ideals, this means that if a and b are coprime in R, then b is a non-zero divisor in R/(a), although R/(a) may no longer be an integral domain. Conversely, if b is a non-zero divisor in R/(a), then gcd(a, b) = 1; for otherwise a/ gcd(a, b) would kill b in R/(a). This set up will be generalized in Chapter II by the notion of M -sequence. Remark I.1.8

The complex (a,b)

0 → R2 → R → R/(a, b) → 0

4

Recollections and perspectives

is a truncation of the following Koszul complex (to be described in Chapter II):  

0→R

−b a →

  (a,b)

R2 → R → R/(a, b) → 0.

Does the latter complex coincide with the resolution of (a, b), a = 0 = b, described in the proof of Corollary I.1.6? Recalling the identiﬁcation K = (a) : b, −b im = (s, t) ∈ R2 | s = −br, t = ar for some r ∈ R a corresponds to (a) ⊆ (a) : b and we are asking whether (a) = (a) : b. We claim that equality holds if and only if gcd(a, b) = 1. For, by the previous remark gcd(a, b) = 1 means that b is a non-zero divisor in R/(a), and so (a) : b vanishes in R/(a). When unique factorization of elements does not hold in our integral domain R, we might relax the condition a bit and ask: what kind of ring allows for the unique factorization of principal ideals into prime ideals? We do not know the answer to that, but we can ask for a generalization of principal ideal domains, namely what kind of ring, R, (besides a PID) may have unique factorization of ideals into products of prime ideals? Actually, there is a name for such a ring: Dedekind domain. It turns out that this condition is equivalent to a combination of other properties, namely that of being noetherian, normal (or integrally closed), and being of dimension one. Some of these terms will be discussed in great detail in Chapter II, but we brieﬂy point out here that one of the many characterizations of a noetherian ring is that every ideal is ﬁnitely generated (another one is that no inﬁnite strictly ascending chain of ideals can exist). This is certainly true of a PID, so every PID is noetherian. To say that the dimension of a ring is equal to one turns out to mean (see Section II.3) that every non-zero prime ideal is maximal, and the observations immediately preceding Deﬁnition I.1.3 imply that in a PID all prime ideals are indeed maximal. Finally, it is clear that every PID (in fact, every UFD) is normal, so that we know every PID is a noetherian, normal domain of dimension one. However, these three properties do not quite characterize a PID; rather, there is the following theorem. Theorem I.1.9

For an integral domain R, the following are equivalent.

(i) R is a noetherian normal domain of dimension 1. (ii) Every proper ideal a of R can be expressed as a product of prime ideals, in a unique way, up to permutations of the factors. Proof

Cf., for example, Reference [87, chapter 5, section 6, theorem 13, p. 275]. 2

So we are led to make the following deﬁnition.

Factorization

5

Deﬁnition I.1.10 An integral domain is called a Dedekind domain if it satisﬁes the equivalent conditions of Theorem I.1.9. The ring of algebraic integers in any algebraic number √ ﬁeld is always a Dedekind domain. Some very accessible ones are the rings Z+Z n with n a squarefree element of Z − {0, 1} such that n is not congruent to √ 1 modulo 4 (this latter condition ensuring that this is the ring of integers in Q( n)). The family of Dedekind domains properly includes the family of principal ideal domains, for Dedekind domains may have ideals which are not principal. √ √ Example I.1.11 Let √ and √ R = Z + Z −5, a = 1 + −5, b = 3. gcd(a, b) exists equals 1, for if s + t −5 divides both a and b, then its norm N (s + t −5) = s2 + 5t2 must divide both 6 and 9, hence their gcd 3; but s2 + 5t2 | 3 forces t = 0 and s√= ±1. If a = (a, b) were a √ principal ideal, gcd(a, b) = 1 would imply a = R. But −5 ∈ / a, since otherwise −5 = αa + βb would give 5 = 6N (α) + 9N (β) and 3 should√divide 5, a contradiction. Finally, notice that lcm(a, b) does not exist: if s + t −5 were a lcm(a, b), s2 +√5t2 would be divisible by both 6 and 9, hence by lcm(6, 9) = 18; moreover, s+t√ −5 should √ divide both 36 and 54, √ hence gcd (36, 54) = 18, since both 6 = (1 + −5)(1 − −5) = 2 · 3 and 3 + 3 −5 are common multiples; thus s2 + 5t2 = 18, which is impossible. If one had simply wanted to prove that this √ a PID, it would have √ ring is not suﬃced to point out that 6 = 2 × 3 = (1 + −5)(1 − −5), show that each of these factors is irreducible, and conclude that this contradicts UFD, hence PID. The rather longer discussion above, though, actually produces a non-principal ideal. While it may be slightly disappointing that there are Dedekind domains that are not principal ideal domains, it is a well-known property of Dedekind domains that all their ideals can be generated by at most two elements (cf., e.g. Reference [87, chapter 5, section 7, theorem 17, p. 279]). So at least we are not too far oﬀ the mark. If R is any commutative ring, the collection of prime ideals of R is called the spectrum of R, written Spec(R). The set of maximal ideals of R is called the maximal spectrum, and is denoted by Max(R). By Theorem I.1.9, given a Dedekind domain R, Spec(R) = {0} ∪ Max(R), Proposition I.1.12 R is a PID.

If R is a Dedekind domain such that |Max(R)| < ∞, then

Proof Let Max(R) = {m1 , m2 , . . . , mt }. For every i = 1, 2, . . . , t there exists / m2i and ai ∈ / mj , j = i (by the Chinese an element ai ∈ mi such that ai ∈ Remainder Theorem, Reference [87], chapter 5, section 7, theorem 17, p. 279). Then (ai ) = mi , and the principality of all maximal ideals implies the principality of every other ideal (by part (ii) of Theorem I.1.9). 2 If we localize a Dedekind domain at a non-zero prime m, Rm is still Dedekind, since condition (i) of Theorem I.1.9 is preserved by localization. (We assume the

6

Recollections and perspectives

reader is familiar with the process of forming rings of quotients with respect to a multiplicative subset. Essentially this is just the “fractions” having arbitrary elements of the ring on top, and elements of the multiplicative subset as denominators. All the bells and whistles of localization are explained in Reference [41], section 2.1). In fact, Rm is a PID (by the last proposition), because 0 and mm are its only prime ideals. If mm = (a) for some a ∈ Rm , then every other ideal of Rm is of type (an ) for some positive n. Notice that since Rp is a PID for every p ∈ Spec(R), unique factorization of elements is locally true for every Dedekind domain. Local Dedekind domains are known as discrete valuation rings. I.1.2 Polynomial and power series rings Given any commutative ring (with 1), say R, a (formal) power series in t indeterminates over R, t ∈ N−{0}, is a function f : Nt → R. Power series can be added and multiplied. Addition is simply addition of functions. Multiplication is deﬁned by (f g)(n1 , . . . , nt ) = mi +li =ni f (m1 , . . . , mt )g(l1 , . . . , lt ). The set of all power series in t indeterminates over R turns out to be a commutative ring (with 1) with respect to the indicated operations. The customary notation for this ring is R[[X1 , . . . , Xt ]], for one identiﬁes f : Nt → R with the formal sum f (n1 , . . . , nt )X1n1 · · · Xtnt . In particular, an X n stands for the function f : N →R such that f (n) = an for every n ∈ N. Clearly, R[[X1 , . . . , Xt ]] = (R[[X1 , . . . , Xt−1 ]])[[Xt ]]. Given R as above, a polynomial in t indeterminates over R is a power series f : Nt → R which is zero almost everywhere. The corresponding symbol f (n1 , . . . , nt )X1n1 · · · Xtnt is usually meant to be restricted to the (ﬁnitely many) non-zero values f (n1 , . . . , nt ), thereby giving a ﬁnite formal sum. Polynomials form a subring of the ring of power series, denoted by R[X1 , . . . , Xt ]. Clearly, R[X1 , . . . , Xt ] = (R[X1 , . . . , Xt−1 ])[Xt ]. Often one writes R[[X]] and R[X] instead of R[[X1 , . . . , Xt ]] and R[X1 , . . . , Xt ], meaning that X = {X1 , . . . , Xt }. The following proposition collects some properties valid when |X| = t = 1. Proposition I.1.13

Let R be a commutative ring (with 1).

(i) f ∈ R[X] is invertible in R[X] if and only if a0 is invertible in and all R n other coeﬃcients are nilpotent in R (as usual, we assume f = i=0 ai X i ; an element of a ring is nilpotent if some power of it is equal to 0). (ii) f ∈ R[[X]] is invertiblein R[[X]] if and only if a0 is invertible in R (as ∞ usual, we assume f = i=0 ai X i ). (iii) R has no zero divisors if and only if R[X] has no zero divisors if and only if R[[X]] has no zero divisors.

Factorization

7

Proof We only prove (ii), not because it is harder, but because ∞ we need it soon. If f is invertible in R[[X]], there exists g ∈ R[[X]], g = i=0 bi X i say, such that f g = 1. Hence f g = a0 b0 + (a0 b1 + a1 b0 )X + (a0 b2 + a1 b1 + a2 b0 )X 2 + · · · = 1 forces a0 b0 = 1, and a0 is a unit in R. Conversely, assume that a0 is a unit in R and look for some g as above, such that f g = 1. The following equalities must be satisﬁed: a0 b0 = 1,

a0 b1 + a1 b0 = 0,

a0 b2 + a1 b1 + a2 b0 = 0,

...

The invertibility of a0 allows us to solve these equations for b0 , b1 , b2 , . . . one after the other, and we are done. 2 The last statement of Proposition I.1.13 hints at a general question: what properties of R are inherited by R[X] and R[[X]]? For instance, if R is a Euclidean domain (i.e. a domain where one has division with remainder), the domain R[X] need not be Euclidean. Proposition I.1.14 noetherian.

If R is noetherian, then both R[X] and R[[X]] are

Proof For R[X], this is the Hilbert basis theorem, (cf., for example, Reference [87], chapter 4, section 1, theorem 1, p. 201). For R[[X]], there is a proof very much in the spirit of the proof of the Hilbert basis theorem, (cf., e.g. Reference [87], chapter 7, section 1, theorem 4, p. 138). 2 Corollary I.1.15 If R is noetherian, R[[X1 , . . . , Xt ]] are noetherian.

then

both

R[X1 , . . . , Xt ]

and

When R is a noetherian domain, both R[X1 , . . . , Xt ] and R[[X1 , . . . , Xt ]] (being noetherian domains) are factorization domains: they cannot contain any strictly ascending inﬁnite chain of principal ideals. This remark leads to the following question: if R is a UFD, is it true that R[X1 , . . . , Xt ] and R[[X1 , . . . , Xt ]] are UFD? Unlike Proposition I.1.14, we cannot give a unique answer: we will prove in a moment that R[X1 , . . . , Xt ] does inherit the property of being a UFD from R; but R[[X1 , . . . , Xt ]] may not be a UFD. The ﬁrst counterexample was given by P. Samuel in 1961 (see [77]). Theorem I.1.16

If R is a UFD, then R[X1 , . . . , Xt ] is a UFD.

Proof By induction on t, it suﬃces to show that R[X] is a UFD. Since we already know that R[X] is a factorization domain, part (ii) of Proposition I.1.5 says that it is enough to prove that a lcm(f, g) exists for any two polynomials f and g in R[X]. Let Q denote the ﬁeld of quotients of R. Since Q is a ﬁeld, Q[X] is a Euclidean domain, hence a PID, hence a UFD. So a lcm(f, g) certainly exists in Q[X].

8

Recollections and perspectives

Call it h. Clearly, h can be expressed as h = c(h) · h , where h ∈ R[X] and has coprime coeﬃcients, while c(h) ∈ Q. Write f = c(f ) · f and g = c(g) · g , where f and g are assumed to have coprime coeﬃcients. Since R is a UFD by hypothesis, a lcm(c(f ), c(g)) exists in R. 2 Call it c. Then c · h is a lcm(f, g) in R[X], as required. Although a similar theorem does not hold for R[[X1 , . . . , Xt ]], we have the following partial result. Proposition I.1.17

If R is a ﬁeld, then R[[X1 , . . . , Xt ]] is a UFD.

Proof We cannot reduce to the case t = 1. But since we already know that R[[X1 , . . . , Xt ]] is a factorization domain, it suﬃces to show that every irreducible element generates a (principal) prime ideal (cf. part (iii) of Proposition I.1.5). This can be accomplished by induction on t, and using the statement of the previous theorem, (cf., e.g. Reference [87], chapter 7, section 1, theorem 6, p. 148). 2 We give another property of K[[X1 , . . . , Xt ]], when K a ﬁeld. Proposition I.1.18 If K is a ﬁeld, then K[[X1 , . . . , Xt ]] is a local ring with maximal ideal m = (X1 , . . . , Xt ). Proof The proof of part (ii) of Proposition I.1.13 works word for word in every ring (R[[X1 , . . . , Xs−1 ]])[[Xs ]]. Since in our case R is a ﬁeld, the non-units of K[[X1 , . . . , Xt ]] are the elements with zero constant term. That is, (X1 , . . . , Xt ) 2 consists of all the non-invertible elements of K[[X1 , . . . , Xt ]]. When t = 1, K[[X]] is in fact a discrete valuation ring (= local Dedekind domain), hence a PID (cf. Proposition I.1.12), for it is not hard to check that every proper ideal in K[[X]] is a power of m = (X). I.2 Linear algebra In this section we deal with linear algebra over a commutative ring, not just over a ﬁeld. In doing so, we review some basics of homological algebra. For more details, we refer the reader to References [15], [33], and [41]. I.2.1 Free modules Let R be a commutative ring (with 1). An R-module is an immediate generalization of a vector space. That is, if K is a ﬁeld and V a vector space over K, we know that V is an abelian group, K acts on V , and this action satisﬁes certain conditions. One notices that the conditions in no way make use of the fact that K is a ﬁeld; thus we may replace K by the commutative ring R, write M for V , and get the deﬁnition for a module M over the ring R.

Linear algebra

9

The usual deﬁnitions of linearly independent subset, linearly dependent subset, generators, submodule, submodule generated by a subset, that are used for vector spaces apply mutatis mutandis to R-modules. The diﬀerence, as we will see, lies in the fact that our base ring is not in general a ﬁeld; thus such things as the existence of a basis for every vector space do not hold true for modules over arbitrary rings. (Recall that a basis of a module is a linearly independent subset which generates the module.) Yet the existence of maximal linearly independent subsets of a module is proved in exactly the same way as is done for vector spaces. It may be, however, that the empty set is a maximal linearly independent subset of a module, but the module is not necessarily the zero module. For example, the abelian group Z/(2), considered as a Z-module, has two elements, but its maximal linearly independent set is the empty set. For {0} is not independent and {1} is not independent because 2 · 1 = 0. Thus, while for vector spaces we have the fact that a maximal independent subset is a basis for (hence generates) the vector space, this is no longer the case for general modules. Deﬁnition I.2.1

An R-module M is called free if it has a basis.

We note immediately that the zero module is free (its basis is the empty set). The following result shows that the basis of a free module can have any cardinality. Proposition I.2.2 Given any non-empty set I, there is a free R-module with basis in one-to-one correspondence with I. Proof Let M be the set {f : I → R | f is zero almost everywhere}. It is an R-module with respect to the operations (f1 + f2 )(i) = f1 (i) + f2 (i) and (rf )(i) = rf (i). Clearly M has an R-basis {fi }i∈I , where fi stands for the map sending i to 1 and all other elements of I to 0. 2 Homomorphisms of R-modules, often called R-maps, are deﬁned as in the case of vector spaces. The free module built in the proof of Proposition I.2.2 is canonically R-isomorphic to ⊕i∈I Ri , where Ri is a copy of the R-module R for every i ∈ I. The basis of ⊕i∈I Ri corresponding to {fi }i∈I in this isomorphism is called the canonical basis of ⊕i∈I Ri . When I is ﬁnite, say |I| = t, we write Rt instead of ⊕i∈I Ri . Remark I.2.3 A free R-module M having a ﬁnite basis B cannot have an inﬁnite basis B . For every element of B can be expressed as a linear combination of ﬁnitely many elements of B , and if C is the (ﬁnite linearly independent) set of all the elements of B involved in the expressions of the elements of B, then every element of M is generated by C, so that C = B .

10

Recollections and perspectives

Proposition I.2.4 If B = {m1 , . . . , mt } and B = {m1 , . . . , ms } are two ﬁnite bases of the same free R-module M , then t = s. Proof Write each mi (respectively, mj ) as an R-linear combination of the elements of B (respectively, B): mi =

s

aij mj ,

mj =

j=1

t

bji mi .

i=1

Call S the t × s matrix (aij ) and T the s × t matrix (bji ). Clearly, ST equals the t × t identity matrix It , and T S = Is . If t = s, say t > s, consider the square matrices of order t: T S = (S | t − s zero columns) and T = . t − s zero rows Their product is still It , but det It = 1, while det(ST) = det S det T = 0: 2 a contradiction. Similarly for t < s, using T S = Is . We remark that the deﬁnition of determinant is the same as for ﬁelds, and that det S det T = det ST is purely formal and does not require that the base ring be a ﬁeld. Deﬁnition I.2.5 A free R-module is called ﬁnite if all its bases have ﬁnitely many elements. A ﬁnite free R-module has rank t if it has a basis consisting of t elements (hence every basis consists of t elements). Remark I.2.6 If a free R-module M happens to have a ﬁnite system of generators, then M must be a ﬁnite free R-module. Just argue as in Remark I.2.3, calling B the system of generators and B any basis. Proposition I.2.7 Let F be a rank t free R-module with basis B = {f1 , . . . , ft }. Let S be a t×t matrix with entries in R. Let C = {m1 , . . . , mt } ⊆ F be deﬁned by     m1 f1  ..   .   .  = S  ..  . mt Then the following are equivalent. (i) det S is a unit in R. (ii) C is another basis of F . (iii) C is a generating system of F .

ft

Linear algebra

11



   m1 f1     Proof (i) ⇒ (ii): S −1  ...  =  ...  shows that C generates F (S −1 mt ft exists for det S invertible allows the use of the customary formula); independence is easy. (ii) ⇒ (iii): Trivial. (iii) ⇒ (i): Call T the matrix expressing B in terms of C; then T S = It , and det S is a unit in R. 2 Corollary I.2.8 (i) Every generating system {m1 , . . . , mt } of Rt is a basis. (ii) Every generating system of Rt has at least t elements. (iii) If ϕ : Rn → Rm is an R-epimorphism, then n ≥ m. Proposition I.2.9 Let ϕ be an R-morphism from Rt to Rt , and let S be its matrix with respect to the canonical basis of Rt . Then ϕ is injective if and only if det S is a non-zero divisor in R. Proof First divisor ⇒ ϕ injective.   part:det Sa non-zero   x1 0 x1       If not, S  ...  =  ...  for some non-zero  ... , x1 = 0 say. Then xt xt 0    x1 0     S  ... | It2 · · · Itt  =  ... | S 2 · · · S t , where Iti stands for the i-th column xt 0 of It , and similarly for S i . Thus det S · x1 = 0, a contradiction. Second part: ϕ injective ⇒ det S a non-zero divisor. It suﬃces to show that if det S is a zero divisor, then ϕ is not injective, that is, the columns of S are not an independent system in Rt . If t = 1, this is obvious. If t ≥ 2, the statement is a corollary of the following more general result. Let m1 , . . . , ms be elements of Rt (t ≥ 2). If a ∈ R − {0} kills all the maximal minors of the matrix (m1 · · · ms ), then {m1 , . . . , ms } is not an independent system. The proof is by induction on the number s of t-tuples. If s = 1, then am1 = 0 with a = 0 prevents m1 from being independent. We now assume that t ≥ s > 1. If a also kills all maximal minors of the matrix (m2 · · · ms ), then {m2 , . . . , ms } is not an independent system (by induction hypothesis) and a fortiori {m1 , . . . , ms } is not either. If a does not kill all maximal minors of (m2 · · · ms ), let b be one of those minors such that ab = 0; let us say that b is given by the last s − 1 rows of the matrix (m2 · · · ms ). We now use the assumption that a kills all maximal minors of (m1 · · · ms ). 

12

Recollections and perspectives

Let T denote the t×t matrix ( It−s 0 | m1 · · · ms ). Since det T equals a maximal minor of (m1 · · · ms ), a det T = 0. If T denotes the companion matrix of T , and Ti is the i-th column of T, then T T = det T · It implies:   1   ..   . 0     1   ,  a T T1 · · · Tt−s aTt−s+1 Tt−s+2 · · · Tt = det T     1     ..   . 0 1 with a in position (t − s + 1, t − s + 1). It follows (since a det T = 0):   0  ..   .     0    ∗  0 = T aTt−s+1 = T   ab  , with ab in row t − s + 1.  ∗     .   ..  ∗ Hence 0 = abm1 + (∗m2 ) + · · · + (∗ms ), so that (since ab = 0) {m1 , . . . , ms } cannot be an independent system. Finally, we consider the case s > t, that is, (m1 · · · ms ) is a t × s matrix with t < s. In that case, its maximal minors are of order t; if a kills all the maximal minors, in particular it kills det(m1 · · · mt ). This implies (case s = t of the induction hypothesis) that {m1 , . . . , mt } cannot be an independent system; a fortiori, {m1 , . . . , ms } cannot be either. This concludes the proof of the more general result on Rt , as well as of the proposition. 2 Corollary I.2.10 (i) If ϕ : Rn → Rm is an R-monomorphism, then n ≤ m. (ii) A non-zero ideal a of R is a ﬁnite free R-module if and only if it is principal, and generated by a non-zero divisor. ϕ

Proof (i) If n > m, then there would be a monomorphism Rn → Rm → S Rm ⊕ Rn−m = Rn associated with the n × n matrix S = (n−m zero rows) , where S is the matrix of ϕ with respect to the canonical bases. But det S = 0 contradicts the proposition. (ii) If a is ﬁnite R-free, say a ∼ = Rt , the inclusion a → R gives a monomorphism t R → R, and t ≤ 1 by part (i). So a = Ra = (a) for some independent (that is, non-zero divisor) a ∈ R. The converse is obvious. 2

Linear algebra

13

I.2.2 Projective modules Every R-module M is the quotient of a free R-module F . For if {mi }i∈I is a generating system of M (if necessary, the generating system may consist of all the elements of M ), then we call F the free module of Proposition I.2.2. If {fi }i∈I is the basis of F deﬁned in that proposition, the R-epimorphism ϕ : F → M sending fi to mi for every i does the job. If N denotes the kernel of ϕ, then there exists another R-epimorphism ψ : E → N with E an R-free module. And one gets the following exact complex: ψ

ϕ

E → F → M → 0,

(∗)

which is called a free presentation of M (0 stands for the zero module and M → 0 is the zero map). We recall that complex means, wherever you have two consecutive arrows, the image of the left arrow is included in the kernel of the right arrow. Exact complex means that the inclusion is always an equality. If |I| < ∞ (i.e. M is ﬁnitely generated), the above F is a ﬁnite free R-module, but E need not be ﬁnite. Yet in some cases (for instance when R is noetherian), E does have a ﬁnite basis, and M is said to be ﬁnitely presented. Then ψ can be expressed by a matrix (relative to some ﬁxed bases of E and F ), carrying information on M = coker(ψ). Let us go back to the R-epimorphism ϕ : F → M and consider the exact complex (a short exact sequence is what it is generally called): 0 → ker(ϕ) → F → M → 0. Does it imply that F ∼ = M ⊕ ker(ϕ)? More generally, does an exact complex of R-modules (∗∗)

β

0 → M → M → M → 0 α

imply that M ∼ = M ⊕ M ? It is clear that the answer cannot be positive, in general (just think of the exact complex of Z-modules α

β

0 → Z → Z → Z/(n) → 0, where α is multiplication by the positive integer n and β is the canonical projection). Deﬁnition I.2.11 The exact complex (∗∗) is called split if it implies M ∼ = M ⊕ M by means of an isomorphism ϕ : M → M ⊕ M such that ϕ−1 | M equals α and the composite M → M ⊕ M → M ϕ

pr2

equals β. Conditions for being split are easily proven.

14

Recollections and perspectives

Proposition I.2.12

The following are equivalent for the exact sequence (∗∗).

(i) (∗∗) is split. (ii) There exists an R-map γ : M → M such that γ ◦ α = idM . (iii) There exists an R-map δ : M → M such that β ◦ δ = idM . β

Sometimes one says that an exact complex M → M → 0 is split, meaning that there exists δ : M → M such that β ◦ δ = idM . Similarly for an exact α complex 0 → M → M . If the module M of (∗∗) happens to be R-free, condition (iii) of Proposition I.2.12 is automatically satisﬁed, and (∗∗) is split. For if {fi }i∈I is a basis of M , it suﬃces to choose one mi ∈ β −1 (fi ) for every i and deﬁne δ by means of δ(fi ) = mi for every i. The above is just an instance of an important property satisﬁed by free modules. π

Proposition I.2.13 Given any exact complex M → N → 0 of R-modules, and any R-map ϕ : F → N with F free, there exists an R-map ψ : F → M (called a lifting of ϕ) such that ϕ = π ◦ ψ. Proof Fix a basis {fi }i∈I of F and choose one mi ∈ π −1 (ϕ(fi )) for every i. 2 Then deﬁne ψ by means of ψ(fi ) = mi for every i. The proposition leads to a well-known generalization of free modules. Deﬁnition I.2.14 An R-module P is called projective if whenever we are π given an exact complex M → N → 0 of R-modules and an R-map ϕ : P → N , there exists an R-map ψ : P → M such that ϕ = π ◦ ψ. Proposition I.2.15 (i) (ii) (iii) (iv)

Given an R-module P , the following are equivalent.

P is projective. π Every exact complex M → P → 0 is split. P is a direct summand of a free R-module. π Every exact complex M → N → 0 induces an exact complex π◦

HomR (P, M ) −→ HomR (P, N ) → 0, where the left arrow sends f to π ◦ f . Proof We just show (iii) ⇒ (iv). Let F = P ⊕ Q with F free. Then π◦

HomR (F, M ) −→ HomR (F, N ) → 0 is exact, thanks to Proposition I.2.13, and we are done because HomR (P ⊕ Q, X) = HomR (P, X) ⊕ HomR (Q, X) for every R-module X.

2

Linear algebra

15

The family of projective R-modules does not always properly include the family of free R-modules. Lemma I.2.16 (Nakayama’s lemma) Let M be a R-module and a an ideal of R such that a ⊆ rad(R). Then

ﬁnitely

generated

(i) if aM = M , then M = 0; (ii) if {m1 , . . . , mt } ⊆ M induces a generating system {m1 , . . . , mt } of the R/a-module M/aM , then {m1 , . . . , mt } is a generating system of M . Proof Cf., for example, Reference [41, corollary 4.8, p. 124]

2

Proposition I.2.17 Let (R, m) be a local ring (not necessarily noetherian) and let M = 0 be a ﬁnitely generated R-module. If M is R-projective, then M is R-free. Proof Part (i) of the lemma implies that M/mM (= M ⊗R R/m) is a non-zero ﬁnitely generated R/m-vector space, thus it has a basis, say {m1 , . . . , mt }. By part (ii) of the lemma, it follows that {m1 , . . . , mt } is a generating system of M , hence there exists an epimorphism ϕ : F → M with F a rank t free module. Since M is projective, ϕ splits and F = M ⊕ N for a suitable N . It follows that F/mF = M/mM ⊕ N/mN as R/m-vector spaces. But F/mF and M/mM have both dimension t, so that N/mN must have dimension 0, that is, N = mN and N = 0 (again part (i) of the lemma). 2 In the proof of Proposition I.2.17, we indicated that M/mM = M ⊗R R/m without explaining the symbol, “⊗R ,” known as the tensor product. We will assume that this operation is well known to the reader; if in doubt, a straightforward treatment may be found in Reference [41] sections 2.2 and A.2.2. We also point out here that in Proposition I.2.18 we will write Mm = M ⊗R Rm , thereby indicating that the localization of a module is the same as taking the tensor product of the module with the ring of quotients of the ring. These are facts that we will use often in what follows. • In connection with the use of tensor products in this book, we oﬀer a caveat to the reader. While we often try to indicate the ring over which a given tensor product is taken, especially when a certain ambiguity exists, there are many times that the base ring is omitted. We believe that when this occurs, the context is such as to leave no question in the reader’s mind what the base ring is.

There are other cases when projective R-modules and free R-modules coincide, for instance when R is a PID (see Corollary I.2.23 below) and when R = K[X1 , . . . , Xn ], K a ﬁeld (Quillen-Suslin Theorem). A case in which the family of projective R-modules properly contains the family of free R-modules is given by R a Dedekind domain: see Remark I.2.21 below.

16

Recollections and perspectives

The property of being projective is a local property, in the sense of the following proposition. Proposition I.2.18 Let M be a ﬁnitely presented R-module. M is projective if and only if Mm is Rm -free for every m ∈ Max(R). Proof The only if part follows from Proposition I.2.17, once we remark that Mm = M ⊗R Rm R-projective implies Mm is Rm -projective (this comes from Proposition I.2.15 (iii) and the fact that ⊗R commutes with ⊕). The if part uses Proposition I.2.15 (iv), coupled with the remark that a map is onto if and only if its localizations at all maximal ideals are onto. The assumption that M is ﬁnitely presented ensures that Rm ⊗R HomR (M, N ) ∼ = HomRm (Mm , Nm ) 2

for every R-module N . We now brieﬂy turn our attention to a special class of rings.

Deﬁnition I.2.19 A nontrivial commutative ring (with 1) R is called hereditary if every ideal of R is a projective R-module. Clearly every PID is hereditary, because of Corollary I.2.10 (ii). Another example is given by Dedekind domains, due to the following result. Proposition I.2.20 projective R-module.

Every non-zero ideal a of a Dedekind domain R is a

Proof Since R is noetherian, a is ﬁnitely presented as an R-module. Hence by the last proposition, a is R-projective if and only if am is Rm -free for every m ∈ Max(R). Now given m ∈ Max(R), am ⊆ Rm satisﬁes am = (an ), where (a) is the maximal ideal of Rm and n is a positive integer (recall the observations after Proposition I.1.12). Being a principal ideal, am is a rank 1 free 2 Rm -module. Remark I.2.21 Dedekind domains are a more interesting example of hereditary rings than principal ideal domains, because in general not all ideals of a Dedekind domain are free (if they were, they should all be principal, by Remark I.2.6 and Corollary I.2.10 (ii)). Theorem I.2.22 If R is hereditary, then every submodule of a free R-module is R-isomorphic to a direct sum of ideals of R. Proof

Cf., for example, Reference [33], chapter I, theorem 5.3, p. 13.

2

Corollary I.2.23 If R is a PID, then every submodule of a free R-module is free; in particular, every projective R-module P is free. Corollary I.2.24 R is hereditary if and only if every submodule of a projective R-module is projective.

Linear algebra

17

Proof If part: since R is R-projective (being R-free of rank 1), every submodule of R (= every ideal of R) is projective. Only if part: every R-projective P is a direct summand of some free R-module; by the theorem, every submodule M of P is a direct sum of ideals, hence of projective modules (by deﬁnition of hereditary ring); but a direct sum of projective modules is obviously projective. 2 I.2.3 Projective resolutions We push the analysis of non-free modules a little further. Given any R-module M , with a generating system {mi }i∈I , we have already constructed a free presentation of M , that is, an exact complex ϕ1

ϕ0

F1 → F0 → M → 0, with each Fj a free R-module. Since ker(ϕ1 ), too, is a quotient of some free R-module F2 , a longer exact complex is obtained: ϕ2

ϕ1

ϕ0

F2 → F1 → F0 → M → 0. Repeating the argument, one gets an a priori inﬁnite exact complex (∗)

ϕn+1

ϕn

ϕ1

ϕ2

ϕ0

· · · → Fn → · · · → F1 → F0 → M → 0

of free R-modules (except M ), which is called a free resolution of M . Sometimes one says that the free resolution of M is just the exact complex ϕn+1

ϕn

ϕ1

ϕ2

· · · → Fn → · · · → F1 → F0 and that M = coker(ϕ1 ). By the same token, we may also consider projective resolutions of a module, M , namely, an exact sequence (∗∗)

dn+1

d

d

d

n · · · →2 P1 →1 P0 → M → 0 · · · → Pn →

where each module Pi is projective. Since free modules are projective, and we have already shown the existence of free resolutions, the existence of projective resolutions is assured. If for some non-negative integer n we have Fn (Pn ) = 0 and Fn+t (Pn+t ) = 0 for every t > 0, we say that M has a ﬁnite free (projective) resolution of length n. A famous theorem due to D. Hilbert states that if R = K[x1 , . . . , xt ], K a ﬁeld, then every ﬁnitely generated R-module has a ﬁnite free resolution of length less than or equal to t. (Note that in the case considered by Hilbert, each of the modules occurring in the resolution is ﬁnitely generated, since R is noetherian.) We can now write down the formal deﬁnition of homological dimension that we promised at the end of the proof of Corollary I.1.6.

18

Recollections and perspectives

Deﬁnition I.2.25 A module, M , has homological (free) dimension less than or equal to n if it has a projective (free) resolution of length equal to n. Otherwise it is said to have inﬁnite homological (free) dimension. Since every free module is projective, if the R-module M has a ﬁnite free resolution of length n, then obviously its homological dimension hdR M cannot be larger than n. However, it is quite possible for the free dimension of a module to exceed its homological dimension, as the module R/(a, b) of Example I.1.11 easily shows. Let M and N be two R-modules. Take a (possibly inﬁnite) projective resolution of M , such as (∗∗) above, and tensor it by N : dn+1 ⊗1

d ⊗1

d ⊗1

d ⊗1

n 2 1 · · · −→ P1 ⊗R N −→ P0 ⊗R N → M ⊗R N → 0 · · · −→ Pn ⊗R N −→

(1 stands for the identity map on N ). We still have exactness at M ⊗R N (for ⊗R preserves surjectivity), but the truncated complex dn+1 ⊗1

d ⊗1

d ⊗1

d ⊗1

n 2 1 · · · −→ Pn ⊗R N −→ · · · −→ P1 ⊗R N −→ P0 ⊗R N

may have nontrivial homology. Deﬁnition I.2.26 For every n ≥ 0, the n-th torsion module, TorR n (M, N ), is deﬁned as follows:   coker(d1 ⊗ 1) if n = 0 (M, N ) = TorR n  ker(dn ⊗1) if n ≥ 1. im(dn+1 ⊗1) (Notice that TorR 0 (M, N ) = M ⊗R N .) It is well known that the deﬁnition of TorR n (M, N ) does not depend on the choice of the projective resolution of M . Furthermore, if one picks a projective resolution of N and tensors it by M , the resulting truncated complex yields the R same homology modules. That is, TorR n (M, N ) = Torn (N, M ). R Clearly, Torn (M, N ) = 0 for every n ≥ 1, whenever either M or N is projective. As one might expect, Tor has some connection with the notion of torsion. If N is an R-module, an element a ∈ N is said to be a torsion element if there is a non-zero divisor r ∈ R such that ra = 0. The subset Ntor = {a ∈ N | a is a torsion element} is clearly a submodule of N , and is called the torsion submodule of N . N is called torsion-free if Ntor = 0; it is called a torsion module if N = Ntor . Example I.2.27 Then

Let R be a commutative ring and r ∈ R a non-zero divisor. r

0 → R → R → R/(r) → 0

Linear algebra

19

is a projective (in fact free) resolution of R/(r), and for every R-module N , we have that TorR 1 (R/(r), N ) = {elements of N killed by r} . If TorR 1 (R/(r), N ) = 0 for every non-zero divisor r ∈ R, then N is a torsion-free R-module. In particular, N free implies N torsion-free (by the remark coming immediately before this example). It is well known that Tor “restores exactness” in the following sense. Given a short exact sequence of R-modules: 0 → M → M → M → 0, tensoring by N may destroy exactness on the left. But one can prove that there is a long exact sequence: ... R R → TorR n (M , N ) → Torn (M, N ) → Torn (M , N ) → ... R R → TorR 1 (M , N ) → Tor1 (M, N ) → Tor1 (M , N ) → M ⊗R N →

M ⊗R N →

M ⊗R N → 0.

Proposition I.2.28 Let R be a PID and N a ﬁnitely generated R-module: N is torsion free if and only if N is free. Proof The if part is in the example above. Let us prove the only if part. Thanks to Corollary I.2.23, Proposition I.2.18, and the fact that localization preserves ﬁnite generation and torsion-freeness, we may assume that R is local, hence a discrete valuation ring with maximal ideal (a), say. If the R/(a)-vector space N ⊗R R/(a) has dimension t, by Lemma I.2.16 (ii) we can ﬁnd a rank t free R-module F such that 0→K→F →N →0 is exact (K simply stands for the kernel of the R-surjection F → N ). Tensoring by R/(a), one gets the long exact sequence: · · · → TorR 1 (R/(a), N ) → K ⊗R R/(a) → F ⊗R R/(a) → N ⊗R R/(a) → 0. But the indicated Tor1 is zero (because N is torsion-free and the last example applies), and the two vector spaces involving F and N have equal dimensions by construction. Thus K ⊗R R/(a) = 0, that is, K/(a)K = 0 and by Lemma I.2.16 (i), K = 0 and F ∼ 2 = N as wished. In the last proposition, the assumption that N is ﬁnitely generated cannot be removed: the Z-module Q is torsion-free, but not free.

20

Recollections and perspectives

Corollary I.2.29 Let R be a commutative ring, and N an R-module. Then N/Ntor is torsion-free. Thus, if R is a PID and N is ﬁnitely generated, then N is the direct sum of a free module and a torsion module. Proof If r ∈ R is a non-zero divisor, and a ∈ N/Ntor is such that ra = 0, then ra ∈ Ntor . Hence there exists a non-zero divisor s ∈ R such that 0 = s(ra) = (sr)a. Since s, r are non-zero divisors, it follows a ∈ Ntor , that is, a = 0 which proves that N/Ntor is torsion-free. But by the last proposition, if R is a PID, N/Ntor is R-free if it is torsion-free. Since N/Ntor is R-free, the short exact sequence 0 → Ntor → N → N/Ntor → 0 splits, and N = Ntor ⊕ N/Ntor .

2

Another example of “exactness restoration” is provided by Ext. Let M and N be two R-modules. Taken a (possibly inﬁnite) projective resolution of M , such as (∗∗) above, the following complex is obtained: ◦d

◦d

1 2 HomR (P1 , N ) −→ ··· , 0 → HomR (M, N ) −→ HomR (P0 , N ) −→

d

n where ◦dn sends a map Pn−1 → N to the composite Pn −→ Pn−1 → N . We still have exactness at HomR (M, N ), just because P0 projects onto M , but the truncated complex

◦d

◦d

1 2 HomR (P1 , N ) −→ ··· , 0 → HomR (P0 , N ) −→

may have nontrivial homology in dimension zero as well as elsewhere. Deﬁnition I.2.30 For every n ≥ 0, the n-th extension module ExtnR (M, N ) is deﬁned as follows:  if n = 0  ker(◦d1 ) n ExtR (M, N ) =  ker(◦dn+1 ) if n ≥ 1. im(◦dn ) (Notice that since Ext0R (M, N ) is the kernel of the map ◦d1 , and since ◦d

1 0 → HomR (M, N ) → HomR (P0 , N ) −→ HomR (P1 , N )

is exact, we have Ext0R (M, N ) = HomR (M, N ).) As in the case of Tor, one can check that the deﬁnition of ExtnR (M, N ) does not depend on the choice of the projective resolution of M . But if one wants to get the same homology modules starting with a resolution of N , it is necessary to pick an injective resolution of N and apply HomR (M, ). It is clear that ExtnR (M, N ) = 0 for every n ≥ 1, whenever M is projective.

Multilinear algebra

21

Moreover, one can prove that given a short exact sequence of R-modules: β

0 → M → M → M → 0, α

there exists a long exact sequence (note the reverse arrows): HomR (M , N ) ← HomR (M, N ) ← HomR (M , N ) ← 0 ← Ext1R (M , N ) ← Ext1R (M, N ) ← Ext1R (M , N ) ← ... · · · ← ExtnR (M , N ) ← ExtnR (M, N ) ← ExtnR (M , N ) ← By deﬁnition, an R-module L is called an extension of M by N if there exists a short exact sequence 0 → N → L → M → 0. If the sequence is split, the extension is trivial, because L = N ⊕ M . As the reader may have guessed, these extensions may be turned into a group which is isomorphic to Ext1R (M, N ), with the zero element being the split extension. Thus Ext1R (M, N ) = 0 if and only if all extensions of M by N are trivial. If part: Any short exact sequence 0 → K → F → M → 0 with F free is split by assumption; hence M is projective and ExtnR (M, N ) = 0 for every n ≥ 1, as already observed. Only if part: Apply HomR ( , N ) to every given short exact sequence β

α

0 → N → L → M → 0. Since Ext1R (M, N ) = 0 by assumption, the long exact sequence of Ext yields the surjectivity of ◦α

HomR (L, N ) → HomR (N, N ). In particular, there exists ϕ : L → N such that ϕ ◦ α = idN , and we are done by Proposition I.2.12 (ii). I.3 Multilinear algebra In this section, we deal with some algebras to be used extensively in the future, namely symmetric, divided power, and exterior algebras. Our approach here is via the structure of Hopf algebra, which involves the notions of algebra, coalgebra, multiplication, comultiplication, and antipode map, among others. We do this in order to make clear the relationship between the symmetric and divided power algebras, and to bring out their many useful properites. For more details, we refer the reader to References [15], [16], and [41].

22

Recollections and perspectives

I.3.1 R[X1 , . . . , Xt ] as a symmetric algebra Let R be a commutative ring (with 1), and let S denote the polynomial ring R[X1 , . . . , Xt ]. S is an R-module with respect to the usual addition of polynomials and to their multiplication by elements of R. Moreover, multiplication of general elements of S deﬁnes a map mS : S × S → S which is R-bilinear. Hence one has an R-morphism S ⊗R S → S, still denoted by mS . If one further considers the R-map uS : R → S deﬁned by means of uS (1) = 1, the fact that multiplication of polynomials is associative and has a neutral element implies that S is an R-algebra, in the sense of the following deﬁnition. Deﬁnition I.3.1 Given a commutative ring R (with 1), an R-algebra is an R-module A endowed with two R-morphisms mA : A ⊗R A → A (multiplication),

uA : R → A (unit)

such that the following diagrams are commutative: m⊗1

A⊗A⊗A → A⊗A 1⊗m↓

↓m m

→

A⊗A 1⊗u

A⊗R → A⊗A ∼ =↓ A

1

→

A

R⊗A

↓m

∼ =↓

A

A

u⊗1

→

A⊗A ↓m

1

→

A

(we have omitted subscripts, and 1 stands for the identity on A). One should remark that an R-algebra A is always a ring, multiplication being given by the composite map mA ◦ χ, where χ stands for the canonical bilinear map A × A → A ⊗R A, and 1 being provided by uA (1R ). It is a commutative ring if the following diagram commutes: τ

A ⊗ A→A ⊗ A m↓ A

↓m 1

→

A

(here τ is the R-linear map sending a1 ⊗ a2 to a2 ⊗ a1 ). By an ideal (right, left, two-sided) of the R-algebra A we always mean an ideal (right, left, two-sided) of the ring A.

Multilinear algebra

23

Given two R-algebras A and B, an R-algebra homomorphism ϕ : A → B is an R-map also compatible with the ring structures of A and B. An important feature of a polynomial is its degree (total degree, that is). If for every i ∈ N, Si stands for the R-linear span of all monomials of total degree i, S decomposes (as an R-module) as the direct sum ⊕i∈N Si . Moreover, the product of f ∈ Si and g ∈ Sj belongs to Si+j . This means that S is a graded algebra, in the sense of the following deﬁnition. Deﬁnition I.3.2 An R-algebra A is called graded if it decomposes (as an R-module) as ⊕i∈N Ai , in such a way that Ai Aj ⊆ Ai+j for every i and j. The (non-zero) elements of Ai are called homogeneous elements of degree i. Let now a be an ideal of S. Clearly the ring S/a is again an R-algebra. But S/a inherits the graduation of S if and only if a has a system of generators which are homogeneous. Remark I.3.3 Let A be any graded R-algebra, a a two-sided ideal of A. The R-module A/a always inherits from A the structure of an R-algebra. But the R-algebra A/a inherits the graduation of A if and only if a has a system of generators which are homogeneous. We now turn to a signiﬁcant property of the R-algebra S. Let F be the rank t free R-module corresponding to the set I = {x1 , . . . , xt } (recall Proposition I.2.2). Denote by {fxi } the canonical basis of F and deﬁne the R-map ϕ : F → S by means of ϕ(fxi ) = xi . Clearly, ϕ(F ) is a set of (commuting) generators for the R-algebra S. Proposition I.3.4 For every R-morphism ψ : F → A, with A an R-algebra, such that the elements of ψ(F ) commute in A, there exists a unique R-algebra morphism χ : S → A verifying ψ = χ ◦ ϕ. Proof If we deﬁne χ c(s1 , . . . , st )xs11 · · · xst t = c(s1 , . . . , st )ψ(fx1 )s1 · · · ψ(fxt )st , everything works.

2

The last proposition says that our polynomial ring S identiﬁes with the symmetric algebra S(F ), in the sense of the following deﬁnition. Deﬁnition I.3.5 Let M be an R-module. We call a symmetric algebra on M any pair (S(M ), ϕ) satisfying the following requirements: S(M ) is an R-algebra, ϕ : M → S(M ) is an R-morphism, the elements of ϕ(M ) commute in S(M ), and for every R-map ψ : M → A, with A an R-algebra, such that the elements of ψ(M ) commute in A, there exixts a unique R-algebra morphism χ : S(M ) → A verifying ψ = χ ◦ ϕ. It is clear that, if symmetric algebras on M do exist, there is a unique isomorphism from one to the other which is the identity on M .

24

Recollections and perspectives

Theorem I.3.6

Every M admits a symmetric algebra.

We sketch a proof of the theorem. The idea is to build ﬁrst an R-algebra T (M ) and an R-map σ : M → T (M ) with the following properties: (1) for every R-map ψ : M → A, with A an R-algebra, there exists a unique R-algebra morphism ρ : T (M ) → A verifying ψ = χ ◦ σ; (2) the elements of σ(M ) generate T (M ). Then one deﬁnes S(M ) as the quotient T (M )/a, where a is the two-sided ideal generated by all the elements σ(m1 )σ(m2 ) − σ(m2 )σ(m1 ) with m1 and m2 in M . The map ϕ is deﬁned as the composite: σ

M → T (M ) → T (M )/a. Clearly, the elements of ϕ(M ) commute in S(M ). If ψ : M → A is given, such that the elements of ψ(M ) commute in A, the map ρ : T (M ) → A verifying ρ ◦ σ = ψ vanishes on a, and we call χ the map T (M )/a → A induced by ρ. Finally, since T (M ) is generated by the elements of σ(M ), ϕ(M ) generates S(M ) and the uniqueness of χ follows. A pair (T (M ), σ), necessarily unique, up to isomorphism, can simply be obtained by taking the R-module ⊕i∈N M ⊗i and by imposing that every product (m1 ⊗ · · · ⊗ mi )(m1 ⊗ · · · ⊗ mj ) be equal to m1 ⊗ · · · ⊗ mi ⊗ m1 ⊗ · · · ⊗ mj . (We mean M ⊗0 = R.) The map σ identiﬁes M with the summand M ⊗1 . One should remark that T (M ) (called the tensor algebra on M ) is graded by construction, and that the ideal a is generated by homogeneous elements. Hence S(M ) inherits the graduation of T (M ). One usually writes S(M ) = ⊕i∈N Si (M ). In particular, ϕ identiﬁes M with S1 (M ), and the elements of S1 (M ) generate S(M ). Thus S(M ) is commutative. The construction of S(M ) is functorial [meaning that every R-map M1 → M2 induces an R-algebra morphism S(M1 ) → S(M2 )] and commutes with base change [meaning that S(M ⊗R R ) = S(M ) ⊗R R for every ring homomorphism R → R ]. These properties follow from those of the tensor algebra, that is, of T (M ). If M is R-free of ﬁnite rank, with basis {m1 , . . . , mt } say, then S(M ) is R-free, and a basis for Si (M ) is given by all distinct elements ms11 · · · mst t such that s1 + · · · + st = i (not surprisingly!). Our polynomial ring S, being a symmetric algebra, is also a graded coalgebra, and in fact a graded Hopf algebra. Deﬁnition I.3.7 Given a commutative ring R (with 1), an R-coalgebra is an R-module A endowed with two R-morphisms cA : A → A ⊗R A (comultiplication),

εA : A → R (counit)

Multilinear algebra

25

such that the following diagrams commute: A⊗A⊗A

c⊗1

←− A ⊗ A

1⊗c↑

↑c

A⊗A 1⊗ε

A ⊗ R ←− A ⊗ A ∼ =↑ A

1

←−

c

←−

A ε⊗1

R ⊗ A ←− A ⊗ A

↑c

∼ =↑

A

A

↑c 1

←−

A

(as before, we have omitted subscripts, and 1 stands for the identity on A). It should be noticed that the diagrams occurring in the last deﬁnition are obtained by reversing arrows in those describing algebra properties. S(M ) is an R-coalgebra with respect to the following counit and comultiplication. ε is the identity on S0 (M ) = R and the zero map on Si (M ) for every i ≥ 1. c is constructed from the diagonal mapping ∆ : m ∈ M → (m, m) ∈ M × M : since ∆ is R-linear, a morphism S(M ) → S(M × M ) of R-algebras is induced; but S(M × M ) ∼ = S(M ) ⊗R S(M ) as R-algebras, and c is deﬁned to be the composite S(M ) → S(M × M ) ∼ = S(M ) ⊗R S(M ). Remark I.3.8 In the above, the following deﬁnition is understood: given two R-algebras A and B, the R-algebra A ⊗R B is the R-module A ⊗R B with multiplication deﬁned by (a1 ⊗ b1 )(a2 ⊗ b2 ) = a1 a2 ⊗ b1 b2 u ⊗u

and unit given by the composition RA −→ R ⊗R R A−→B A ⊗R B. ∼ S(M ) ⊗R S(M ) is easily seen if The fact, stated above, that S(M × M ) = one takes, instead of M × M the direct sum M ⊕ M . It is clear that M ⊕ M is the degree one part of both S(M ⊕ M ) and of S(M ) ⊗R S(M ). The universality property of the functor S(M ) does the rest. As for an explicit description of c in the case of S(M ), since the isomorphism S(M × M ) ∼ = S(M ) ⊗R S(M ) is induced by (m1 , m2 ) → m1 ⊗ 1 + 1 ⊗ m2 , and c is a morphism of algebras, one gets c(m1 · · · mk ) =

k i=1

(mi ⊗ 1 + 1 ⊗ mi ) =

(mi1 · · · mih ) ⊗ (mj1 · · · mjk−h ),

26

Recollections and perspectives

the sum ranging over all pairs of strictly increasing sequences i1 < · · · < ih and j1 < · · · < jk−h such that {i1 , . . . , ih } ∪ {j1 , . . . , jk−h } = {1, . . . , k} (including the cases in which one of the two sequences is empty). One should remark that S(M ) is a graded coalgebra, in the sense that all maps are compatible with the given gradings. Remark I.3.9 Since the comultiplication of S(M ) is obtained from the diagonal map M → M × M , it is customary to call diagonal map the comultiplication of any coalgebra, replacing the letter c by ∆. In order to explain what a (graded) Hopf algebra is, we need the deﬁnition of the product of coalgebras. Let A and B be two given R-coalgebras. The R-coalgebra A ⊗R B is the R-module A ⊗R B with counit εA⊗R B (a ⊗ b) = εA (a)εB (b) and with diagonal map (ai ⊗ bj ) ⊗ (ai ⊗ bj ), ∆A⊗R B (a ⊗ b) =

ai

i,j

where i ai ⊗ = ∆A (a) and j bj ⊗ bj = ∆B (b). We also need the following: given R-coalgebras A and B, an R-coalgebra homorphism ϕ : A → B is an R-module map such that the diagrams ∆

A A⊗A A −→

ϕ↓

↓ϕ⊗ϕ ∆

B B⊗B B −→

ε

A R A −→

ϕ↓

↓1 ε

B B −→ R

are commutative. Deﬁnition I.3.10 A Hopf algebra over R is an R-module A, which is both an algebra and a coalgebra over R, satisfying the following properties: (1) the multiplication m and the unit u are coalgebra morphisms; (2) the diagonal map ∆ and the counit ε are algebra morphisms; (3) there exixts an R-module map sA : A → A (called the antipode) such that the diagrams 1⊗s

A ⊗ A −→ A ⊗ A ∆↑ A are commutative.

↓m u◦ε

−→

A

s⊗1

A ⊗ A −→ A ⊗ A ∆↑ A

↓m u◦ε

−→

A

Multilinear algebra

27

We leave as an exercise the veriﬁcation that S(M ) is indeed a graded Hopf algebra, that is, a Hopf algebra with all structure maps preserving the graded structure. The antipode s is deﬁned as 1 on all Si (M ) with i even, and as −1 on all Si (M ) with i odd. In fact, S(M ) is a commutative graded Hopf algebra, meaning that it is both a commutative graded R-algebra and a cocommutative graded R-coalgebra. A little bit of care is necessary here, for the general deﬁnition of (co-) commutativity in the graded case does not suit the intuitive ideas suggested by our polynomial ring. Deﬁnition I.3.11 Let A = ⊕i∈N Ai and B = ⊕i∈N Bi be two graded R-modules. Let τB,A : B ⊗R A → A ⊗R B be the map deﬁned on homogeneous elements by means of τB,A (b ⊗ a) = (−1)deg(b) deg(a) a ⊗ b. If A has the structure of a graded R-algebra, we say that it is a commutative graded R-algebra if the following diagram τA,A

A ⊗ A −→ A ⊗ A m↓ A

↓m 1

−→

A

is commutative. If B has the structure of a graded R-coalgebra, we say it is a cocommutative graded R-coalgebra if the following diagram τB,B

B ⊗ B ←− B ⊗ B ∆↑ B

↑∆ 1

←−

B

is commutative. For these notions of commutativity and cocommutativity to suit a symmetric algebra ⊗iεN Si (M ), one usually assumes that the elements of Si (M ) have degree 2i, not i. Remark I.3.12 The last deﬁnition also has an impact on the construction of the graded R-algebra A ⊗R B. The multiplication deﬁned on A ⊗R B in Remark I.3.8 is the composition A⊗B⊗A⊗B

1A ⊗τ ⊗1B

−→

A⊗A⊗B⊗B

mA ⊗mB

−→

A⊗B

(with τ (b, a)=(a, b)) and applies to non-graded algebras, as well as to graded algebras concentrated in even degrees (such as the symmetric algebras). But in the

28

Recollections and perspectives

general graded case, it is replaced by the following, slightly diﬀerent, composition A⊗B⊗A⊗B

1A ⊗τB,A ⊗1B

−→

A⊗A⊗B⊗B

mA ⊗mB

−→

A ⊗ B,

with τB,A as in the last deﬁnition. That is, (a1 ⊗ b1 )(a2 ⊗ b2 ) = (−1)deg(b1 ) deg(a2 ) a1 a2 ⊗ b1 b2 . In the future, unless otherwise stated, we always adopt for graded algebras the speciﬁed graded versions of (co-)commutativity and tensor product. We also assume that the elements of S(Mi ) have degree 2i. I.3.2 The divided power algebra We again let F be a rank t free R-module, and let a basis of F be {x1 , . . . , xt }. We use E (rather than F ∗ ) to denote the R-module HomR (F, R), the dual of F . Since ⊕ti=1 commutes with HomR , E is a rank t free R-module. We indicate by 1 if i = j (the {y1 , . . . , yt } the basis of E such that for every i, yi (xj ) = 0 otherwise so-called dual basis of {x1 , . . . , xt }). For every n ∈ N, the symmetric group Sn acts on the n-fold tensor product Tn (E) = E ⊗R · · · ⊗R E by means of σ(e1 ⊗ · · · ⊗ en ) = eσ(1) ⊗ · · · ⊗ eσ(n) . The elements of Tn (E) ﬁxed by such an action are called symmetric tensors of order n and form an R-submodule of Tn (E), denoted by Dn (E). There is an R-map, sym : ⊕n∈N Tn (E) −→ ⊕n∈N Dn (E), called symmetrization, deﬁned by means of σ(z) sym(z) = σ∈Sn

for every z ∈ Tn (E) and for every n ∈ N. If z ∈ Dn (E), then sym(z) = n!z. Let D(E) stand for the graded R-module ⊕n∈N Dn (E). Proposition I.3.13

D(E) is a graded R-algebra.

Given z ∈ Di (E) and z ∈ Dj (E), then zz ∈ Di+j (E) is deﬁned to be σ∈Si,j σ(z ⊗ z ), where

Proof

Si,j = {σ ∈ Si+j | σ(1) < · · · < σ(i) The remainder of the proof is standard.

and σ(i + 1) < · · · < σ(i + j)} . 2

Remark I.3.14 If e1 , . . . , en belong to E, then their product e1 · · · en in D(E) equals sym(e1 ⊗ · · · ⊗ en ). Deﬁnition I.3.15 R-module E.

D(E) is called the divided power algebra on the free

If we assume that the elements of Dn (E) have degree 2n, then D(E) is commutative as a graded R-algebra.

Multilinear algebra

29

The construction of D(E) is functorial and commutes with base change. We now give a reason for the name assigned to D(E). It is clear that for every e ∈ E and for every n ∈ N, the n-fold tensor product e ⊗ · · · ⊗ e belongs to Dn (E): we denote it by e(n) . Since sym(e(n) ) = n!e(n) , by Remark I.3.14 it follows that n!e(n) equals the product in D(E) of n copies of e (=e(1) ), that is, the n-th power of e. If n! happens to be invertible in R, then e(n) coincides with the n-th power of e divided by n! Deﬁnition I.3.16 For every e ∈ E and every n ∈ N, the element e(n) of Dn (E) is called the n-th divided power of e. Remark I.3.17 (1) e(i) e(j) =

It is not hard to check that divided powers verify: (i+j)! (i+j) i!j! e

(2) (e1 +e2 )(n) =

n

(i) (n−i) i=0 e1 e2

(no binomial coeﬃcients in the summation!).

Recalling the basis {y1 , . . . , yt } of E, one shows that every Dn (E) is R-free and has a basis given by all elements (s1 )

y1

(st )

· · · yt

with s1 + · · · + st = n.

D(E) is in fact a commutative graded Hopf algebra, with diagonal map deﬁned by: (s ) (s ) (u ) (u ) (v ) (v ) ∆ y1 1 · · · yt t = y1 1 · · · yt t ⊗ y 1 1 · · · y t t , where the sum ranges over all pairs (u1 , . . . , ut ) and (v1 , . . . , vt ) of elements of Nt , such that ui + vi = si for every i in {1, . . . , t}. We leave all the details to the reader. The indicated basis of D(E) leads one to think that D(E) is a symmetric algebra (a polynomial ring). This is indeed the case if R contains a copy of the (s ) (s ) rationals. For then every basis element y1 1 · · · yt t can be replaced by (s1 )

s1 ! · · · st !y1

(st )

· · · yt

= y1s1 · · · ytst

and D(E) ∼ = R[y1 , . . . , yt ]. In general, however, D(E) is not a symmetric algebra. What is true, though, is that D(E) is the graded dual of a symmetric algebra. Remark I.3.18

It is well known that, given R-modules Mi , i ∈ N, and L, Mi , L ∼ HomR (Mi , L). HomR = i∈N

i∈N

In particular, the ordinary dual of a graded R-algebra A = ⊕i∈N Ai is i∈N A∗i . Hence, in order to make sure that the dual of a graded algebra is still a graded algebra, one replaces A∗ with another object, namely ⊕i∈N A∗i , which is called the graded dual of A and is denoted by A∗gr .

30

Recollections and perspectives

Theorem I.3.19 D(E) is the graded dual of the symmetric algebra S(F ), and S(F ) is the graded dual of the divided power algebra D(E). If one uses the notation F ∗ in place of E, the above says that D(F ∗ ) = S(F )∗gr and S(F ) = D(F ∗ )∗gr . We sketch a proof of the theorem, by giving an idea of why D(E)∗gr equals S(F ). Essentially, it all amounts to showing that the diagonal map of D(E) induces on D(E)∗gr a graded R-algebra structure corresponding to that of S(F ). Let us take f ∈ Di (E)∗ and g ∈ Dj (E)∗ , and use the diagonal map of D(E) to deﬁne an element f g ∈ Di+j (E)∗ . Since f ∈ HomR (Di (E), R) and g ∈ HomR (Dj (E), R), we may set f g equal to the composite map f ⊗g

∆

Di+j (E) −→ Di (E) ⊗R Dj (E) −→ R ⊗R R −→ R, an element of HomR (Di+j (E), R), as required. One checks that this multiplication gives D(E)∗gr (which equals S(F ) as an R-module) an R-algebra structure. We claim that such a structure is the ordinary multiplication in S(F ). Clearly, it suﬃces to prove that ∗ ∗ ∗ (s ) (s ) (r ) (r ) (s +r ) (s +r ) y 1 1 · · · yt t = y1 1 1 · · · yt t t , (∗) y1 1 · · · y t t where

(s1 )

y1

(s )

(st )

· · · yt

∗

stands for the element of (Ds1 +···+st E)∗ = Ss1 +···+st F

(s )

dual to y1 1 · · · yt t (and due to correspond to xs11 · · · xst t ). But (∗) is obvious, in view of the deﬁnition of ∆ in D(E) recorded above. Example I.3.20

Let us take t = 2 and show that y1∗ (y1 y2 )∗ = (y1 y2 )∗ . (2)

f ⊗g

∆

We consider D3 (E) −→ D1 (E) ⊗R D2 (E) −→ R ⊗RR −→ R, where f andg (3) (2) (2) (3) are y1∗ and (y1 y2 )∗ , respectively. A basis for D3 (E) is y1 , y1 y2 , y1 y2 , y2 . (2)

We must prove that the composite map assigns 1 to y1 y2 and 0 to all other basis elements. (2) (2) The action of y1∗ ⊗(y1 y2 )∗ on ∆(y1 y2 ) = y1 ⊗y1 y2 +y2 ⊗y1 yields 1⊗1+0⊗0, hence 1. (2) (2) (2) ∗ (3) ) = y1 ⊗ y1 , on ∆(y1 y2 ) = y1 ⊗ y2 + The action of y1∗ ⊗(y1 y 2 ) on ∆(y (3)

(2)

y2 ⊗ y1 y2 , and on ∆ y2 = y2 ⊗ y 2 and 0 ⊗ 0; hence 0 all the times.

yields (respectively) 1 ⊗ 0, 1 ⊗ 0 + 0 ⊗ 1,

Concerning the example, one should notice that the product of y1 and y1 y2 in (2) D(E) equals 2y1 y2 (by part 1 of Remark I.3.17), while x1 times x1 x2 in S(F ) gives x21 x2 . I.3.3 The exterior algebra Deﬁnition I.3.21 Let M be an R-module. We call an exterior algebra on M a pair (Λ(M ), ϕ) satisfying the following requirements: Λ(M ) is an R-algebra,

Multilinear algebra

31

ϕ : M → Λ(M ) is an R-morphism, every z ∈ ϕ(M ) gives z 2 = 0, and for every R-map ψ : M → A, with A an R-algebra, such that every a ∈ ψ(M ) gives a2 = 0 in A, there exixts a unique R-algebra morphism χ : Λ(M ) → A verifying ψ = χ ◦ ϕ. Clearly, if exterior algebras on M exist, there is a unique isomorphism from one to the other which is the identity on M . Theorem I.3.22

Every M admits an exterior algebra.

As in the case of S(M ), one takes the tensor algebra T (M ) and deﬁnes Λ(M ) as the quotient T (M )/b, where b is the two-sided ideal generated by all elements m ⊗ m, with m ∈ M . The map ϕ is given by the composition: M → T (M ) → T (M )/b. Clearly, ϕ(m)2 = 0 for every m ∈ M . If ψ : M → A is given, such that every a ∈ ψ(M ) gives a2 = 0 in A, then the map T (M ) → A extending ψ vanishes on b, and induces the required χ : T (M )/b → A. Finally, since T (M ) is generated by the elements of M , ϕ(M ) generates Λ(M ) and the uniqueness of χ follows. One should remark that since b is homogeneous, Λ(M ) inherits the graduation of T (M ): Λ(M ) = ⊕i∈N Λi (M ) is the customary notation. In particular, ϕ identiﬁes M with Λ1 (M ), and the elements of Λ1 (M ) generate Λ(M ). The construction of Λ(M ) is functorial and commutes with base change. Multiplication in Λ(M ) is called exterior multiplication. The usual notation for it is m1 ∧ m2 , and we stick to this convention. Regarding the commutativity of Λ(M ), some considerations are in order. Since m1 , m2 ∈ M implies 0

= (m1 + m2 ) ∧ (m1 + m2 ) = m1 ∧ m1 + m1 ∧ m2 + m2 ∧ m1 + m2 ∧ m2 = m 1 ∧ m2 + m 2 ∧ m 1 ,

we get m1 ∧ m2 = −m2 ∧ m1 . Hence (m1 ∧ · · · ∧ mi ) ∧ (mi+1 ∧ · · · ∧ mi+j ) =(−1)ij (mi+1 ∧ · · · ∧ mi+j ) ∧ (m1 ∧ · · · ∧ mi ). But Λ(M ) generated by Λ1 (M ) then says z ∧ w = (−1)deg(z) deg(w) w ∧ z for every choice of homogeneous elements z and w. We conclude that Λ(M ) is a commutative graded algebra in the sense of Deﬁnition I.3.11. In fact more is true. Λ(M ) is a commutative graded Hopf algebra. We do not go into the details, but record here the diagonal map: ∆(m1 ∧ · · · ∧ mk ) = (−1)s (mi1 ∧ · · · ∧ mih ) ⊗ (mj1 ∧ · · · ∧ mjk−h ),

32

Recollections and perspectives

where s stands for the number of ordered pairs (u, v) verifying jv < iu , and the sum ranges over all pairs of strictly increasing sequences i1 < · · · < ih and j1 < · · · < jk−h such that {i1 , . . . , ih } ∪ {j1 , . . . , jk−h } = {1, . . . , k} (including the cases in which one of the two sequences is empty). We now focus our attention on Λ(F ), where F is a ﬁnite free R-module, and reobtain some results of Subsection I.2.1, namely Propositions I.2.4, I.2.7, and I.2.9. Lemma I.3.23 Let M be a ﬁnitely generated R-module and let {m1 , . . . , mt } be a generating system. Then Λn (M ) = 0 for every n > t. Proof For every n ≥ 1, Λn (M ) has a generating system given by all elements 2 of type mi1 ∧ · · · ∧ min , and if n > t, some mi occurs more than once. Proposition I.3.24 Let F be a free R-module t with basis {f1 , . . . , ft }. Then Λ(M ) is R-free with a basis of cardinality i=0 ti = 2t . Proof Write F = F1 ⊕ · · · ⊕ Ft , where Fi = Rfi for every i = 1, . . . , t. Since Fi has one generator, Lemma I.3.23 says that Λ(Fi ) = R ⊕ Fi ; hence Λ(Fi ) is R-free with basis {1, fi }. It follows that Λ(F ) = Λ(F1 ) ⊗ · · · ⊗ Λ(Ft ) has a basis given by all elements m1 ∧ · · · ∧ mt , with mi ∈ {1, fi }. In particular, for every s ∈ {0, 1, . . . , t}, the elements m1 ∧ · · · ∧ mt with s factors diﬀerent from 1 are a basis, Bs , of Λs (F ), with cardinality st . Explicitly: Bs = {fi1 ∧ · · · ∧ fis with i1 < · · · < is }. 2 Remark I.3.25 (Proposition I.2.4) If B = {f1 , . . . , ft } and B = {f1 , . . . , fs } are two ﬁnite bases of the same free R-module F , then t = s. For both t < s and t > s give a contradiction. For instance, if t < s, then |B| = t implies Λs (F ) = 0 (by Lemma I.3.23), while |B | = s implies Λs (F ) = 0, since Λs (F ) has a basis s of cardinality s = 1 (by Proposition I.3.24). In order to state the next lemma, we need the notion of an alternating n-linear function on M . It is an n-linear function ψ : M × · · · × M → N , with N any R-module, such that ψ(m1 , . . . , mn ) = 0 whenever two of the arguments coincide. Clearly, ϕ : M × · · · × M → Λn (M ),

(m1 , . . . , mn ) → m1 ∧ · · · ∧ mn

is an alternating n-linear function. It is not hard to see that every ψ as above factors through ϕ, that is, ψ = χ ◦ ϕ for some linear map χ : Λn (M ) → N . Lemma I.3.26 Let F be a ﬁnite free R-module and {m1 , . . . , mt } ⊆ F . Then, m1 ∧· · ·∧mt = 0 if and only if there exists a t-linear function ψ : F ×· · ·×F → R such that ψ(m1 , . . . , mt ) = 0.

Multilinear algebra

33

Proof If part. Write ψ as the composite χ ◦ ϕ, where χ : Λt (F ) → R sends m1 ∧ · · · ∧ mt to ψ(m1 , . . . , mt ). Then m1 ∧ · · · ∧ mt = 0 would give χ(m1 ∧ · · · ∧ mt ) = 0, a contradiction. Only if part. If not, all elements of (Λt (F ))∗ would vanish on m1 ∧ · · · ∧ mt , that is, m1 ∧ · · · ∧ mt (as an element of (Λt (F ))∗∗ = Λt (F )) would vanish on 2 (Λt (F ))∗ , that is, m1 ∧ · · · ∧ mt = 0 in Λt (F ). Proposition I.3.27 Let F be a ﬁnite free R-module and {m1 , . . . , mt } ⊆ F . Then, m1 , . . . , mt are linearly dependent if and only if there exists a ∈ R − {0} such that am1 ∧ · · · ∧ mt = 0. Proof Only if part. If a1 m1 + · · · + at mt = 0 with not all coeﬃcients equal to zero, a1 = 0 say, then a1 m1 equals −a2 m2 − · · · − at mt and the properties of exterior multiplication give a1 m1 ∧ m2 ∧ · · · ∧ mt = 0. If part. By induction on t, case t = 1 being trivial. Assume am1 ∧· · ·∧mt = 0; if also am2 ∧· · ·∧mt = 0, then m2 , . . . , mt are dependent by inductive hypothesis, and a fortiori m1 , . . . , mt are dependent. If am2 ∧ · · · ∧ mt = 0, Lemma I.3.26 says that there is a (t − 1)-linear function ψ such that ψ(am2 , . . . , mt ) = b = 0. Since m1 ∧ am2 ∧ · · · ∧ mt = 0, every t-linear function vanishes on (m1 , am2 , m3 , . . . , mt ) (again by Lemma I.3.26). In particular, this happens for (m1 , . . . , mt ) →

t

(−1)i+1 ψ(m1 , . . . , m i , . . . , mt )mi ,

i=1

where means omitted. Thus ψ(am2 , . . . , mt )m1 = bm1 equals a linear 2 combination of m2 , . . . , mt , and m1 , . . . , mt turn out to be dependent. Corollary I.3.28 Let F be a free R-module with basis {f1 , . . . , ft }, and let {m1 , . . . , mt } ⊆ F . Then {m1 , . . . , mt } is a basis of F if and only if m1 ∧ · · · ∧ mt = af1 ∧ · · · ∧ ft with a invertible in R. Proof Only if part. By Proposition I.3.24, {f1 , . . . , ft } a basis of F implies f1 ∧ · · · ∧ ft a basis of Λt (F ) ∼ = R; hence m1 ∧ · · · ∧ mt = af1 ∧ · · · ∧ ft for some a in R. Similarly, {m1 , . . . , mt } a basis of F implies f1 ∧ · · · ∧ ft = bm1 ∧ · · · ∧ mt for some b in R. It follows f1 ∧ · · · ∧ ft = baf1 ∧ · · · ∧ ft , hence ba = 1. If part. Deﬁne χ : Λt (F ) → R by χ(f1 ∧ · · · ∧ ft ) = a−1 . Call ψ the composite χ ◦ ϕ : F × · · · × F → R, where ϕ is as deﬁned before Lemma I.3.26. Clearly,

34

Recollections and perspectives

χ(m1 ∧ · · · ∧ mt ) = aa−1 = 1; hence ψ(m1 , . . . , mt ) = 1. For every m ∈ F , m ∧ m1 ∧ · · · ∧ mt = 0 since Λt+1 (F ) = 0. By Lemma I.3.26, it must vanish on (m, m1 , . . . , mt ) the (t + 1)-linear function (m0 , m1 , . . . , mt ) →

t

(−1)i ψ(m0 , . . . , m i , . . . , mt )mi .

i=0

Hence ψ(m1 , . . . , mt )m =

t

(−1)i+1 ψ(m, m1 , . . . , m i , . . . , mt )mi .

i=1

Since ψ(m1 , . . . , mt ) = 1, m turns out to be a linear combination of m1 , . . . , mt and we have shown that {m1 , . . . , mt } is a generating system. It is in fact a basis, 2 thanks to Proposition I.3.27, because m1 ∧ · · · ∧ mt = 0 by hypothesis. Corollary I.3.29 Let F be a free R-module with basis {f1 , . . . , ft }. Let ϕ : F → N be an injective R-map. Then the induced R-map Λ(ϕ) : Λ(F ) → Λ(N ) is also injective. (By deﬁnition, for every n ∈ N, Λ(ϕ) sends each m1 ∧· · ·∧mn ∈ Λn (F ) to ϕ(m1 ) ∧ · · · ∧ ϕ(mn ).) Proof If not, let the kernel of Λ(ϕ) contain a linear combination of distinct basis elements ai1 ,...,is fi1 ∧ · · · ∧ fis , where every ai1 ,...,is is = 0. Choose a summand having minimum number of factors in the exterior product, say the one corresponding to the indices i1 < · · · < is0 . Call I the subset of {1, . . . , t} having empty intersection with i1 , . . . , is0 , but containing all the other indices occurring in the sum. Say I = {j1 , . . . , jk }, with j1 < · · · < jk . Ley fI denote the basis element fj1 ∧· · ·∧fjk . Then ai1 ,...,is fi1 ∧ · · · ∧ fis ∧ fI simply equals the exterior product of the chosen summand and fI . Since it is still an element of the two-sided ideal ker(Λ(ϕ)), if we apply Λ(ϕ) to it, we get: 0 = ai1 ,...,is ϕ(fi1 ) ∧ · · · ∧ ϕ(fis ) ∧ ϕ(fj1 ) ∧ · · · ∧ ϕ(fjk ). 0

By Proposition I.3.27, it follows that ϕ(fi1 ), . . . , ϕ(fis ), ϕ(fj1 ), . . . , ϕ(fjk ) are 0 linearly dependent: a contradiction, because fi1 , . . . , fis , fj1 , . . . , fjk are inde0 pendent by hypothesis, and the injective map ϕ preserves independence. 2 Remark I.3.30

(Proposition I.2.7)

1. Let F be a free R-module with basis {f1, . . . ,ft }. Let {m1, . . . ,mt } ⊆ F be deﬁned by means of (m1 , . . . , mt )T = S(f1 , . . . , ft )T , where S is a t × t matrix

Multilinear algebra

35

with entries in R, and T denotes transposition. Using the properties of exterior multiplication, one checks that m1 ∧ · · · ∧ mt = af1 ∧ · · · ∧ ft

with a = det S.

Hence by Corollary I.3.28, {m1 , . . . , mt } is another basis of F if and only if det S is invertible in R. 2. Let F be a free R-module with basis {f1 , . . . , ft }. Let {m1 , . . . , mt } ⊆ F be a generating system. We claim that {m1 , . . . , mt } is in fact a basis. For (f1 , . . . , ft )T = U (m1 , . . . , mt )T for some appropriate t × t matrix U with entries in R. Hence f1 ∧ · · · ∧ ft = det U m1 ∧ · · · ∧ mt couples with m1 ∧ · · · ∧ mt = af1 ∧ · · · ∧ ft , yielding a det U = 1. It follows that a is a unit, and {m1 , . . . , mt } is a basis by Corollary I.3.28. Remark I.3.31 (Proposition I.2.9) Let F be a free R-module with basis {f1 , . . . , ft }. Let ϕ : F → F have matrix S with respect to the given basis. Consider the induced map Λt (ϕ) : Λt (F ) → Λt (F ). This is a map R → R, identiﬁed by the value assigned to 1, say a. Then Λt (ϕ)(f1 ∧ · · · ∧ ft ) = af1 ∧ · · · ∧ ft . In fact, a = det S (using the properties of exterior multiplication). Thus if ϕ is injective by assumption, then Λt (ϕ) is injective as well (by Corollary I.3.30), and det S cannot be a zero divisor. Conversely, if det S is not a zero divisor, then t ϕ must be injective. Otherwise, ϕ(f ) = 0 for some non-zero f ∈ F implies ϕ( i=1 ai fi ) = 0 with some non-zero t ai , that is, i=1 ai ϕ(fi ) = 0. By Proposition I.3.27, there exists a ∈ R − {0} such that aϕ(f1 ) ∧ · · · ∧ ϕ(ft ) = 0, that is, aΛt (F )(f1 ∧ · · · ∧ ft ) = 0, that is, a det Sf1 ∧ · · · ∧ ft = 0. Thus a det S = 0, and det S is a zero divisor. We call the reader’s attention to the fact that if F is R-free of ﬁnite rank t, say, then Λ(F ) = Λ0 (F ) ⊕ · · · ⊕ Λt (F ) implies (Λ(F ))∗ = (Λ(F ))∗gr = (Λ0 (F ))∗ ⊕ · · · ⊕ (Λt (F ))∗ = Λ0 (F ∗ ) ⊕ · · · ⊕ Λt (F ∗ ) = Λ(F ∗ ), where the penultimate equality is a consequence of the ﬁrst part of the following remark.

36

Recollections and perspectives

Remark I.3.32 1. Let F be R-free with basis {f1 , . . . , ft }. Let {g1 , . . . , gt } be the basis of F ∗ dual to {f1 , . . . , ft }. Then for every k ∈ {1, · · · t}, we can deﬁne an isomorphism Λk (F ∗ ) → (Λk (F ))∗ by sending each basis element gi1 ∧· · ·∧gik to the application . , ik ) ∈ HomR (Λk (F ), R) deﬁned by means of ϕ(i1 , . . . , ik )(fj1 ∧ · · · ∧ ϕ(i1 , . . 1 if ih = jh ∀h fjk ) = . The indicated isomorphism is independent of the choice 0 otherwise of the basis {f1 , . . . , ft }. 2. Let F be a rank t free R-module. If a non-zero element of Λt (F ) is chosen, for example, an isomorphism ψ0 : Λt (F ) → R is ﬁxed, then further isomorphisms ψk : Λt−k (F ) → Λk (F ∗ ) are obtained for k ∈ {1, . . . , t}. Namely, given z ∈ Λt−k (F ) and w ∈ Λk (F ), one sets ψk (z)(w) = ψ0 (z ∧ w). This means that an isomorphism Λ(F ) ∼ = Λ(F ∗ ) also exists. However, it is not independent of the choice of the basis in F (= the choice of the element of Λt (F )). Also the minors of a matrix can be described in terms of exterior multiplication. For let E and F be ﬁnite free R-modules, with bases {e1 , . . . , es } and {f1 , . . . , ft }, respectively. Let ψ ∈ HomR (E, F ) have t × s matrix S with respect to the indicated bases, and let Λk (ψ), 1 ≤ k ≤ min{t, s}, have kt × ks matrix U relative to the corresponding bases of Λk (E) and Λk (F ). Then the entry of U , which is located at the intersection of the row indexed by the basis element fi1 ∧ · · · ∧ fik and the column indexed by the basis element ej1 ∧ · · · ∧ ejk , is precisely the k × k minor of S with rows indexed by fi1 , . . . , fik and columns indexed by ej1 , . . . , ejk . Such a minor is often denoted by (i1 , . . . , ik | j1 , . . . , jk ). It is customary to denote by Ik (ψ) the ideal generated by the entries of U , that is, by the k × k minors of S. The ideal Ik (ψ) is independent of the choice of bases because the isomorphism HomR (Λk (E), Λk (F )) ∼ = (Λk (E))∗ ⊗R Λk (F ), coupled with (Λk (E))∗ ⊗R Λk (F ) ∼ = (Λk (E) ⊗R Λk (F ∗ ))∗ = HomR (Λk (E) ⊗R Λk (F ∗ ), R), identiﬁes Λk (ψ) with a map Λk (E) ⊗R Λk (F ∗ ) → R, whose image is precisely Ik (ψ). This suggests that we deﬁne Λk (ψ)

Ik (ψ) = im(Λk (E) ⊗R Λk (F ∗ ) −→ R) ∀k ≥ 0, which, among other things, implies that the ideal is 0 for k > min{t, s} (no minors exist), and that the 0 × 0 minors generate R (a 0 × 0 matrix has determinant 1).

II LOCAL RING THEORY

The development of the ﬁrst ﬁve sections of this chapter is guided by the desire to establish a string of inequalities for local rings (always assumed to be noetherian here) which lead to the result, in Section II.5, that a local ring is regular if and only if its global dimension is ﬁnite. The development here is very close to that used in the notes of Reference [19] and rests on the work found in References [7] and [8]. In the ﬁrst section we deﬁne and develop the properties of the Koszul complex, which will play a fundamental role throughout the book. The next section, on local rings, ends with a statement of a fundamental inequality whose proof rests almost completely on these properties. This inequality states that the global dimension of a local ring always exceeds the minimal number of generators of its maximal ideal. While Section II.3 on Hilbert–Samuel polynomials does not use the complex itself, it does rely on the notion of “regular sequence” for some of its fundamental results, and this notion is intimately connected with that tool. In that section we establish the fact that the dimension of any local ring is always exceeded by the minimum number of generators of its maximal ideal (Krull’s Principal Ideal Theorem). The results of Section II.4 on codimension and ﬁnitistic global dimension are inextricably tied up with the Koszul complex; there we establish the facts that the ﬁnitistic global dimension of a local ring is equal to its codimension, and that its codimension is never greater than its dimension. Section II.5 on regular local rings puts together all of the above to achieve the result mentioned in our ﬁrst paragraph. Another main result there is the Cohen–Macaulay Theorem; its main new “homological” point is that the “unmixedness” part of the classical theorem is sharpened by the observation that the Koszul complex associated to the generators of the ideal is acyclic, which then yields the result. The sixth section of this chapter contains a full proof of the unique factorization theorem for regular local rings. This theorem, too, was one of the prime objectives of the application of homological methods to commutative ring theory. In the penultimate section, we take up the notion of multiplicity, which is again intimately tied up with the Koszul complex. While a detailed study of this theory is beyond the scope of our book, it would be remiss to completely ignore it. In this section, therefore, we give a deﬁnition of multiplicity and establish some of its properties by means of the Koszul complex. Connected with this topic is the notion of Cohen–Macaulay ring, and we will see that for such rings the two

38

Local ring theory

invariants of an ideal, its height and its depth, are equal. As the depth of an ideal is a number that in a certain sense (that of relative primeness) measures the “arithmetic” size of an ideal, while the height is a number that measures its “geometric” size, their coincidence in the case of Cohen–Macaulay rings suggests why the use of ideal-theoretic methods in algebraic geometry, when the variety is Cohen–Macaulay, is so eﬀective. From multiplicity in Section II.7, we move to the Serre deﬁnition of intersection multiplicity in Section II.8 and we close the chapter with a description of the socalled homological conjectures. This book does not deal with these conjectures, but we feel that no book that purports to present the interaction of local ring theory and homology can omit at least a brief mention of these fundamental questions and the work that has been done on them. II.1 Koszul complexes In the beginning, many mathematicians wondered how and how come homological methods could be related and applied to problems in local ring theory. That homology is tied up naturally with algebra can be seen easily by looking at the following example: Let R be a commutative ring (all rings are assumed to have an identity element), and let x be an element of R. Then we may consider the rather trivial complex x

· · · → 0 → 0 → R → R → 0, x

where R → R means multiplication by the element x. If we call this complex K(x), we see that H0 (K(x)) = R/(x), and H1 (K(x)) = 0 : x = {r ∈ R| rx = 0}. Thus in a sense H0 measures how divisible R is by x, while H1 tells us whether or not x is a zero divisor. For an arbitrary module, M, we may also consider the complex K(x) ⊗R M : x

··· → 0 → 0 → M → M → 0 and observe that H0 (K(x)⊗R M ) = M/xM and H1 (K(x)⊗R M ) = 0 : x = {m ∈ M | xm = 0}. Again we may use these homology groups to tell us how divisible the module M is by the element x, and whether or not x is a zero divisor for M. Now let us suppose that we have two elements x, y in R. Then we have the y map K(x) → K(x) induced by multiplication by y: x

···→0→0→R →R →0 ↓y ↓y x ···→0→0→R →R →0 and it is natural for homologists to consider the “mapping cone” of this map of complexes: f

g

··· → 0 → 0 → R → R ⊕ R → R → 0

Koszul complexes

39

where g(r1 , r2 ) = r1 x + r2 y and f (r) = (−ry, rx). Calling this complex K(x, y), we have H0 (K(x, y)) = R/(x, y); H2 (K(x, y)) = 0 : (x, y) = {r ∈ R| rx = 0 = ry}. H1 (K(x, y)) = {(r1 , r2 )| r1 x+r2 y = 0}/{(−ry, rx)}. If we suppose that R is an integral domain, and that neither x nor y is zero, we see that H2 (K(x, y)) = 0 and that {(r1 , r2 )| r1 x + r2 y = 0} is mapped isomorphically onto (x) : y = {r ∈ R| ry ∈ (x)} by the map sending (r1 , r2 ) to r2 . Under this map, the submodule {(−ry, rx)} is sent onto the ideal (x) and thus H1 (K(x, y)) = (x) : y/(x). If R were a unique factorization domain, then H1 (K(x, y)) would be zero if and only if x and y were relatively prime. Therefore, we again see that a certain amount of homological information is intimately connected with arithmetical questions. We could proceed step by step in this way, building up complexes K(x, y, z, . . .), but it is more eﬃcient to consider the following general construction. Let f : A → R be a morphism from the R-module A to R. Then we may deﬁne the Koszul complex, K(f ), by setting K(f )p = Λp A and deﬁning df : Λp A → Λp−1 A by ∧ df (a1 ∧ · · · ∧ ap ) = (−1)i+1 f (ai ) a1 ∧ · · · ∧ ai ∧ · · · ∧ ap , ∧

where ai means we omit ai . It is well known, and easy to check, that (df )2 = 0. Also, if α ∈ Λp A and β ∈ Λq A, then df (α ∧ β) = df (α) ∧ β + (−1)p α ∧ df (β). To see how this subsumes the sort of thing we have been discussing, consider a sequence, x1 , . . . , xs , of elements of our ring R. We can deﬁne a morphism f : Rs → R, where Rs denotes the direct sum of R with itself s times, and f is deﬁned by f (1, . . . , 0) = x1 , . . . , f (0, 0, . . . , 1) = xs . In this case, we denote the complex K(f ) by K(x1 , . . . , xs ). For s = 1 or s = 2, we obtain the complexes K(x) and K(x, y) described above. Before we return to the general case of a morphism f : A → R, let us review the mapping cone construction for a map between complexes. First, a convention about complexes. Since in most of our applications our complexes will be left complexes, we will assume that a complex C is given by {Cn ; dn } with the boundary map d = {dn } of degree −1. That is, for each n, dn : Cn → Cn−1 , and we will usually write the complex: d

d

d

d

d−1

d−2

· · · →3 C2 →2 C1 →1 C0 →0 C−1 → C−2 → · · · . Thus, our complex K(f ) above has all its negative parts equal to zero, K(f )0 = R, K(f )1 = A, and so on. Deﬁnition II.1.1 Let F : C → D be a map of complexes. The mapping cone of the map F, denoted by M(F ), is deﬁned as follows M (F )n = Dn ⊕ Cn−1 ;

40

Local ring theory

and the boundary map δn : M (F )n → M (F )n−1 is deﬁned by δn (xn , yn−1 ) = (dn (xn ) + (−1)n−1 Fn−1 (yn−1 ), dn−1 (yn−1 ) . (Here we are writing the boundary map for the complex D as d = {dn }.) Notice that, module-theoretically, the complex M(F ) is the direct sum of the complexes C and D, except for one thing: the degree of the component coming from C is oﬀ by one. If we denote by C the complex {C n ; dn }, where C n = Cn−1 and dn = dn−1 , then our mapping cone is, in each degree, the direct sum of C and D. Proposition II.1.2 Let F : C → D be a map of complexes. Then we have the short exact sequence of complexes: (∗)

0 → D → M(F ) → C → 0

and the long exact sequence of homology: · · · → Hn+1 (C) → Hn (D) → Hn (M(F )) → Hn (C) → · · · where the maps Hn+1 (C) → Hn (D) are the maps in homology induced by ±F. Proof The exactness of the short exact sequence (∗) is clear on the moduletheoretic level. What has to be checked is that the obvious injection and projection maps are maps of complexes. But this is quite simple. Once the exactness of (∗) is established, the long exact homology sequence is an immediate consequence; what must be explained is the remark that the maps Hn+1 (C) → Hn (D) are the maps in homology induced by ±F. From the deﬁnition of C, it is clear that Hn+1 (C) = Hn (C), so that Hn+1 (C) → Hn (D) is really Hn (C) → Hn (D). But the deﬁnition of the boundary map of M(F ) involves the map F with a sign. A careful diagram chase to describe the “connecting homorphism” makes it clear that this morphism is induced by the map F in homology, with the sign determined in each dimension according to the deﬁnition of δn . 2 Returning to the general case of a morphism f : A → R, we state the fundamental properties of the complex K(f ) : Proposition II.1.3 p ≥ 0.

If a0 is an element of A, then f (a0 )Hp (K(f )) = 0 for all

Proposition II.1.4 If x is any element of R, and f : A → R a morphism, we have the map of complexes x : K(f ) → K(f ) : f

· · · → Λ3 A → Λ2 A → A → R ↓x ↓x ↓x ↓x . f

· · · → Λ3 A → Λ 2 A → A → R

Koszul complexes

41

The mapping cone of this map of complexes is denoted by K(f ) where f : A⊕R → R is deﬁned by f (a, r) = f (a) + rx. Proof The proof of Proposition II.1.3 is obtained by writing a homotopy s : Λp A → Λp+1 A deﬁned by s(a1 ∧ · · · ∧ ap ) = a0 ∧ a1 ∧ · · · ∧ ap , and showing that multiplication by f (a0 ) is chain homotopic, via s, to the zero morphism. That is, one must show that df s(a1 ∧ · · · ∧ ap ) + sdf (a1 ∧ · · · ∧ ap ) = f (a0 )(a1 ∧ · · · ∧ ap ). for all p. Proposition II.1.4 is proved by observing that for any module A, Λp (A⊕ R) ∼ = Λp A ⊕ Λp−1 A, and then checking that under this identiﬁcation the map 2 df is the boundary map of the mapping cone. From Proposition II.1.4 and Proposition II.1.2 we obtain the following. Corollary II.1.5 If f : A → R is a morphism, x an element of R, and f : A ⊕ R → R deﬁned as in Proposition II.1.4, we have the exact sequence: ±x

· · · → Hp (K(f )) → Hp (K(f )) → Hp−1 (K(f )) −→ Hp−1 (K(f )) → · · · · · · → H0 (K(f )) → H0 (K(f )) → 0. In particular, if x1 , . . . , xs+1 is a sequence of elements of R, we have the exact sequence: · · · → Hp (K(x)) → Hp (K(x, xs+1 )) → Hp−1 (K(x))

(E)

±xs+1

−→ Hp−1 (K(x)) → Hp−1 (K(x, xs+1 )) → · · · ,

where x stands for the sequence x1 , . . . , xs . We also get, for an arbitrary Rmodule M, the exact sequence: (E(M ))

· · · → Hp (M(x)) → Hp (M(x, xs+1 )) → Hp−1 (M(x)) ±xs+1

−→ Hp−1 (M(x)) → Hp−1 (M(x, xs+1 )) → · · · ,

where M(x) is the complex K(x) ⊗R M (still called a Koszul complex). Proof All but the last statement about the exactness of (E(M )) follows immediately from the propositions cited. The exactness of (E(M )) is due to the fact that the short exact sequence (∗) applied to the complexes C = D =K(x1 , . . . , xs ) and K(x1 , . . . , xs+1 ) = M(F ), where F is multiplication by xs+1 , splits. Thus tensoring by M preserves exactness, and the long homology sequence (E(M )) is the one associated to the short one. 2 From the remarks made about the meaning of H1 (K(x, y)), we are led to make the following deﬁnition.

42

Local ring theory

Deﬁnition II.1.6 Let M be an R-module, and x1 , . . . , xs a sequence of elements of R. Then x1 , . . . , xs is said to be an M -sequence if (a) M/(x1 , . . . , xs )M = 0; (b) for every i, with i = 1 . . . , s, the element xi is not a zero divisor for M/(x1 , . . . , xi−1 )M. For i = 1, this means that x1 is not a zero divisor for M. Sometimes an M -sequence is called an M -regular sequence, or just regular sequence when M = R. Proposition II.1.7 0 for all p > 0.

If x1 , . . . , xs is an M -sequence, then Hp (M(x1 , . . . , xs )) =

The proof proceeds by induction on s, the case s = 1 being self-evident. To go from s to s + 1, one merely uses the exact sequence (E(M )) to show that x Hp (M(x1 , . . . , xs )) = 0 for p > 1, and the fact that H0 (M(x1 , . . . , xs−1 )) →s H0 (M(x1 , . . . , xs−1 )) is a monomorphism (because H0 (M(x1 , . . . , xs−1 )) equals M/(x1 , . . . , xs−1 )M ) to show that H1 (M(x1 , . . . , xs )) = 0. Remark II.1.8 It is not true in general that Hp (M(x1 , . . . , xs )) = 0 for all p > 0 implies that x1 , . . . , xs is an M -sequence. For example, let R = k[X, Y, Z]/Y (X − 1, Z), with k a ﬁeld. If we write X and Z for the residue classes of X and Z in R, we see easily that X, Z is an R-sequence, so that Hp (K(X, Z)) = 0 for p > 0. Therefore we obviously have that Hp (K(Z, X)) = 0 for p > 0 (since for any ring R, any sequence of elements x1 , . . . , xs in R, and any R-module M, we have M(x1 , . . . , xs ) ∼ = M(xπ(1) , . . . , xπ(s) ), where π is a permutation of the set {1, . . . , s}). However, it is clear that Z is a zero divisor in R, so that Z, X fails to be an R-sequence. As a result of the above remark, it is natural to ask when we can prove the converse of Proposition II.1.7. If we now use our assumptions that R is noetherian and M ﬁnitely generated, we conclude that Hp (M(x1 , . . . , xs )) is a ﬁnitely generated R-module for any sequence of elements x1 , . . . , xs in R. Moreover, if xi is in the radical of R, we can then conclude that the map ∧

∧

x

Hp (M(x1 , . . . , xi , . . . , xs )) →i Hp (M(x1 , . . . , xi , . . . , xs )) ∧

is an epimorphism if and only if Hp (M(x1 , . . . , xi , . . . , xs )) = 0. Using these facts we can prove Proposition II.1.9 Let R be a noetherian ring, M a ﬁnitely generated nonzero R-module, and x1 , . . . , xs a sequence of elements in the radical of R. Then (1) if Hp (M(x1 , . . . , xs )) = 0 for some p, we have Hp+k (M(x1 , . . . , xt )) = 0 for all k ≥ 0 and all t, 1 ≤ t ≤ s; (2) if H1 (M(x1 , . . . , xs )) = 0, then x1 , . . . , xs is an M -sequence.

Local rings

43

Thus, under these hypotheses, we have x1 , . . . , xs is an M -sequence if and only if Hp (M(x1 , . . . , xs )) = 0 for all p > 0 (or for p = 1). Proof The proof of part (1) proceeds by induction on s. For s = 1 and p = 1, there is nothing to be said. If s = 1 and p = 0, this means that M/x1 M = 0. But by Nakayama’s Lemma (cf. Lemma I.2.16), this implies that M = 0, and the rest follows. Now assume that s > 1, and assume the statement true for all smaller values. From the exactness (using (E(M ))) of ±x

Hp (M(x1 , . . . , xs−1 )) →s Hp (M(x1 , . . . , xs−1 )) → Hp (M(x1 , . . . , xs )), we conclude that multiplication by xs is an epimorphism. But this says that Hp (M(x1 , . . . , xs−1 )) = 0, and thus that Hp+k (M(x1 , . . . , xt )) = 0 for all k ≥ 0 and all 1 ≤ t ≤ s − 1. So it only remains to show that Hp+k (M(x1 , . . . , xs )) = 0 for all k > 0. But again using the exactness of (E(M )), this follows from the vanishing of Hp+k (M(x1 , . . . , xs−1 )) for all k ≥ 0. The proof of part (2) follows easily by induction on s, and from part (1). For if H1 (M(x1 , . . . , xs )) = 0, then H1 (M(x1 , . . . , xs−1 )) = 0, and our induction hypothesis tells us that x1 , . . . , xs−1 is an M -sequence. We again apply the exactness of (E(M )) to conclude that x

H1 (M(x1 , . . . , xs )) → H0 (M(x1 , . . . , xs−1 )) →s H0 (M(x1 , . . . , xs−1 )) is exact. Since H0 (M(x1 , . . . , xs−1 )) is equal to M/(x1 , . . . , xs−1 )M, and since H1 (M(x1 , . . . , xs )) = 0, we may conclude that xs is not a zero divisor on 2 M/(x1 , . . . , xs−1 )M. Thus we have established part 2. Corollary II.1.10 The hypotheses being as above, we have that x1 , . . . , xs is an M -sequence if and only if xπ(1) , . . . , xπ(s) is an M -sequence for every permutation π of {1, . . . , s}. Corollary II.1.11 If R is a noetherian ring, M a ﬁnitely generated R-module, p a prime ideal of R such that Mp = M ⊗R Rp = 0 (i.e. p is in Supp(M )), and x1 , . . . , xs a sequence of elements in p which is an M -sequence, then x1 , . . . , xs , considered as elements in Rp , is an Mp -sequence. II.2 Local rings We shall now assume that R is a local ring, with maximal ideal m. The residue ﬁeld R/m will be denoted by k; all R-modules will be assumed ﬁnitely generated. In this case, the radical of R is m, and all proper ideals of R are contained in m. If E is an R-module and I an ideal of R, we have E/IE = 0 if and only if E = 0 because of Nakayama’s Lemma. Furthermore, if e1 , . . . , et are elements of E such that e1 , . . . , et generate E/mE as a k-vector space, then e1 , . . . , et generate E as an R-module. Remark II.2.1 Any generating set of an R-module E contains a minimal generating set of E; any two minimal generating sets of E have the same number of

44

Local ring theory

elements. This number is equal to the dimension of the vector space E/mE over k, and is denoted by [E/mE : k]. Moreover, any subset of E linearly independent modulo mE may be extended to a minimal generating set of E. Lemma II.2.2 If E is an R-module, there exists a free module F and an epimorphism g : F → E such that ker(g) is contained in mF. Proof The proof depends on nothing deeper than the fact that an epimorphism of a vector space onto another of the same dimension is an isomorphism. One then chooses F to be free on the number of generators in a minimal generating set of E. 2 Lemma II.2.3 Proof

If E is an R-module such that TorR 1 (k, E) = 0, then E is free.

We use the above remark to obtain an exact sequence 0→K→F →E→0

with F free and K ⊂ mF. Tensoring this exact sequence with k = R/m, we get the exact sequence TorR 1 (k, E) → K/mK → F/mF → E/mE → 0. Since the map F/mF → E/mE is an isomorphism, the map TorR 1 (k, E) → (k, E) = 0 to K/mK is surjective. But here we can invoke the fact that TorR 1 conclude that K/mK = 0 and, by Nakayama, K itself must be zero. 2 Lemma II.2.4

For every R-module E, hdR (E) ≤ hdR (k).

Lemma II.2.5

If R is a local ring, gldimR = hdR k.

Recall that the global dimension of R, gldimR, is deﬁned to be sup hdR (M ) M

as M runs through all R-modules. Lemmas II.2.4 and II.2.5 are easy consequences of Lemma II.2.3. The main result we are heading for now is that [m/m2 : k] ≤ gldimR, that is, that the global dimension of R is never less than the minimal number of generators of the maximal ideal of R. The idea of the proof is the following. We take a minimal generating set x1 , . . . , xn of the maximal ideal m, and form the complex K(x1 , . . . , xn ). We then show that we can ﬁnd a free resolution of k : d

d

d

· · · → X3 →3 X2 →2 X1 →1 R → k → 0 such that di (Xi ) ⊂ mXi−1 and such that K(x1 , . . . , xn ) is a subcomplex of this resolution. Tensoring this resolution with k makes the boundary maps zero, so that TorR p (k, k) = Xp /mXp . Since K(x1 , . . . , xn ) is a subcomplex of the resolution, and K(x1 , . . . , xn ) is non-zero in dimension n, we see that TorR n (k, k) = 0. Thus hdR k ≥ n and we have the desired result using the facts that gldimR = hdR k and n = [m/m2 : k].

Local rings

45

The complex K(x1 , . . . , xn ) is of the form: δ

f

δ

n 2 0 → Λn R n → · · · → Λ2 R n → Rn → R.

We may therefore choose X1 of our resolution to be Rn with d1 : X1 → R deﬁned to be the morphism f. Assume now that X1 , . . . , Xp have been deﬁned such that 1. Xl = Xl ⊕ Λl Rn ; 2. the morphism dl : Xl → Xl−1 restricted to Λl Rn is δl ; 3. ker(dl ) ⊂ mXl ; dp

d

d

4. Xp → · · · → X2 →2 X1 →1 R → k → 0 is exact. If p < n, we will show that we can ﬁnd a free module Xp+1 such that 1–4 are true with l replaced by p + 1. Our conditions 1 and 2 tell us that the composition δp+1

dp

Λp+1 Rn → Λp Rn → Xp → Xp−1 is zero. Thus δp+1 (Λp+1 Rn ) is contained in Zp = ker(dp ). We will show that if {εi1 ∧ · · · ∧ εip+1 } are a basis of Λp+1 Rn , then ·∧εip+1 )} are linearly independent modulo mZp . That is, we want to {δp+1 (εi1 ∧· · m. But show that if γi1 ...ip+1 δp+1 (εi1 ∧ · · · ∧ εip+1 ) ∈ mZp , then each γi1 ...ip+1 ∈ this is the same as showing that if γ = γi1 ...ip+1 εi1 ∧ · · · ∧ εip+1 in Λp+1 Rn is such that δp+1 (γ) is in mZp , then γ is in mΛp+1 Rn . Suppose, then, that δp+1 (γ) is in mZp . Since Zp ⊂ mXp , we have mZp ⊂ m2 Xp = m2 Λp Rn ⊕ m2 Xp . Since δp+1 (γ) is in Λp Rn , we must have δp+1 (γ) ∈ m2 Λp Rn . Now δp+1 : Λp+1 Rn → Λp Rn induces a map δ p+1 :Λp+1 Rn /mΛp+1 Rn → mΛp Rn /m2 Λp Rn since δp+1 (Λp+1 Rn ) ⊂ mΛp Rn . If we can show that δ p+1 is a monomorphism, we are done. For to say that δp+1 (γ) is in m2 Λp Rn is to say that δ p+1 (γ) = 0 and thus that γ = 0 or γ ∈ mΛp+1 Rn . But Λp+1 Rn /mΛp+1 Rn = R/m⊗R Λp+1 Rn , and mΛp Rn /m2 Λp Rn = m/m2 ⊗R Λp Rn and the map δ p+1 sends ∧ the basis element 1 ⊗ εi1 ∧ · · · ∧ εip+1 to (−1)j+1 xij ⊗ εi1 ∧ · · · ∧ εij ∧ · · · ∧ εip+1 . Clearly this map is a monomorphism. Since we have shown that {δp+1 (εi1 ∧ · · · ∧ εip+1 )} are linearly independent modulo mZp , we know that these elements may be chosen as part of a minimal generating set for Zp , that is, we have a minimal generating set {δp+1 (εi1 ∧ · · · ∧ be the free module on q generators, we have εip+1 )} ∪ {z1, . . . , zq }. Letting Xp+1 = Xp+1 mapping onto Zp . The kernel of the map Xp+1 → Λp+1 Rn ⊕ Xp+1 Zp → Xp is clearly in mXp+1 and dp+1

d

d

Xp+1 → Xp → · · · → X2 →2 X1 →1 R → k → 0 is exact. Thus we have completed the inductive step, and we have proved Theorem II.2.6

If R is a local ring, then [m/m2 : k] ≤ gldimR.

Our next objective is to deﬁne the dimension (or Krull dimension) of a local ring R, and show that dim R ≤ [m/m2 : k]. To do this, we shall use the method

46

Local ring theory

of Hilbert–Samuel polynomials which we introduce in a slightly more general setting now. II.3 Hilbert–Samuel polynomials We let Z+ denote the set of positive integers, and let G be an abelian group. Generally, G will be the group of integers or the Grothendieck group of some category of modules. Deﬁnition II.3.1 Let f : Z+ → G be a function. f is called a polynomial + such that function if there are elements a0 , . . . , ad in G for all nn∈ Z , n d n suﬃciently large (i.e. n >> 0), we have f (n) = i=0 i ai where i denotes the binomial coeﬃcient n!/(i!(n − i)!). If f : Z+ → G is any function, we deﬁne ∆f : Z+ → G by ∆f (n) = f (n + 1) − f (n). ∆f is called the ﬁrst diﬀerence (function) of f. For any integer s > 1, we deﬁne ∆s f = ∆(∆s−1 f ). Lemma II.3.2 A function f : Z+ → G is a polynomial function if and only if ∆f is a polynomial function. d n Proof If f is a polynomial function, we have f (n) = i=0 i ai for some a0 , . . . , ad in G, and all n >> 0. Consequently, for all n >> 0 we have ∆f (n) = f (n + 1) − f (n) is equal to d n+1 i=0

i

ai −

d n i=0

i

ai =

d n+1 i=0

i

d n n − ai . ai = i i−1 i=1

Thus ∆f is a polynomial function. d n Conversely, if ∆f (n) = i=0 i bi for n >> 0, deﬁne g(n) = f (n) − d n i=0 i+1 bi . Then for n >> 0, ∆g(n) = 0 so that g(n) is a constant a0 . Thus, d+1 2 letting ai = bi−1 for i > 0, we have f (n) = i=0 ni ai for n >> 0. We have proved Lemma II.3.2 in detail, since this deﬁnition of polynomial function is slightly more general than the usual one. However, with Lemma II.3.2 established, we shall only state three facts we need and omit the proofs. d d Lemma II.3.3 If f (n) = i=0 ni ai = i=0 ni ai is a polynomial function, then d = d and ai = ai for all i. Thus, the degree, d, of f is a well-deﬁned integer, and the coeﬃcients ai of f are uniquely determined. Now consider a commutative noetherian ring, R, a set of indeterminates, {X1 , X2 , . . .}, and let A be a full abelian subcategory of the category of Rmodules. For each integer s = 0, 1, 2, . . . , let As be the category of ﬁnitely

Hilbert–Samuel polynomials

generated graded R[X1 , . . . , Xs ]-modules E =

ν≥0

47

Eν such that

1. if s = 0, Eν is in A for all ν; 2. if s> 0, Eν is in A for all ν and the graded R[X1 , . . . , Xs−1]-modules Xs Xs and coker are in ker ν≥0 Eν → ν≥0 Eν ν≥0 Eν → ν≥0 Eν As−1 . Finally, let f0 be a function from the objects of A to an abelian group G which factors through the Grothendieck group, K(A), of A, that is, f0 is additive with respect to exact sequences in A. Then Theorem II.3.4 Let E = ν≥0 Eν be an object in As and deﬁne fE : Z+ → G by fE (ν) = f0 (Eν ). Then fE is a polynomial function of degree less than or equal to s − 1. An example of such a set-up occurs when R is a local ring, and A is the full subcategory of R-modules of ﬁnite length. In that case, an R[X1 , . . . , Xs ]-module E = ν≥0 Eν is in As if it is ﬁnitely generated, and if each Eν is an R-module of ﬁnite length. We may then choose G to be the group, Z, of integers and f0 (E) equals the length of E. This is of course the most usual example. To see that it is not the only example, we may choose A to be the category of all ﬁnitely generated R-modules, and G to be K(A) itself with f0 the usual map. Corollary II.3.5 If R is a local ring, E a ﬁnitely generated R-module and q an ideal of R containing some power mn of the maximal ideal m, then E/qν E is an R-module of ﬁnite length, and the function χq (E) : Z+ → Z deﬁned by χq (E; ν) = length(E/qν E) is a polynomial function whose degree is less than or equal to [q/mq : k]. To see why this is a corollary of Theorem II.3.4, we observe that we have exact sequences 0 → qν E/qν+1 E → E/qν+1 E → E/qν E → 0 and therefore ∆χq (E; ν) equals the length of qν E/qν+1 E. The graded module qν E/qν+1 E is a module over R[X1 , . . . , Xs ] where s = [q/mq : k], and ν each q E/qν+1 E is an R-module of ﬁnite length. Thus ∆χq (E) is a polynomial function of degree ≤ s − 1, and hence χq (E) is what we claimed it to be. Remark II.3.6 If q1 and q2 are two ideals containing some powers of the maximal ideal m, that is, if q1 ⊃ mn1 and q2 ⊃ mn2 , then the degrees of χq1 (E) and χq2 (E) are equal. In particular, they are all equal to the degree of χm (E). This is seen easily by observing that for any integer n, χmn (E) and χm (E) have the same degree. Thus, if mn ⊂ q ⊂ m, we must have deg(χmn (E)) ≤ deg(χq (E)) ≤ deg(χm (E)), and hence equality. Deﬁnition II.3.7 If R is a local ring, the dimension of R is the degree of the polynomial function χm (R). The dimension of an R-module E is the degree

48

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of the polynomial function χm (E). If R is a noetherian ring, and p is a prime ideal of R, then the height of p is the dimension of Rp . We immediately obtain the following proposition. Proposition II.3.8 Let R be a local ring, and let s be the smallest number of elements required to generate an ideal q of R which contains some power of the 2 maximal ideal m. Then dim R ≤ s. In particular, dim R ≤ [m/m : k]. Although we will not be using this fact immediately, it is important to note that for a local ring R, the following integers (a) the dimension of R; (b) the smallest number of elements required to generate an ideal containing a power of the maximal ideal; (c) the length of the longest chain of prime ideals in R, where the length of the chain p0 ⊃ p1 ⊃ · · · ⊃ ph is h are equal. The proof that these three integers are equal is not completely trivial. If we denote these integers by d, s and h, respectively, the procedure is to show that d ≤ h ≤ s ≤ d. We point out that an ideal which contains a power of the maximal ideal is called an ideal of deﬁnition. It is the same thing as being an m-primary ideal. A set of elements, x1 , . . . , xd , with d = dimR, is called a system of parameters if the ideal they generate, q, is an ideal of deﬁnition. The reader is referred to D. Eisenbud’s book, [41], for those details about noetherian rings which are mentioned here but not proved. For various reasons, it is useful to be able to compare the dimension of a module E over a local ring R with that of E/xE if x is an element of R. From our point of view, it is extremely helpful to know what happens to dim E/xE when x is not a zero divisor for E. To handle this situation, we quote (without proof) the Artin–Rees Theorem: Theorem II.3.9 Let R be a noetherian ring, M a ﬁnitely generated R-module, M a submodule of M, and I an ideal of R. Then there exists an integer h > 0 such that for all n ≥ h, we have (I n M ) ∩ M = I n−h (I h M ∩ M ). Corollary II.3.10 If R is a local ring and x is an element of the maximal ideal m which is a non-zero divisor for an R-module E, there is an integer h > 0 such that for all ν ≥ h, mν E : x ⊂ mν−h E where mν E : x = {e ∈ E| xe ∈ mν E}. The proof depends on the Artin–Rees Theorem and the easy observation that x(mν E : x) = mν E ∩ (x)E.

Hilbert–Samuel polynomials

49

For then we have x(mν E : x) = mν E ∩ (x)E = mν−h (mh E ∩ (x)E) = xmν−h (mh E : x) ⊂ xmν−h E, from which we conclude: mν E : x ⊂ mν−h E. Proposition II.3.11 Let R be a local ring, E an R-module, and x an element of m. Then dim E/xE ≥ dim E − 1. If x is not a zero divisor for E, we have dim E/xE ≤ dim E − 1 and hence dim E/xE = dim E − 1. Proof If we let E = E/xE, and m = m/(x), we are interested in the length of E/mν E. But E/mν E = E/(mν , x)E, and we have an exact sequence: 0 → (mν , x)E/mν E → E/mν E → E/(mν , x)E → 0. Thus length(E/(mν , x)E) = length(E/mν E) − length((mν , x)E/mν E). Since (mν , x)E/mν E ∼ = xE/mν E ∩ xE ∼ = xE/x(mν E : x), we have length(mν , x) ν ν E/m E ≤ length(E/(m E : x)), with equality holding if x is not a zero divisor for E. Since x ∈ m, we know that mν−1 E ⊂ mν E : x so that length(E/(mν E : x)) ≤ length(E/mν−1 E). Thus length(E/(mν , x)E) = length(E/mν E) − length((mν , x)E/mν E) ≥ length(E/mν E) − length((E/mν+1 E)), so that dim E ≥ dim E − 1. If, however, x is not a zero divisor for E, we have length((mν , x)E/mν E) = length(E/(mν E : x)) and by Corollary II.3.10, length(E/(mν E : x)) ≥ length(E/mν−h E) for suitable ﬁxed h and all ν ≥ h. Thus length(E/(mν , x)E) = length(E/(mν E)) − length(mν , x)E/mν E ≤ length(E/(mν E)) − length(E/mν−h E), which immediately implies that dim E ≤ dim E − 1.

2

In the above proof, we said we were interested in the length of E/mν E instead of the length of E/mν E to emphasize the fact that dim E as an R-module is the same as dim E as an R-module, where R = R/(x). This is mainly to ensure that when E = R, dim R as an R-module is seen to be the dimension of the local ring R. Of course, E/mν E ∼ = E/mν E. As an immediate consequence of Proposition II.3.11 we have Corollary II.3.12 E-sequence. Then

Let R be a local ring, E an R-module, and x1 , . . . , xs an dim(E/(x1 , . . . , xs )E) = dim E − s.

In particular, s ≤ dim E.

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Local ring theory

II.4 Codimension and ﬁnitistic global dimension We will now let R be a noetherian ring (i.e., not necessarily local), E a ﬁnitely generated R-module, and I an ideal in R such that E/IE = 0. Since E/IE = 0, there is some prime ideal p in R such that E/IE ⊗R Rp = 0, and this prime ideal p obviously contains I. From Corollary II.1.10, we see that if x1 , . . . , xs is an E-sequence contained in I, then x1 , . . . , xs is also an E ⊗R Rp -sequence in IRp and thus, by Corollary II.3.12, s ≤ dim E ⊗R Rp . Hence the number of elements in an E-sequence contained in I is bounded, and any E-sequence in I may be extended to a maximal E-sequence in I. We shall prove that any two maximal E-sequences in I have the same number of elements, but ﬁrst we give a quick review of the notion of associated prime ideals. The Deﬁnitions II.4.1 and II.4.5, the Lemmas II.4.2 and II.4.3, Proposition II.4.4, Theorem II.4.6 and Remark II.4.7 can all be found in Reference [41]. We simply write them here for easy reference. Deﬁnition II.4.1 If E is an R-module, a prime ideal p is said to be associated to E if there is some monomorphism R/p → E. The associator of E, denoted by Ass(E), is the set of all primes associated to E. Using the fact that R is noetherian, we have Lemma II.4.2

If E is an R-module, then Ass(E) = ∅ if and only if E = 0.

Another useful lemma is the following: Lemma II.4.3

Let 0 → E → E → E → 0

be an exact sequence of R-modules. Then Ass(E) ⊂ Ass(E ) ∪ Ass(E ). Proposition II.4.4 If R is a noetherian ring and E is a ﬁnitely generated R-module, then Ass(E) is a ﬁnite set. Deﬁnition II.4.5 If R is a noetherian ring and E a ﬁnitely generated Rmodule, we deﬁne the height of E to be min{height p} where p runs through all primes in Ass(E). If I is an ideal of R, we generally abuse notation and call the height of R/I the height of the ideal I. An important result relating the height of an ideal with the number of generators of the ideal, is the Krull Principal Ideal Theorem. We state this separately as a theorem. Theorem II.4.6 (Krull) If R is a noetherian ring, and I is an ideal generated by elements x1 , . . . , xr , then any minimal prime of Ass(R/I) has height less than or equal to r. In particular, height(I) ≤ r.

Codimension and ﬁnitistic global dimension

51

Remark II.4.7 (1) For any R-module E, the set of zero divisors of E is equal to the union of all primes in Ass(E). (2) Thus, if E is a ﬁnitely generated R-module (with R noetherian) and if I is an ideal such that every element of I is a zero divisor of E, then there is a non-zero element e of E such that Ie = 0. For if I is as above, then I ⊂ ∪p where p runs through Ass(E) and hence I ⊂ p for some p ∈ Ass(E) (since Ass(E) is ﬁnite). We therefore have the epimorphism R/I → R/p and a monomorphism R/p → E. The image of 1 under the composite map R → R/I → R/p → E is the desired element e ∈ E. (3) If the ideal I in 2 above is a maximal ideal, then I ∈ Ass(E). We are now ready to prove the main theorem of this section. Theorem II.4.8 Let R be a noetherian ring, E an R-module, and I an ideal of R generated by elements x1 , . . . , xn such that E/IE = 0. Let y1 , . . . , ys be a maximal E-sequence in I. Then s+q = n, where q is the dimension of the highest non-vanishing homology of the complex E(x1 , . . . , xn ). Furthermore, Hq (E(x1 , . . . , xn )) ∼ = ((y1 , . . . , ys )E : I)/(y1 , . . . , ys )E. Proof The proof proceeds by induction on s. When s = 0, it means that every element of I is a zero divisor for E so that by 2 of Remark II.4.7, there is a nonzero element e in E such that Ie = 0. Since Hn (E(x1 , . . . , xn )) = 0 : I = 0, we see that q = n and Hn (E(x1 , . . . , xn )) = (y1 , . . . , ys )E : I/(y1 , . . . , ys )E = 0 : I. When s > 0, we consider the exact sequence y1

0→E→E→E→0 and get the long exact sequence y1

Hq+1 (E(x1 , . . . , xn )) → Hq (E(x1 , . . . , xn )) → Hq (E(x1 , . . . , xn )) → Hq (E(x1 , . . . , xn )) → · · · where q is the dimension of the highest non-vanishing homology group of E(x1 , . . . , xn ). Thus, Hq+1 (E(x1 , . . . , xn )) = 0, Hq (E(x1 , . . . , xn )) = 0, and multiplication by y1 on Hq (E(x1 , . . . , xn )) and Hq−1 (E(x1 , . . . , xn )) is zero since y1 is in I. Thus Hq (E(x1 , . . . , xn )) = 0 while Hq−1 (E(x1 , . . . , xn )) ∼ = Hq (E(x1 , . . . , xn )). Noting that y2 , . . . , ys is a maximal E-sequence in I, and using induction, we have (s−1) +q = n. Since, however, we have just shown that q = q − 1, we have s + q = n. Finally, since Hq (E(x1 , . . . , xn )) ∼ = (y2 , . . . , ys )E : I/(y2 , . . . , ys )E ∼ = (y1 , . . . , ys )E : I/(y1 , . . . , ys )E, and since, in addition, Hq (E(x1 , . . . , xn )) = Hq−1 (E(x1 , . . . , xn )) ∼ = Hq (E(x1 , . . . , xn )), we have ∼ Hq (E(x1 , . . . , xn )) = (y1 , . . . , ys )E : I/(y1 , . . . , ys )E. 2

52

Local ring theory

Corollary II.4.9 If R, E and I are as in Theorem II.4.8, then any two maximal E-sequences in I have the same length. Deﬁnition II.4.10 The length of a maximal E-sequence in I is called the I-depth of E, denoted by depth(I; E). If R is a local ring, the codimension of a module, E, is deﬁned as codim(E) = depth(m; E) where m is the maximal ideal of R. If E = R, we simply write depth(I) for depth(I; R), and call it the depth of I. Finally, we call the ideal I perfect if hdR (R/I) = depth(I). Remark II.4.11 height(I).

If I is any ideal of a noetherian ring R, then depth(I) ≤

An important consequence of Theorem II.4.8 is Theorem II.4.12 hdR E < ∞. Then

Let R be a local ring, and E an R-module such that codim(R) = hdR E + codim(E).

Proof Let m = (x1 , . . . , xn ), where m is the maximal ideal of R. Denote by q the dimension of the highest non-vanishing homology group of K(x1 , . . . , xn ), and by qE the corresponding integer for the complex E(x1 , . . . , xn ). Since codim(R) = n − q and codim(E) = n − qE , we want to show that qE − q = hdR E. Now when hdR E = 0, E is free and clearly qE = q. Suppose that hdR E ≥ 1. Then we have an exact sequence 0→L→F →E→0 with F a free module, and hdR E = 1 + hdR L. This gives us an exact sequence: H1+qL (E(x1 , . . . , xn )) → HqL (L(x1 , . . . , xn )) → HqL (F(x1 , . . . , xn )) → HqL (E(x1 , . . . , xn )) → HqL −1 (L(x1 , . . . , xn )). By induction on hdR E, we know that qL − q = hdR L so what must be shown is that 1 + qL = qE . If qL − q > 0, then HqL (F(x1 , . . . , xn )) = 0 and thus H1+qL (E(x1 , . . . , xn )) = 0, while clearly Ht+qL (E(x1 , . . . , xn )) = 0 for all t > 1. Thus in this case, we would have 1 + qL = qE and we would be done. Our problem, then, is to resolve the case when qL = q, that is, when hdR L = 0 or L is free. In this case, we may assume that L and F have been chosen so that L ⊂ mF. If y1 , . . . , ys is a maximal R-sequence, it is also a maximal F - and L-sequence, and Theorem II.4.8 tells us that Hq (L(x1 , . . . , xn )) ∼ = (y1 , . . . , ys )L : m/(y1 , . . . , ys )L, ∼ (y1 , . . . , ys )F : m/(y1 , . . . , ys )F, Hq (F(x1 , . . . , xn )) = and the map Hq (L(x1 , . . . , xn )) → Hq (F(x1 , . . . , xn )) is the natural map of (y1 , . . . , ys )L : m/(y1 , . . . , ys )L → (y1 , . . . , ys )F : m/(y1 , . . . , ys )F. If we show that this map is not a monomorphism, we are done.

Codimension and ﬁnitistic global dimension

53

/ (y1 , . . . , ys ) with zm ⊂ Since (y1 , . . . , ys ) : m (y1 , . . . , ys ), there is a z ∈ (y1 , . . . , ys ). Since L is free, zL is not included in (y1 , . . . , ys )L. But zL ⊂ zmF ⊂ (y1 , . . . , ys )F. Choosing an element w ∈ zL − (y1 , . . . , ys )L, we have / (y1 , . . . , ys )L and thus w = 0 in (y1 , . . . , ys )L : w ∈ (y1 , . . . , ys )L : m but w ∈ m/(y1 , . . . , ys )L. However, w is mapped to 0 in (y1 , . . . , ys )F : m/(y1 , . . . , ys )F so that our map Hq (L(x1 , . . . , xn )) → Hq (F(x1 , . . . , xn )) is not a monomorphism, and the proof is complete. 2 A particular consequence of Theorem II.4.12 is that if R is a local ring, and E an R-module such that hdR E < ∞, then hdR E ≤ codimR. Deﬁnition II.4.13 If R is a commutative ring, we deﬁne the ﬁnitistic global dimension of R, written fgldimR, to be sup{hdR E} where E ranges over all ﬁnitely generated R-modules of ﬁnite homological dimension. Lemma II.4.14 If R is a local ring, E a ﬁnitely generated R-module, and x ∈ m an element which is not a zero divisor for E, then hdR E/xE = 1+hdR E. Proof Applying TorR (−, R/m) to the short exact sequence x

0 → E → E → E/xE → 0, it is easy to see that E and E/xE both have ﬁnite homological dimension,or both have inﬁnite homological dimension. In the latter case, the lemma is clearly true. In the former, the long exact Tor sequence exhibits the equality. 2 It may be amusing to look at a slightly diﬀerent proof of this result. One can choose a minimal resolution, X, of E, and consider the map Fx : X → X given by multiplication by x. (A resolution is called minimal if the image of Xp is contained in mXp−1 for all p.) Since x ∈ m, it is easy to see that the mapping cone of Fx is a minimal resolution of E/xE. Now a simple look at the mapping cone gives the result. This other proof can be looked on as an extension of the ideas behind the use of the Koszul complex. It was inspired by a more special theorem found in W. Gr¨ obner’s work [46], and was one of the early results that indicated that homological methods could be eﬀectively applied to certain kinds of problems in commutative algebra. Proposition II.4.15

If R is a local ring, then fgldimR = codimR.

Proof We have already seen that fgldimR ≤ codimR. However, if y1 , . . . , ys is an R-sequence with s = codimR, we have (by Lemma II.4.14) hdR R/(y1 , . . . , ys ) = s = codimR and hence the equality.

2

Putting together Theorem II.2.6, Proposition II.3.8, Corollary II.3.12, and Proposition II.4.15, we obtain

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Local ring theory

Theorem II.4.16

If R is a local ring, we have 2

fgldimR = codimR ≤ dim R ≤ [m/m : k] ≤ gldimR. II.5 Regular local rings We are now ready to apply what we have done to the study of some important properties of regular local rings. Deﬁnition II.5.1

2

A local ring R is regular if dim R = [m/m : k].

The following is a well-known property of regular local rings. Proposition II.5.2 Let R be a regular local ring, and x1 , . . . , xn a minimal generating set of the maximal ideal m. Then for each integer i ≥ 0, the ideal (x1 , . . . , xi ) is a prime ideal. In particular, R is a domain and x1 , . . . , xn is a (maximal) R-sequence. Proof That the ideals (x1 , . . . , xi ) are prime for i ≥ 0, can be found in Reference [41], section 10.3. That x1 , . . . , xn is a maximal R-sequence is then an easy consequence. 2 The prototypical example of a regular local ring is the formal power series ring over a ﬁeld: K[[X1 , . . . , Xn ]]. Our ﬁrst main result about regular local rings is: Theorem II.5.3 A local ring R is regular if and only if gldimR < ∞. If R is regular, gldimR = dim R. Proof If R is regular, Proposition II.5.2 tells us that hdR k = hdR R/ (x1 , . . . , xn ) equals n = dim R, and hence gldimR = n < ∞. Conversely, if gldimR < ∞, then gldimR = fgldimR and by Theorem II.4.16, we have 2 2 dim R = [m/m : k]. The utility of Theorem II.5.3 will be seen in the subsequent theorems. The result that comes immediately below was actually the prime motivation for one of the authors (prompted by a conversation with Emil Artin) to establish this characterization of regular local rings. Theorem II.5.4 If R is a regular local ring and p is a prime ideal of R, then Rp is a regular local ring. Proof

gldimRp = hdRp Rp /pRp ≤ hdR R/p < ∞. Thus Rp is regular.

2

Lemma II.5.5 If R is a regular local ring, E a ﬁnitely generated R-module, and p ∈ Ass(E), then hdR E ≥ height p. Proof We have hdR E ≥ hdRp Ep . Since pRp is obviously in Ass(E), and pRp is the maximal ideal of Rp , we have codimEp = 0 as an Rp -module. Thus hdRp Ep = codimRp = dim Rp = heightp. 2

Regular local rings

55

A straightforward application of the Krull Principal Ideal Theorem (Theorem II.4.6) yields the following. Lemma II.5.6 If R is a noetherian ring, I an ideal of R, and x an element of R such that x is not a zero divisor for R/I, and R/(I, x) = 0, then height(I, x) ≥ 1 + heightI. In particular, if x1 , . . . , xs is an R-sequence, then height(x1 , . . . , xs ) = s. Lemma II.5.7 heightI.

Let I be an ideal in a regular local ring R. Then depth(I; R) =

Proof By localizing we know that depth(I; R) ≤ heightI. Suppose heightI = s and let x1 , . . . , xt be a maximal R-sequence in I, that is, t = depth(I; R). Since every element of I is a zero divisor for R/(x1 , . . . , xt ), we must have I contained in some prime ideal p ∈ Ass(R/(x1 , . . . , xt )). But, by Lemma II.5.5, if p ∈ Ass(R/(x1 , . . . , xt )) we have heightp ≤ hdR R/(x1 , . . . , xt ) = t. Thus, since I ⊂ p implies heightI ≤ heightp, we have s ≤ t and thus s = t. 2 It is not hard to see the following result. Lemma II.5.8 Let R be a noetherian ring, E a ﬁnitely generated R-module, and x a non-zero divisor for E. If p is in Ass(E) and p is a prime ideal containing (p, x), then there is a prime ideal p ∈ Ass(E/xE) such that p ⊃ p ⊃ p. Deﬁnition II.5.9 We say a chain of prime ideals p0 ⊂ p1 ⊂ · · · ⊂ pn is saturated if for each i there is no prime lying strictly between pi and pi+1 . We say a ring satisﬁes the saturated chain condition for prime ideals (s.c.c.) if any two saturated chains of primes between any two given primes p ⊂ p have the same length. Theorem II.5.10 Every regular local ring satisﬁes the saturated chain condition for prime ideals. Proof It obviously suﬃces to show that if p ⊂ p is saturated, then heightp = 1 + heightp. Suppose, then, that s = heightp. Then there is an R-sequence x1 , . . . , xs which is maximal in p. Clearly there is an element xs+1 in p − p such that x1 , . . . , xs+1 is an R-sequence. Using Lemma II.5.8, there is a prime p ∈ Ass(R/(x1 , . . . , xs+1 )) such that p ⊃ p ⊃ p. Since heightp = s + 1 > heightp, 2 we cannot have p = p. Thus p = p and heightp = s+1 = 1+heightp. Remark II.5.11 1. By constructing a local ring that does not satisfy the s.c.c., M. Nagata was able to show that not every local ring is a factor ring of a regular local ring. For clearly, a factor ring of a ring satisfying the s.c.c. must also satisfy the s.c.c. 2. We see that if R is a regular local ring, then dim R/p+heightp = n = dim R. 3. It can be shown that if R satisﬁes the s.c.c., if I is an ideal of R, and x an element of R, then height(I, x) ≤ 1 + heightI. We now come to the Cohen–Macaulay Theorem.

56

Local ring theory

Theorem II.5.12 (Cohen–Macaulay) Let R be a regular local ring, and I = (x1 , . . . , xs ) an ideal of height s. Then I is unmixed (i.e., every prime p ∈ Ass(R/I) has height s), and x1 , . . . , xs is an R-sequence. Proof The main point of the proof is to show that x1 , . . . , xs is an R-sequence, for then the rest follows easily. To show that x1 , . . . , xs is an R-sequence, we proceed by induction. Certainly when s = 1 we are done (since a regular local ring is an integral domain). Assuming s > 1, we have s = height(x1 , . . . , xs ) ≤ 1+ height(x1 , . . . , xs−1 ). But then s − 1 ≤ height(x1 , . . . , xs−1 ) ≤ s − 1, so by induction we have x1 , . . . , xs−1 is an R-sequence. It is then trivial to show that xs is not a zero divisor for R/(x1 , . . . , xs−1 ) (by using a height argument), and so we have the result. 2 The deﬁnition of unmixed given in the statement of Theorem II.5.12 holds for any noetherian ring, R. II.6 Unique factorization In this section we outline a proof of the fact that every regular local ring is a unique factorization domain. In Section IV.2 we will give a second proof of this theorem. The fact that regular local rings are factorial was another one of the early major results about local rings obtained using homological techniques, or at least the homological characterization of regularity. In order to discuss unique factorization in regular local rings, we ﬁrst establish a few results that extend our discussion of Section I.1. Proposition II.6.1 If R is a noetherian local ring, then R is a unique factorization domain if and only if hdR R/(x, y) ≤ 2 for every x, y ∈ R. Proof In Section I.1 we have proved the implication one way (Corollary I.1.6). It remains to prove that if hdR R/(x, y) ≤ 2 for every x, y ∈ R, then R is a unique factorization domain. First we see from Corollary I.2.10, that R is a domain, since if we pick x = y = 0, we have the ﬁniteness of the homological dimension of R/(x), which guarantees that x is not a zero divisor. But then, as in the proof of Corollary I.1.6, we see that the sequence 0 → (x) : y → R2 → R → R/(x, y) → 0 is exact, which tells us that hdR ((x) : y) = 0. Hence this ideal is projective, indeed free, hence principal, and we are done. 2 An immediate corollary of this proposition is the following. Corollary II.6.2 If R is a regular local ring of dimension less than or equal to 2, then R is a unique factorization domain.

Unique factorization

57

Proof If R is regular of dimension less than or equal to 2, then its global dimension is at most 2. Hence the condition of the preceding proposition is satisﬁed, and R is a unique factorization domain. 2 Proposition II.6.3 equivalent:

Let R be a noetherian domain. Then the following are

(1) R is a unique factorization domain. (2) Every minimal prime ideal of R is principal. (3) Every unmixed ideal of height one is principal. Proof (1) ⇒ (2): If R is a unique factorization domain, then every irreducible element is prime. Let p be a minimal prime ideal, and let (x) be maximal among the principal ideals contained in p. Clearly x is an irreducible element. But then it is prime. Since (x) ⊆ p, and p is a minimal prime, the two prime ideals are equal. (2) ⇒ (1): We must show that irreducible elements are prime. If x is irreducible, let (x) ⊆ p for some minimal prime p. We are assuming that p is principal, say p = (y). From the irreducibility of x, we see that x must be a unit multiple of y, and we are done. (1) ⇒ (3): Since we are assuming factoriality, we know that all minimal primes are principal. We ﬁrst show that if q is p-primary for some minimal prime ideal p, then q is principal. Since p is principal, we have p = (a). Suppose that q ⊆ (an ), but q (an+1 ). Set q = q : an . It is easy to see that q = q (an ). Now, if q = R, then it must be contained in p. But q ⊆ p, implies that an+1 ∈ q, and this is a contradiction. Hence q = R, and we see that q = (an ). Since every unmixed ideal of height one is the intersection of primaries belonging to minimal primes, every such ideal is the intersection of principal ideals. We know from Section I.1 that in a unique factorization domain, the intersection of principal ideals is principal, and so we have the result. Clearly (3) implies (2), so we have proven the proposition. 2 We now begin a series of propositions to prove that every regular local ring of dimension 3 is a unique factorization domain. Following that, we shall use a proof of M. Nagata [67] to show that this is suﬃcient to conclude that all regular local rings are factorial. Proposition II.6.4 Let (R, m) be a local ring. Let x be in m and y in R such that a = (x) : y satisﬁes the following conditions: (a) hdR a ≤ 1, and (b) x is not in ma. Then a = (x) and x is not a zero divisor . Proof Suppose x does not generate a. Since x is not in ma, there exist a1 , . . . , an in a (n > 0) such that x, a1 , . . . , an is a minimal generating set for a. Let Rn+1

58

Local ring theory

denote the direct sum of n+1 copies of R, deﬁne f : Rn+1 → a by f (r0 , . . . , rn ) = n r0 x + i=1 ri ai , and let K = kerf . Since x, a1 , . . . , an is a minimal generating set for a, we have that f is an epimorphism and K is contained in mRn+1 . From the exact sequence 0 → K → Rn+1 → a → 0 and the fact that hdR a ≤ 1, we have that K is R-free. Since a is not principal, we have that hdR a = 1 and thus K = 0. Let Ni = (ti0 , . . . , tin ) be a free basis for K over R (i = 1, . . . , m). Since a1 is in a, we have that ya1 = −vx for some v in R. Let V =(v, y, 0, . . . , 0) m and T = (a1 , −x, 0, . . . , 0). Then V and T are in K. Let V = j=1 rj Nj and m m T = j=1 sj Nj . Now xV = −yT. Therefore we have that j=1 xrj Nj = m j=1 −ysj Nj . Since the {Nj } are a free basis for K over R, we have that xrj = −ysj for all j = 1, . . . , m. But (x) : y = a. Therefore we have that each sj is m m in a. Since T = (a1 , −x, 0, . . . , 0) = j=1 sj Nj , it follows that −x = j=1 sj tj1 . Therefore x is in ma (since K is contained in mRn+1 ) which contradicts the fact that x is not in ma. Thus a = (x). The fact that x is not a zero divisor follows from the fact that hdR a ≤ 1. (Recall the proof of Proposition II.6.1) 2 Corollary II.6.5 Suppose p is a prime ideal in R such that dimRp = 1 and hdR/p ≤ 2. Then p is a principal ideal. Proof Since hdR R/p ≥ hdRp Rp /pRp = gldimRp , it follows that the gldimRp is ﬁnite. Therefore Rp is a regular local ring of dimension 1. Let x1 , . . . , xt be a minimal generating set for p. Then the x1 , . . . , xt , considered as elements in Rp , generate pRp . Since Rp is a regular local ring of dimension 1, we have that pRp = xj Rp for some j. Let (xj ) = q1 ∩ · · · ∩ ql be an irredundant, primary decomposition for (xj ). From pRp = xj Rp it follows that one of the qi is p. Let us say q1 = p. Then for y in (q2 ∩ · · · ∩ ql ) − p we have that (xj ) : y = p. Since xj is not in mp and hdR p ≤ 1, it follows from the previous proposition that 2 p = (xj ). Theorem II.6.6 Let R be a local domain of dimension ≤ 3 such that hdR R/p is ﬁnite for all minimal primes p. Then R is a unique factorization domain. Proof Since R is a noetherian domain, it follows from our Proposition II.6.3 that it suﬃces to show that each minimal prime ideal is principal in order to show that R is a unique factorization domain. But by Corollary II.6.5, it will follow that a minimal prime ideal p is principal if we can show that hdR R/p ≤ 2. Since hdR R/p < ∞ we have that hdR R/p + codimR/p = codimR ≤ dimR. But codimR/p ≥ 1 and dimR ≤ 3. Thus hdR R/p ≤ 2, which completes the proof. 2 Since every module has ﬁnite homological dimension over a regular local ring, we have established Corollary II.6.7 ization domain.

Every regular local ring of dimension ≤3 is a unique factor-

Multiplicity

Theorem II.6.8

59

Every regular local ring is a unique factorization domain.

Proof We want to prove that if R is a regular local ring of dimension d ≥ 4, then it is factorial (recall that “factorial” is the same as “unique factorization domain”). We assume, by induction, that all regular local rings of dimension less than d are factorial. Following M. Nagata [67], we proceed in two steps. In the ﬁrst, we take a minimal prime ideal p, write it (the way we did in Corollary II.6.5) as p = (x) : y, and show that hdR R/(x, y) ≤ d − 1. We then show that, with the inductive assumption, if elements x, y ∈ R are such that hdR R/(x, y) ≤ d − 1, then the ideal (x) : y is principal. We let the integer t be such that (x, y) : mt = (x, y) : mt+1 , and set J = / m2 , (x, y) : mt . Then J : m = J. We may therefore choose an element u ∈ m, u ∈ ¯ such that J : u = J. Since R = R/(u) is regular local of dimension d − 1, it is factorial, and therefore hdR¯ R/(x, y, u) ≤ 2. From the fact that codimR¯ R/(x, y, u) + hdR¯ R/(x, y, u) = d − 1, we see that codimR¯ R/(x, y, u) ≥ d − 3 > 0. Since codimR¯ R/(x, y, u) = codimR R/(x, y, u) is positive, we see that (x, y, u) : m = (x, y, u) so that (x, y, u) ⊇ J. (To see this, note that (x, y) ⊆ (x, y, u) so that J = (x, y) : mt ⊆ (x, y, u) : mt = (x, y, u).) But this tells us that (x, y) ⊆ J ⊆ (x, y, u) and, since J : u = J, it follows easily that (x, y) = J. Therefore we have shown that (x, y) : m = (x, y) which implies that hdR R/(x, y) ≤ d − 1. We have therefore completed the ﬁrst step of our proof. We now want to show that hdR R/(x, y) ≤ d−1 implies that (x) : y is principal. From the exact sequence used in the proof of Proposition II.6.1, we see that hdR R/((x) : y) ≤ d − 2, which implies that we can ﬁnd an element v ∈ m − m2 such that ((x) : y) : v = (x) : y and ((x) : y, v) : m = ((x) : y, v). Let q be an associated prime of ((x) : y, v). Since q = m, the local ring Rq has dimension less than d, so it is factorial. Thus, in Rq the ideal (x) : y is principal, so the ideal ((x) : y, v)Rq is generated by two elements which form a regular Rq -sequence. Consequently q is of height 2 (by the Cohen–Macaulay Theorem), and we see then that ((x) : y, v)/(v) ⊆ R/(v) is unmixed of height 1 in the regular local ring R/(v) (which is of dimension d − 1). But this tells us that ((x) : y, v)/(v) is a principal ideal, that is, ((x) : y, v)/(v) = (c, v)/(v) for some c ∈ (x) : y. We therefore have, in the ring R, that (c, v) = ((x) : y, v) and (c) ⊆ (x) : y ⊆ ((x) : y, v) = (c, v), from which it follows that (c) = (x) : y. Recalling that our original ideal p = (x) : y, we see that we have proved that p is a principal ideal. This completes the proof of our theorem. 2 II.7 Multiplicity In Section II.3, we proved that if R is a local ring and q an ideal of deﬁnition of R (i.e., an ideal which contains a power of the maximal ideal), then for any ﬁnitely generated R-module, E, the function χq (E) deﬁned by χq (E; n) = length(E/qn E) is a polynomial function. In fact, we deﬁned the dimension of E (Deﬁnition II.3.7) to be the common degree of this polynomial for any ideal

60

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of deﬁnition, q, and we saw that it was always less than or equal to the dimension, d, of the ring, R. In this section, we will take a closer look at the leading coeﬃcient of these polynomials or, more precisely, the coeﬃcient of the term of degree d. First, let us expand a bit on what we did in Section II.3 on polynomial functions. Deﬁne the polynomials Xi in Q[X] by X X(X − 1) · · · (X − i + 1) = . i i! It is easy to show that these polynomials form a linear basis for the polynomial ring, Q[X], over the rationals, Q. They also help to explain why the term “polynomial” enters into the deﬁnition in Section II.3: we of “polynomial function” think of the binomial coeﬃcient ni as the polynomial Xi evaluated at n. If f (n) is an integer-valued polynomial function, then this means that there are integers a0 , . . . , ad such that n n f (n) = a0 + a1 + · · · + ad 1 d for all n suﬃciently or that, for suﬃciently large n, the function, f , and d large the polynomial, i=0 ai Xi , agree. Notice that while the term ad is an integer, the coeﬃcient of X d is ad /d!. We will use this observation in our deﬁnition of multiplicity. While we will not be making use of many of the convenient properties of the polynomials, Xi , we point out two of them (without proof) that the reader may explore if moved by curiosity. (a) −X = (−1)i X+i−1 ; i i X+Y i X Y = j=0 j i−j . (b) i Employing these, a seemingly complicated identity such as n+i−j−1 n n+i−1 (−1)j =− , i−j j i j>0 becomes trivial to prove. We return now to a deﬁnition of multiplicity. Deﬁnition II.7.1 Let R be a local ring with maximal ideal, m, E a ﬁnitely generated R-module, and q an ideal of deﬁnition of R. We deﬁne the multiplicity of q with respect to E, written eE (q), by eE (q) = d! c, where c is the coeﬃcient of X d in the polynomial χq (E). We deﬁne the multiplicity of R to be eR (m). This multiplicity is also sometimes called the Samuel multiplicity.

Multiplicity

61

We see immediately that, since the degree of χq (E) ≤ d, we have eE (q) = 0 if and only if dim(E) < d. It is also evident, from the deﬁnition of the Hilbert– Samuel polynomial, that eE (q) ≥ 0. Our next goal is to tie up the multiplicity just deﬁned with the Euler–Poincar´e characteristic of a certain Koszul complex. To do this, we ﬁrst outline a proof of the following theorem. The full proof can be found in Reference [10], p. 395. In fact, the theorem proven in Reference [10] is a bit more general than the one stated here; as we do not need the more general statement, we will just give the more restrictive result. Theorem II.7.2 Let E be a ﬁnitely generated module over the local ring, R, and let q be an ideal of deﬁnition generated by x1 , . . . , xs . Then χ(H(E(x1 , . . . , xs ))) = ∆s χq (E; n) for n suﬃciently large, where χ(H(E(x1 , . . . , xs ))) denotes the Euler–Poincar´e characteristic of the homology of the Koszul complex. Before we outline a proof of this result, we point out that the number, s, must be at least equal to d, the dimension of R (Section II.3). Therefore, the result is most signiﬁcant when s = d, that is, when x1 , . . . , xs is a system of parameters, and dim(E) = d. Proof (Outline) We will denote by C the complex E(x1 , . . . , xs ): C : 0 → Cs → Cs−1 → · · · → C1 → C0 , where Ci = Λi Rs ⊗R E. We now consider the graded ring, R, deﬁned by R = R/q ⊕ q/q2 ⊕ · · · = qn /qn+1 . n≥0

If we let x ¯1 , . . . , x ¯s be the cosets of x1 , . . . , xs in q/q2 , we see that R is isomorphic ¯s ], that is, it is the image of a polynomial ring to the graded ring R/q [¯ x1 , . . . , x in s linear variables over R/q. Our strategy is to relate our Koszul complex over R to another graded Koszul complex over this graded ring, R. This requires a few more deﬁnitions and constructions. Given our ﬁnitely generated R-module, obtain a ﬁnitely generated E, we n n+1 q E/q E. Finally, given our graded R-module, E, by setting E = n≥0 complex, C, we note that the complex C : 0 → C s → C s−1 → · · · → C 1 → C 0 is none other than the (graded) Koszul complex E(¯ x1 , . . . , x ¯s ).

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Since E is a ﬁnitely generated R-module, we see that Hi (C) is ﬁnitely gener¯s ), so that it is a ﬁnitely ated. However, this homology is annihilated by (¯ x1 , . . . , x generated module over R/q, and hence of ﬁnite length (since R/q is). But Hi (C) is graded; it is the direct sum (l) Hi (C). Hi (C) = l≥0 (l)

Therefore, for suﬃciently large l, we must have all Hi (C) = 0. If we do this for each i, we arrive at an integer l0 with the property that for all l ≥ l0 , we have (l) Hi (C) = 0 for all i. It is this fact that we exploit in order to obtain our result. And we exploit it by writing explicitly what this means, namely, the following sequence is exact for suﬃciently large l: (∗)

0 → ql Cs /ql+1 Cs → ql+1 Cs−1 /ql+2 Cs−1 → · · · → ql+s C0 /ql+s+1 C0 → 0.

The exactness of (∗) allows us to prove the exactness of the complex C(l) : 0 → ql Cs → ql+1 Cs−1 → · · · → ql+s C0 → 0, which in turn implies that Hi (C) = Hi (C/C(l) ) for all suﬃciently large l. Thus these two complexes have equal Euler–Poincar´e characteristics. However, it is clear that the terms of the complexes C/C(l) are of ﬁnite length over R, so that we have χ(H(C)) = χ(H(C/C(l) )) = χ(C/C(l) ), and this is clearly seen to be equal to ∆s χq (E; l). The argument that the exactness of (∗) implies the exactness of the complex C(l) is found, in more generality, in Reference [10]. 2 From the above result follow a number of others of interest. We will list some of them, and refer the reader to Reference [9], sections 4–6, and Reference [78] for proofs and more detail. This next result is an immediate corollary of the above theorem, and is the aim of the foregoing discussion. Theorem II.7.3 If R is a local ring of dimension d, q an ideal of deﬁnition generated by a system of parameters, x1 , . . . , xd , and E a ﬁnitely generated R-module, then χ(H(E(x1 , . . . , xd )) = eE (q). From this, it is relatively straightforward to prove the following result ([9], corollary 4.3). Theorem II.7.4 If R is a local ring of dimension d, q an ideal of deﬁnition generated by a system of parameters, x1 , . . . , xd , and E a ﬁnitely generated

Multiplicity

63

R-module, then eE (q) = length(E/qE)−length(qd−1 E : xd /qd−1 E) −

d−1

e(ql−1 E:xl /ql−1 E) (q/ql )

l=1

where ql = (x1 , . . . , xl ), l = 0, . . . , d − 1. This theorem shows us that the multiplicity, eE (q), always exceeds the length of E/qE. Further investigation into what accounts for this diﬀerence led to the study of Macaulay modules and rings, a notion that we now deﬁne. Deﬁnition II.7.5 An R-module, E, is called a Cohen–Macaulay module (C–M for short) if codimE = dimR. The ring, R, is Cohen–Macaulay (C–M for short) if, when considered as a module over itself, it is a C–M module. The following two results provide us with some insight into the relationship between the property of being C–M, unmixedness, and systems of parameters. These are theorem 5.6 and proposition 5.7 of Reference [9]. Theorem II.7.6 If E is a C–M module, and x1 , . . . , xs s ≥ 0 is a set of elements in m such that height(E/(x1 , . . . , xs )E) = s, then the submodule (0) in E/(x1 , . . . , xs )E is unmixed (that is, every prime ideal belonging to (0) in E/(x1 , . . . , xs )E is of height s) and x1 , . . . , xs is an E-sequence if s ≥ 1. If E is an R-module having the properties that dimE = dimR and for every set of elements x1 , . . . , xs in m such that height(E/(x1 , . . . , xs )E) = s, the submodule (0) in E/(x1 , . . . , xs )E is unmixed, then E is a C–M R-module. Theorem II.7.7 Let E be an R-module (dimR > 0). Then the following statements are equivalent: (a) E is a C–M module. (b) If x1 , . . . , xd is a system of parameters for R, then Hi (E(x1 , . . . , xd )) = 0 for all i > 0. (c) If x1 , . . . , xd is a system of parameters for R, then x1 , . . . , xd is an Esequence. (d) There exists a system of parameters for R which is an E-sequence. Finally, we have the fact that the equality of length and multiplicity characterizes the property of being C–M ([9], theorem 5.10). Theorem II.7.8 Let R be a local ring (dimR > 0) with maximal ideal m, let E be an R-module, and let x1 , . . . , xd be a system of parameters for R. Then the following statements are equivalent: (a) E is a C–M module. (b) The system of parameters x1 , . . . , xd is an E-sequence. (c) If (y1 , . . . , yd ) = (x1 , . . . , xd ), then codimE/(y1 , . . . , yd−1 )E > 0. (d) If q = (x1 , . . . , xd ), then eE (q) = length(E/qE).

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Local ring theory

Moreover, if E is a C–M module, then any system of parameters for R satisﬁes conditions (b), (c), (d). II.8 Intersection multiplicity and the homological conjectures Certainly one of the most interesting threads intertwining homology and commutative algebra is that of the so-called homological conjectures which, to a large degree, emerged from the deﬁnition, proposed by J.-P. Serre, of intersection multiplicity. Here is the deﬁnition [78]. Deﬁnition II.8.1 Let R be a regular local ring of dimension d, and p, q two prime ideals of R such that R/p ⊗R R/q has ﬁnite length. Then Serre deﬁnes the intersection multiplicity, χ (R/p, R/q), by χ(R/p, R/q) =

d

(−1)i length(TorR i (R/p, R/q));

i=0

more generally, for ﬁnitely generated R-modules, M and N, such that M ⊗R N have ﬁnite length, he deﬁnes χ(M, N ) =

d

(−1)i length(TorR i (M, N )).

i=0

Without going into detail, suﬃce it to say that his choice of deﬁnition was informed by the desire to be able to prove a “B´ezout Theorem.” The classical conjectures that immediately emerged are: 1. dim(M ) + dim(N ) ≤ dim(R). 2. (Non-negativity) χ(M, N ) ≥ 0. 3. χ(M, N ) > 0 if and only if dim(M ) + dim(N ) = dim(R). Equivalently, one can write: 1. dim(M ) + dim(N ) ≤ dim(R). 2. (Vanishing) If dim(M ) + dim(N ) < dim(R), χ(M, N ) = 0. 3. (Positivity) If dim(M ) + dim(N ) = dim(R), χ(M, N ) > 0. In 1985, P. Roberts and H. Gillet-C. Soul´e, using K-theory methods, local Chern classes [74, 75] and Adams operations on Grothendieck groups of complexes [44], succeeded in proving Vanishing. (J.-P. Serre had already proven that dim(M )+dim(N ) ≤ dim(R).) In 1997, O. Gabber proved it is always the case that χ(M, N ) ≥ 0. (The best reference for this is the expository paper by M. Hochster [52], and may be obtained on his web page. His references are to references [11] and [36].) Therefore, what remains is to prove the Positivity. Almost every method of attack on this problem is tied to the following fundamental fact.

The homological conjectures

65

Theorem II.8.2 Let x1 , . . . , xs be a sequence of elements of R, and I the ideal they generate. Suppose that M/IM has ﬁnite length. Write K for the Koszul complex associated to the sequence x1 , . . . , xs , and deﬁne χ(K ⊗ M ) =

s

(−1)i length (Hi (K ⊗ M )) .

i=0

Then χ(K ⊗ M ) = eM (I), where eM (I) is the Samuel multiplicity, that is, a slight generalization of the multiplicity deﬁned in the last section (Section II.7), and discussed in detail in References [9] and [78]. Usually the proof requires the use of a spectral sequence argument, or else the type of argument we used in the last section. However, a particular case that is very suggestive is the following: Consider, as usual, R, p and q, with length(R/p ⊗ R/q) < ∞. Suppose that q is generated by a regular sequence x1 , . . . , xs . Then dim(R/q) = dim(R) − s, and therefore dim(R/p) ≤ dim(R) − dim(R/q) = s and dim(R/p) + dim(R/q) = dim(R) if and only if dim(R/p) = s. Since the sequence x1 , . . . , xs is regular, the Koszul complex K is a free resolution of R/q. Therefore TorR i (R/q, R/p) is the homology Hi (K ⊗ R/p). From the above theorem, we see χ(R/p, R/q) = eR/p (q). Since the Samuel multiplicity is always non-negative, and positive if and only if dim(R/p) = s, the conjectures are true in this special case. In the equicharacteristic case, that is, in the case that both the local ring, R, and its residue ﬁeld, k, have the same characteristic, J.-P. Serre proved the multiplicity conjecture by means of a “reduction to the diagonal” method, one that pretty much follows the classical discussions of intersection multiplicity in algebraic geometry. If R is a formal power series ring over a ﬁeld, K[[X1 , . . . , Xd ]], and M and N R-modules such that M ⊗R N is of ﬁnite length, he introduces a new set of indeterminates, Y1 , . . . , Yd , and considers N as a module over K[[Y1 , . . . , Yd ]]. K N over K, as a module over He then deﬁnes a “complete” tensor product, M ⊗

66

Local ring theory

the ring K[[X1 , . . . , Xd , Y1 , . . . , Yd ]] and proves that K[[X,Y ]] ∼ K N, K[[X, Y ]]/(X1 − Y1 , . . . , Xd − Yd ) . M⊗ TorR i (M, N ) = Tori Since X1 − Y1 , . . . , Xd − Yd form a regular sequence, we obtain in this way the result for formal power series rings, and the conjectures for the general equicharacteristic case may be reduced to the special case by means of completion and the Cohen structure theorems of complete local rings. The proof of non-negativity due to Gabber uses a theorem of A. de Jong on the existence of “regular alterations” to reduce the conjectures to questions on regular immersions into projective space on R. Clearly all of these proofs are beyond the scope of this book. However, the Serre conjectures have led to a large number of related conjectures: the so-called homological conjectures. They have also led to an industry within commutative algebra with branches that spread almost everywhere. For the reader who is curious to see what these conjectures look like, we produce a list of them; for the interested reader, we suggest a number of more detailed references: [42, 49, 52, 53, 74–76, 78]. A few paragraphs earlier, we introduced the term “equicharacteristic.” Another term, “mixed characteristic,” comes up often in the context of the homological conjectures. This simply means that the local ring, R, has characteristic zero, while its residue ﬁeld, k, is of characteristic p = 0. The Conjectures. Throughout, R is a local ring with maximal ideal, m, and all modules are ﬁnitely generated unless otherwise indicated. 1. Zero divisor theorem. If M = 0 has ﬁnite homological dimension, and r ∈ R is a non-zero divisor for M, then r is a non-zero divisor for R. Current status: true in all cases. 2. Bass’ question. If M = 0 has a ﬁnite injective resolution, then R is a C–M ring. Current status: true in all cases. 3. Intersection theorem. If M ⊗R N = 0 is of ﬁnite length, then the Krull dimension of N (i.e., dim(R/ann(N ))) ≤ hdR (M ). Current status: true in all cases. 4. New intersection theorem. Let G : 0 → Gn → · · · → G0 → 0 be a ﬁnite complex of free modules such that ⊕i Hi (G) is of ﬁnite length, but not zero. Then the Krull dimension of R is less than or equal to n. Current status: true in all cases. 5. New improved intersection theorem. Let G : 0 → Gn → · · · → G0 → 0 be a ﬁnite complex of free modules such that Hi (G) is of ﬁnite length for i > 0 and H0 (G) possesses a minimal generator annihilated by a power of the maximal ideal. Then dimR ≤ n. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. 6. Direct summand conjecture. If R ⊆ S is a ring extension with R regular (not necessarily local) and S is ﬁnitely generated as an R-module, then R

The homological conjectures

7.

8.

9.

10.

11.

12.

67

is a direct summand of S as an R-module. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. Canonical element conjecture. Let x1 . . . , xd be a system of parameters for R, let G be a projective resolution of the residue ﬁeld of R with G0 = R, and let us denote by K the Koszul complex of R with respect to x1 . . . , xd . Lift the identity R = K0 → G0 = R to a map of complexes. Then, whatever the choice of system of parameters or lifting, the last map R = Kd → Gd is not 0. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. Conjecture on existence of big balanced C–M modules. There is an R-module W (not necessarily ﬁnitely generated) such that mW = W and every system of parameters for R is a regular sequence on W. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. Conjecture on a direct summand being C–M. If R is a direct summand of a regular local ring, S, as an R-module, then R is C–M (R not necessarily local). Current status: true for rings that contain a ﬁeld; open in mixed characteristic. Conjecture on the vanishing of Tor maps. Let A ⊆ R → S be morphisms, where R is not necessarily local, with A, S regular and R a ﬁnitely generated A-module. Let W be an arbitrary A-module. Then the A map TorA i (W, R) → Tori (W, S) is zero for every i ≥ 1. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. The strong direct summand conjecture. Let A ⊆ R be a map of complete local domains, and let q be a prime ideal of R of height 1 that lies over xA, where A and A/xA are regular. Then xA is a direct summand of q as an A-module. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. Conjecture on the existence of weakly functorial C–M algebras. Let R → S be a local morphism of complete local domains. Then there exist an R-algebra BR and an S-algebra BS such that BR is a big balanced C–M algebra for R, BS equally for S, and a map BR → BS such that BR ↑ R

→ →

BS ↑ S

is commutative. Current status: true for rings that contain a ﬁeld; open in mixed characteristic. 13. Serre multiplicity conjectures. Suppose R regular of dimension d, and M ⊗R N of ﬁnite length. Then we have χ(M, N ) = (−1)i length(TorR i (M, N )) = 0

68

Local ring theory

if dimM +dimN < d, and positive if the sum equals d. Current status: true for rings that contain a ﬁeld; open in mixed characteristic (question of positivity open: non-negativity is always true). 14. The small C–M modules conjecture. If R is complete, then there exists a ﬁnitely generated R-module, M = 0, such that some (or, equivalently, every) system of parameters for R is a regular sequence for M . Current status: true in dimensions ≤ 2; other cases, nothing known, not even if the ring contains a ﬁeld, and no matter what the characteristic. The following is a diagram of implications: 13 14

1 8

12

→

→ 7, 6, 5

11, 10

→

4

→3

2

9

III GENERALIZED KOSZUL COMPLEXES In the preceding chapter, we saw the fundamental role that the Koszul complex played in the study of certain aspects of local ring theory. One of the main ideas tied up with that complex is that of regular sequence, with the Cohen–Macaulay Theorem one of the major results obtained through its use. There is another theorem, known as the Generalized Cohen–Macaulay Theorem [35], which deals with the heights of ideals generated by the minors of an m × n matrix. (More precisely, the classical form of this theorem states that if R is a regular local ring, then the ideal generated by the p × p minors of an m × n matrix with entries in R, has height at most equal to (m − p + 1)(n − p + 1).) We can regard the “old” Cohen–Macaulay Theorem, then, as having to do with the 1 × 1 minors of an m × 1 matrix, and the Koszul complex as a complex that “approximates” the resolution of a cyclic module (that is, a module generated by a single element). With this point of view, it is natural to ask if one can associate a complex to an arbitrary m×n matrix (m ≥ n) which generalizes the Koszul complex, and which may be used to prove the generalized theorem mentioned above. This chapter will focus on this question; in fact, we will develop a whole class of “generalized Koszul complexes” that will also play a role in Chapter IV. In our last section of this chapter we take a look at the connection between leading coeﬃcients of certain Hilbert–Samuel polynomials and the Euler– Poincar´e characteristics of our generalized Koszul complexes. It was natural at the time that the papers [18, 27] were being written, to try to generalize the notions of Hilbert–Samuel polynomials and multiplicity to the situation of ﬁnitely generated modules rather than just cyclic modules, that is, to modules of the form coker(f : Rm → Rn ), m ≥ n, rather than those of the form coker(f : Rm → R). In the last sections of Reference [27], these generalizations were introduced, but the new multiplicity was not pursued further. However, in the mid-1990s, a ﬂurry of work on these notions began (see [43, 58–61]), which includes more geometric approaches to the subject. (In the more recent literature, this multiplicity is often referred to as “Buchsbaum-Rim multiplicity.”) As in Chapter II, we have to say that a detailed study of this topic would lead us much too far beyond the scope of this book, so we merely indicate what the original deﬁnition was and its relation to the generalized Koszul complex. III.1 A few standard complexes Before we start to develop generalized complexes, it is worthwhile to look a little more closely at the Koszul complex and some of its immediate relatives. This

70

Generalized Koszul complexes

section will devote itself almost entirely to these considerations. The free modules we deal with are always assumed to be nontrivial and ﬁnitely generated. III.1.1 The graded Koszul complex and its “derivatives” If F is a free R-module, with basis x1 , . . . , xn , we know from Chapter I that the symmetric algebra, S(F ), is isomorphic to the polynomial ring R[x1 , . . . , xn ]. In that ring, x1 , . . . , xn is a regular sequence, so that the Koszul complex associated to the ideal, J, generated by these elements is acyclic. (Lest there be any confusion about the meaning of the expression acyclic complex, we will take it to mean that the positive-dimensional homology groups of the complex are all zero.) If we denote the ring S(F ) by S, and the free S-module S ⊗R F ∼

∼

∼

by F , then the terms of this Koszul complex are 0 → Λn F → Λn−1 F → ∼

· · · → Λ1 F → S → 0. But all of these modules are graded S-modules; in fact, ∼ ∼ S ⊗R Λk F = S ⊗ Λk F, and its grading is given by that of S Λk F = p≥0 p ∼ itself, namely, Λk F = Sp ⊗ Λk F. The boundary map of this complex, which p maps p Sp ⊗ Λk F to p Sp ⊗ Λk−1 F, is easily seen to be the sum of the maps Sp ⊗ Λk F → Sp+1 ⊗ Λk−1 F where each of these maps is the composition 1⊗∆

m⊗1

Sp (F ) ⊗ Λk F −→ Sp (F ) ⊗ F ⊗ Λk−1 F −→ Sp+1 (F ) ⊗ Λk−1 F, ∆ is the diagonal map from Λk F to F ⊗ Λk−1 F, and m is the multiplication map from Sp (F ) ⊗ F to Sp+1 (F ). In fact, we see that the Koszul complex is the direct sum of complexes of R-modules: 0 → Λq F → S1 ⊗ Λq−1 F → · · · → Sq−l ⊗ Λl F → · · · → Sq−1 ⊗ Λ1 F → Sq → 0 which are exact for all choices of q > 0. When q = 0, the complex is nothing other than 0→R→0 and its homology in dimension zero is R itself. Since R = S/J, this is precisely the statement that the original Koszul complex is acyclic, and its zero-dimensional homology is S/J. Deﬁnition III.1.1 For each free R-module, and each integer q ≥ 0, we deﬁne Λq (F ) to be the complex 0 → Λq F → S1 ⊗ Λq−1 F → · · · → Sq−l ⊗ Λl F → · · · → Sq−1 ⊗ Λ1 F → Sq → 0. • An important observation to make here (it will be used later in this chapter) is that for q ≥ n = rankF , the length—or dimension—of this complex is equal to n, and the last term on the left is Sq−n F ⊗ Λn F.

A few standard complexes

71

Deﬁnition III.1.2 If ϕ : G → F is a map of free R-modules, and q ≥ 0, we deﬁne Λq (ϕ) to be the complex 0 → Λq G → S1 (F ) ⊗ Λq−1 G → · · · → Sq−l (F ) ⊗ Λl G → · · · → Sq−1 (F ) ⊗ Λ1 G → Sq (F ) → 0. The complex Λq (F ) is the same as Λq (id), where id is the identity map on F. The boundary maps in the complexes Λq (ϕ) are the evident analogs of those in Λq (F ) : 1⊗∆

Sp (F ) ⊗ Λk G → Sp (F ) ⊗ G ⊗ Λk−1 G

1⊗ϕ⊗1

→

m⊗1

Sp (F ) ⊗ F ⊗ Λk−1 G → Sp+1 (F ) ⊗ Λk−1 G, Of course, while the complexes Λq (F ) are exact for all q > 0, that is not necessarily the case for arbitrary Λq (ϕ). As we saw in Chapter I, the graded dual of the symmetric algebra is the divided power algebra. Thus, if we were to take the linear dual of the complex Λq (F ) above, we would obtain, for each q > 0, the exact sequence (exact because Λq (F ) is split exact) Dq (F ) : 0 → Dq → Dq−1 ⊗ Λ1 F → · · · → Dq−l ⊗ Λl F → · · · → D1 ⊗ Λq−1 F → Λq F → 0. For q = 0, we again just obtain the trivial complex 0 → R → 0. This observation leads us to make the corresponding deﬁnition: Deﬁnition III.1.3 If ϕ : G → F is a map of free R-modules, and q ≥ 0, we deﬁne Dq (ϕ) to be the complex 0 → Dq (G) → Dq−1 (G) ⊗ Λ1 F → · · · → Dq−l (G) ⊗ Λl F → · · · → D1 (G) ⊗ Λq−1 F → Λq F → 0. The complex Dq (F ) is the same as Dq (id), where id is the identity map on F . Again, in the case of general ϕ we can make no assertion about exactness or acyclicity. To be sure that there is no confusion about the boundary maps in the complex Dq (ϕ), we point out that each boundary map, ∂pq , is the composition ∆⊗1

Dp (G) ⊗ Λk F → Dp−1 (G) ⊗ G ⊗ Λk F 1⊗m

1⊗ϕ⊗1

→

Dp−1 (G) ⊗ F ⊗ Λk F → Dp−1 (G) ⊗ Λk+1 F.

72

Generalized Koszul complexes

III.1.2 Deﬁnitions of the hooks and their explicit bases Given the complexes Λq (F ) and Dq (F ) deﬁned above, it is reasonable to ask whether we can describe their cycles in some convenient way. That is, if we are given a basis x1 , . . . , xn of F, can we in some succinct way describe a basis for each of the modules of cycles of these complexes? For example, we know that power monomials: a basis of Dp (F ) may be described as the set of all divided (α1 ) (α2 ) (αt ) αj = p, and αj > 0 where 1 ≤ i1 < i2 < · · · < it ≤ n, xi1 xi2 · · · xit for j = 1, . . . , t. (This is ofcourse equivalent to the description as the set of all (α ) (α ) (α ) divided power monomials x1 1 x2 2 · · · xn n with αj = p and αj ≥ 0 for j = 1, . . . , n, but it will be seen soon why we are using the ﬁrst description.) We also know that a basis for Λk F may be described as the set of all elements {xj1 ∧ xj2 ∧ · · · ∧ xjk } where 1 ≤j1 < · · · < jk ≤ n. Thus we often describe the (α ) (α )

(α )

basis of Dp (F )⊗ Λk F as the set xi1 1 xi2 2 · · · xit t ⊗ xj1 ∧ xj2 ∧ · · · ∧ xjk . A convention among combinatorialists is to write these elements out in tableau form in the following way:   α1 α2 αt # $% & # $% & # $% & xi1 xi1 · · · xi1 xi2 xi2 · · · xi2 · · · xit xit · · · xit    x  j   1 .  x   j2   .   .. xjk

That is, the repeats in the row, when they occur, are to be read as the appropriate divided power of the repeated element, rather than as simply the product of the elements (hence our choice of the word “combinatorialists”) and the product of these divided powers is the element of the divided power algebra that the total row “represents”. Notice that when the tableau is written as above, it has the property that the indices of the basis elements are weakly increasing in the rows (in this case, just the top row is involved), while the column indices are strictly increasing. So we see that we may describe the basis of Dp (F )⊗ Λk F as the set of all tableaux of the form   xu1 xu2 · · · xup   xj1     xj2 T(u,j) =    ..   .  xjk with 1 ≤ u1 ≤ u2 ≤ · · · ≤ up ≤ n, 1 ≤ j1 < j2 < · · · < jk ≤ n, where u = {u1 , u2 , . . . , up } and j = {j1 , j2 , . . . , jk }. The rest of this subsection will be devoted to obtaining a description of a basis of the cycles of our complexes in a tableau form that is equally suggestive. The reader may ask how we can assume that there is a basis for the cycles; after all, we have not directly established that the cycles are free modules. That

A few standard complexes

73

they are projective is clear, since the complexes Λq (F ) and Dq (F ) are split exact; hence the cycles are direct summands of free modules and therefore projective. To see that they are indeed free, let us observe that if R is any commutative ring, and F a free R-module, then F = R ⊗Z F0 where F0 is a free Z-module (of the same rank as F ). Since Λk F = R ⊗Z Λk F0 , Dp (F ) = R ⊗Z Dp (F0 ), Sp (F ) = R ⊗Z Sp (F0 ), and since the complexes (over any ring) are split exact, we see ﬁrst of all that the cycles of Λq (F0 ) and Dq (F0 ) are free (because projective abelian groups, being summands of free abelian groups, are torsion-free, hence free by Proposition I.2.28), and then that the cycles of Λq (F ) and Dq (F ) are free, because they are just the tensor product with R of their integral counterparts. This property of a functor, that is, that is R-free and can be obtained by tensoring its integral counterpart by the ring R, is called universal freeness. With the freeness of these cycles established, how do we go about ﬁnding a “convenient” basis for them? To this end, we will use a standard procedure in homology: we will construct a splitting homotopy for the complex Dq (F ), q > 0; this will pick out the basis for the cycles that we are looking for. One proceeds almost identically for the complexes Λq (F ); we will omit the discussion of the homotopy in that case, but simply give a description of the resulting basis. To construct the desired homotopy, we have to deﬁne maps sqp : Dp (F ) ⊗R Λq−p F → Dp+1 (F ) ⊗R Λq−p−1 F

for − 1 ≤ p ≤ q

such that q sqp + sqp−1 ∂pq = 1 (i) ∂p+1

(ii)

sqp+1 sqp

sq−1

for 0 ≤ p ≤ q;

= 0 for 0 ≤ p ≤ q − 1.

sqq

and are the zero maps. Of course, the maps Since all of the modules we are considering are free, it suﬃces to deﬁne these maps on the basis elements. For p > 0, we will use the tableau description of basis elements that we discussed above, and deﬁne   xu1 xu2 · · · xup  xj1      sqp  xj2   ..   .  xjq−p  0             =               

if u1 < j1 ; xj1 xj2 xj3 .. . xjq−p

xu1

···

xup

      

if u1 ≥ j1 .

74

Generalized Koszul complexes

For p = 0, we deﬁne



  sq0 (xj1 ∧ xj2 ∧ · · · ∧ xjq ) =  

xj1 xj2 .. .

   . 

xjq Now we have to check the conditions (i) and (ii) above. First of all, it is clear that   xj1  xj2    ∂1q sq0 (xj1 ∧ xj2 ∧ · · · ∧ xjq ) = ∂1q  .  = xj1 ∧ xj2 ∧ · · · ∧ xjq ,  ..  xjq so (i) is veriﬁed for p = 0. For p > 0, we have   xu1 xu2 · · · xup  xj1     q q  xj2 ∂p+1 sp    ..   .  xjq−p  0          xj1 xu1 · · · xup    xj2 =    xj3 q  ∂  p+1     ..    .    xjq−p while

 sqp−1 ∂pq

     

xu1 xj1 xj2 .. .

xu2

···

xup

if u1 < j1       

if u1 ≥ j1 ,

  m   q Th ,  = sp−1  h=1 

xjq−p where h runs, in order, through the m distinct elements of {u1, . . . , up } and   xu1 xu2 · · · x h · · · xup   xh     xj1 Th =  ,   ..   . xjq−p where x h indicates that the element xh is to be omitted.

A few standard complexes

75

Now let us consider the two cases: u1 < j1 and u1 ≥ j1 . In must show that  xu1 xu2 · · · xup m  xj1   Th =  xj2 sqp−1  .. h=1  .

the ﬁrst case, we     .  

xjq−p Since u1 ≤ · · · ≤ up , we see that except when h = u1 , the tableau Th always has the property that u1 is less than all the subscripts in the ﬁrst column, so that sqp−1 (Th ) = 0. Thus, we see that   xu1 xu2 · · · xup  xj1     q q  xj2 sp−1 ∂p    ..   .  xjq−p  =

sqp−1

     

x u1

···

xu2



xup

   .  

xu1 xj1 .. . xjq−p 

   Since this latter term is equal to   

xu1

xu2

···

xj1 xj2 .. .

xup

    , we are done.  

xjq−p In the second case, we must calculate the sum   xj1 xu1 · · · xup  xj2     xj3  q ∂p+1    ..   .  xjq−p

+

m h=1

sqp−1

      

xu1 xh xj1 .. . xjq−p

xu2

···

x h

···

xup

    .  

76

Generalized Koszul complexes

Since we are now assuming that u1 ≥ j1 , we see that in each tableau Th for which h = u1 , h > j1 so that  sqp−1

xu1

     

···

xu2

x h

···

xup

xh xj1 .. . xjq−p 

   = −  

xj1

xu1

···

xu2

···

x h

···

      

xup

xh xj2 .. .

      

xjq−p for h = u1 , while for h = u1 we have  sqp−1

     

x u1

xup

xu1 xj1 .. .

    =0  

xjq−p

if u1 = j1 , while for u1 > j1 we have  sqp−1

     

x u1

xu2

···

xup

xu1 xj1 .. .

      

xjq−p     = −  

xj1 xu1 xj2 .. . xjq−p

x u1

xu2

···

xup

    .  

A few standard complexes

    q As for the other summand, ∂p+1   

xj1

77

···

xu1

xj2 xj3 .. .

xup

    , we see that  

xjq−p it is equal to 

xj1

 xh m   xj2 h=1   ..  . xjq−p

      

···

xu1

xu1

xu2

···

xup

xj1 xj2 .. . xjq−p



 m   +   h=1 

xj1

x h

xup

      

if u1 = j1 ;

      

···

xu1

···

x h

···

xup

xh xj2 .. .

      

if u1 > j1 .

xjq−p Comparing terms (including signs), we see that we obtain our desired result. The veriﬁcation of condition (ii) is straightforward; in fact it is trivial. With our homotopy in hand, how do we describe the bases we are looking for? We claim that if we have a basis of Dp (F )⊗Λq−p F, and consider the subset {Tβ } q sqp (Tβ )} generate consisting of those elements for which sqp (Tβ ) = 0, then {∂p+1 q−p the p-cycles of Dq (F ). For if z ∈ Dp (F ) ⊗ Λ F is a p-cycle, write z= cα Tα + d β Tβ α

β

where the Tα are the basis elements that map to zero, while the Tβ are those already described. Since q sqp (z) + sqp−1 ∂pq (z), z = ∂p+1

and z is a cycle, we have 

q q z = ∂p+1 sqp (z) = ∂p+1 sqp 

β

 dβ Tβ  =

β

q dβ ∂p+1 sqp (Tβ ).

78

Generalized Koszul complexes

q Therefore, if the elements {∂p+1 sqp (Tβ )} are linearly independent, they form a basis for the p-cycles. That these elements are linearly independent is easy to see in the following way. If we have a linear combination q dβ ∂p+1 sqp (Tβ ) = 0, β

q this means that the term z = β dβ sqp (Tβ ) ∈ ker(∂p+1 ). That is, z is a (p + 1)q q cycle, and as such we know z = ∂p+2 sp+1 (z). But since sqp+1 sqp = 0, this tells us that z = 0. However, we know that each tableau sqp (Tβ ) is a basis element of Dp+1 (F ) ⊗ Λq−p−1 F, so that if z = 0, we must have that each dβ = 0. This q proves the linear independence we are after. (It also shows us that the map ∂p+1 q q restricted to the image of sp is one-to-one.) Now the set of tableaux {sp (Tβ )} is precisely the subset of the basis tableaux of Dp+1 (F ) ⊗ Λq−p−1 F of the form   xu1 xu2 · · · xup+1   xj1     xj2 T(u,j) =     ..   . xjq−p−1 with u1 ≤ u2 ≤ · · · ≤ up+1 and u1 < j1 < j2 < · · · < jq−p−1 . Since it is clear that every element sqp (Tβ ) is of the form just described, it suﬃces to show that every tableau T(u,j) of that form is sqp (Tβ ) for some tableau Tβ . But we see that   xu2 xu3 · · · xup+1  xu1      T(u,j) = sqp  xj1  , and so we are done.  ..   .  xjq−p−1 q Now, though, we are after a good way to describe the elements ∂p+1 (T(u,j) )   xu1 xu2 · · · xup+1   xj1    xj2  for such tableaux T(u,j) =   . The convention  ..   .  xjq−p−1 generally adopted and to denote this boundary   is to again use tableau notation, xu1 xu2 · · · xup+1   xj1     xj2 by the tableau   , where we see that the rows are   ..   . xjq−p−1 weakly increasing, and the column is strictly increasing. Such a tableau has

A few standard complexes

79

a special name; it is called a standard tableau. Notice, then, that this tableau stands for the sum of a number of terms in Dp (F ) ⊗ Λq−p F obtained essentially by diagonalizing the terms in the ﬁrst row, and multiplying them with the term in ΛF represented by the product of the terms in the ﬁrst  column starting with x1 x1 x2 x2  represents the element xj1 . Thus, for example, the tableau  x3 x4     x1 x1 x2 x1 x2 x2  x1   x2   +  in D3 (F ) ⊗ Λ3 F.  x3   x3  x4 x4 The corresponding discussion for the complexes Λq (F ) considers basis elements of Sq−l ⊗ Λl F which are tableaux of the form   xu1 xu2 · · · xul   xj1    xj2   ,  ..   .  xjq−l where now we have u1 < u2 < · · · < ul and j1 ≤ j2 ≤ · · · ≤ jq−l , and no relation assumed between u1 and j1 . This means that we are regarding the elements of the ﬁrst row as xu1 ∧ xu2 ∧ · · · ∧ xul ∈ Λl F, and xj1 · · · xjq−l as an element of Sq−l (F ). One then shows that the l-cycles of Λq (F ) can be parametrized by the set of tableaux   xu1 xu2 · · · xul+1  xj1      xj2     ..   . xjq−l−1 with u1 < u2 < · · · < ul+1 , j1 ≤ j2 ≤ · · · ≤ jq−l−1 and u1 ≤ j1 . Again, to read oﬀ what this element is, we diagonalize the element represented by the ﬁrst row, and multiply the ﬁrst entry with the element of S(F ) represented by the product of the terms in the ﬁrst column starting with xj1 . (This procedure is a bit more straightforward than for the ﬁrst case as we have here no repeats in the top row.) Deﬁnition III.1.4 The module of p-cycles of the complex Dq F is denoted by K(p+1,1q−p−1 ) (F ); the module of p-cycles of the complex Λq (F ) is denoted by L(p+1,1q−p−1 ) (F ). Remark III.1.5 It should be noted immediately, and it will be used later in this chapter, that for any free module, F, we have K(ν+1,1q−1 ) F ∗ ∼ = (L(q,1ν ) F )∗ .

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Generalized Koszul complexes

Notice that when p = 0, the basis of K(1,1q−1 ) (F ) is the set of tableaux   x   u1          xj1    xj2    with u1 < j1 < · · · < jk−1 . But this, as we would expect, is   .     ..        xjq−1 exactly what parametrizes the basis of Λq (F ). A similar observation applies to the 0-cycles of Λq (F ). The modules K(p+1,1q−p−1 ) (F ) are called hook Weyl modules; the modules L(p+1,1q−p−1 ) (F ) are called hook Schur modules. Later, when we make a more systematic study of the representations of the general linear group (Chapter VI), we will see that these modules are special instances of more general representations. At that later time, it will become clear why we use these particular subscripts to designate these modules. III.2 General setup In order to construct Koszul-type complexes in a much more general form, we ﬁrst take a look at the familiar bar complex, which is even more classical than the Koszul complex. It arises in the following way: If Λ is an algebra over the commutative ring, R, and M a Λ-module, then one can form the following complex: ··· → Λ ⊗ ··· ⊗ Λ ⊗ M → Λ ⊗ ··· ⊗ Λ ⊗ M → ··· → Λ ⊗ M → M % % &# $ &# $ l

l−1

where the map Λ ⊗ M → M is simply the action of Λ on M and, in general, the map Λ ⊗ · · · ⊗ Λ ⊗ M → Λ ⊗ · · · ⊗ Λ ⊗ M is deﬁned by % % &# $ &# $ l

λ1 ⊗ · · · ⊗ λl ⊗ m →

l−1 l−1

(−1)l−j λ1 ⊗ · · · ⊗ λj λj+1 ⊗ · · · ⊗ λl ⊗ m

j=1

+ λ1 ⊗ · · · ⊗ λl−1 ⊗ λl m. In the case of a graded algebra, Λ, one usually restricts the degrees of the λj to be positive and, in that case, the complex is referred to as the normalized bar complex (it is usually assumed that all the λj are homogeneous and that the module, M, is graded also). We will always make this convention when there is a grading. Observe, too, that in the graded case, we get a subcomplex by stipulating that the degree of the λ1 should always be greater than or equal to some ﬁxed integer, s. Another useful observation is that we may always take as our module, M, the algebra Λ itself. (It will always be assumed, unless speciﬁed otherwise, that the tensor product is taken over the ground ring, R.) If B is an arbitrary module over R, we may consider its dual, B ∗=HomR (B,R), and the two exterior algebras: ΛB and ΛB ∗ . We have seen, in our original

General setup

81

discussion of the Koszul complex, that every element, β, of B ∗ induces an endomorphism, dβ (of degree −1) on ΛB. Thus we have a morphism, β → dβ of the module B ∗ into the endomorphism ring of ΛB, or EndR (ΛB), with (dβ )2 = 0. By the universal mapping property of the exterior algebra, this morphism extends to a unique algebra map of ΛB ∗ into EndR (ΛB), and this establishes ΛB as a ΛB ∗ -module. Now suppose that we have a map f : A → B. This induces a map f ∗ : B ∗ → A∗ as well as an algebra map Λf ∗ : ΛB ∗ → ΛA∗ . Since, by the above remarks, ΛA is a ΛA∗ -module, we immediately see that ΛA becomes a ΛB ∗ -module via the map Λf ∗ . Consequently, we can form the bar complex, with ΛB ∗ as our ring Λ, and with ΛA as our graded Λ-module: · · · → ΛB ∗ ⊗ · · · ⊗ ΛB ∗ ⊗ ΛA → ΛB ∗ ⊗ · · · ⊗ ΛB ∗ ⊗ ΛA → % &# $ % &# $ l

l−1 ∗

∗

· · · → ΛB ⊗ ΛB ⊗ ΛA → ΛB ∗ ⊗ ΛA → ΛA. We remarked above that if our ring Λ is graded, and our module M is a graded Λ-module, then we can consider graded strands of the bar complex. In particular, suppose that we are given two integers p ≥ 0 and s > 0. Then instead of the complex we have just written, we could consider

··· →

k1 ≥s ki >0

··· →

Λk1 B ∗ ⊗ · · · ⊗ Λkl B ∗ ⊗ Λp+k1 +···+kl A → % &# $

l

Λ B ∗ ⊗ Λk2 B ∗ ⊗ Λp+k1 +k2 A → k1

k1 ≥s k2 >0

Λk B ∗ ⊗ Λp+k A → Λp A.

k≥s

Notice that if we choose s = 1, and sum for all p, we get the full (graded) complex. We can also do the same thing using ΛB as our ΛB ∗ -module and, given our map Λf : ΛA → ΛB induced from f : A → B, it is not hard to see that we get a commutative diagram Λk1 B ∗ ⊗ Λk2 B ∗ ⊗ Λp+|k| A → Λk B ∗ ⊗ Λp+k A → Λp A ··· → k1 ≥s,k2 >0

··· →

k1 ≥s,k2 >0

k≥s

↓ ∗

∗

Λ B ⊗Λ B ⊗Λ k1

k2

p+|k|

B→

↓ ∗

Λ B ⊗Λ k

↓ p+k

B→Λ B p

k≥s

where |k| denotes the sum of the indices ki and the vertical map is the map Λf tensored with the identity on the appropriate sum of tensor products of copies of ΛB ∗ .

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Generalized Koszul complexes

III.2.1 The fat complexes As in the case of Chapter II, when we see a map of complexes, we automatically take the mapping cone to obtain yet another complex. But instead of doing this in general, let us keep in mind that we are interested in studying the case when A and B are free modules over a commutative ring, R, of ranks m and n respectively. In that case, we see that if we choose our integers p and s above so that p+s > n, then all the terms but Λp B in the bottom row of our commutative map of complexes will vanish. In fact, we will take p + s = n + 1 to accomplish this purpose. When one does this, we see that the mapping cone of our map of complexes is simply the top row augmented by Λp B, that is, we have the complex Λk1 B ∗ ⊗ · · · ⊗ Λkl B ∗ ⊗ Λp+|k| A → · · · ··· → % &# $ k1 ≥s ki >0

→

l

Λ B ∗ ⊗ Λp+k A → Λp A → Λp B. k

k≥s

It is worth remarking that the freeness of the modules A and B does not enter into the above discussion; it is enough to assume that our module B is such that Λn+1 B = 0 to achieve the same result. It may also be amusing to think of the fact that the composition k≥s Λk B ∗ ⊗ Λp+k A → Λp A → Λp B is zero as the generalization of Cramer’s Rule from linear algebra. Although we will continue to use the letters A and B, what we should really keep in the back of our minds is that A = Rm , and B = Rn , with m ≥ n. So, for example, if m = n, our complex is simply · · · → 0 → 0 → Λp R n → Λ p R n , while for m = n + 1, we have 0 → Λn−p+1 Rn∗ ⊗ Λn+1 Rn+1 → Λp Rn+1 → Λp Rn . Up to this point our complexes look fairly svelte, but as soon as the diﬀerence m − n becomes 2 or larger, we get a great many terms that seem extraneous. We will soon see how this problem is addressed in general, but for now, let us just look at the case m = n + 3 to see how “fat” our complexes can get to be. 0 → Λn−p+1 Rn∗ ⊗ Rn∗ ⊗ Rn∗ ⊗ Λn+3 Rn+3 →

Λn−p+1 Rn∗ ⊗ Λ1 Rn∗ ⊗ Λn+2 Rn+3 ⊕ Λn−p+1 Rn∗ ⊗ Λ2 Rn∗ ⊗ Λn+3 Rn+3 ⊕ Λn−p+2 Rn∗ ⊗ Λ1 Rn∗ ⊗ Λn+3 Rn+3

Λn−p+1 Rn∗ ⊗ Λn+1 Rn+3 ⊕ → Λn−p+2 Rn∗ ⊗ Λn+2 Rn+3 ⊕ → Λp Rn+3 → Λp Rn . Λn−p+3 Rn∗ ⊗ Λn+3 Rn+3 Despite this obesity, the complex always has dimension m − n + 1, the number suggested (at least in the case when p = n) by the generalized Cohen–Macaulay

General setup

83

Theorem. But it would be more eﬃcient if we could reduce the fat while keeping the correct dimension. To this end, let us take a closer look at what is going on and what it is we want.

III.2.2 Slimming down Ideally, what we want is a generically acyclic complex, which means that at each step of our construction we want to cover the kernel of our preceding map. So the ﬁrst question we can ask is: Do the images of Λn−p+2 Rn∗ ⊗ Λn+2 Rn+3 and Λn−p+3 Rn∗ ⊗ Λn+3 Rn+3 in Λp Rn+3 above help in covering elements of the kernel of the map Λp Rn+3 → Λp Rn that the map Λn−p+1 Rn∗ ⊗ Λn+1 Rn+3 → Λp Rn+3 missed? And the answer is, No. For in general we have a commutative diagram:

Λk B ∗ ⊗ Λl B ∗ ⊗ Λt+l A

Λk+l B ∗ ⊗ Λt+l A

Λk B ∗ ⊗ Λ t A

Λt−k A ,

where the map Λk B ∗ ⊗ Λl B ∗ ⊗ Λt+l A → Λk+l B ∗ ⊗ Λt+l A is given by the multiplication in ΛB ∗ , and the map Λk B ∗ ⊗ Λl B ∗ ⊗ Λt+l A → Λk B ∗ ⊗ Λt A is given by the action of ΛB ∗ on ΛA. The other two maps are given by the action of ΛB ∗ on ΛA. Since the ﬁrst of the maps (the one given by multiplication in ΛB ∗ ) is a surjection, we see that the image of Λk+l B ∗ ⊗ Λt+l A in Λt−k A is contained in the image of Λk B ∗ ⊗ Λt A in Λt−k A. This tells us that, from the point of view of covering the kernel of Λp Rn+3 → p n Λ R , we do not need the two extra terms above; we needed them in order to use the bar complex construction to proceed; that is, we needed them in order to catch terms coming from, say, Λn−p+1 Rn∗ ⊗ Λ1 Rn∗ ⊗ Λn+2 Rn+3 , under the boundary map of the bar complex. For under this map, a term in Λn−p+1 Rn∗ ⊗ Λ1 Rn∗ ⊗ Λn+2 Rn+3 will have its image in the direct sum of Λn−p+1 Rn∗ ⊗ Λn+1 Rn+3 and Λn−p+2 Rn∗ ⊗ Λn+2 Rn+3 . But the component of the boundary that falls into Λn−p+2 Rn∗ ⊗ Λn+2 Rn+3 is there due to the multiplication in ΛRn∗ . Suppose that, instead of taking an arbitrary element of Λn−p+1 Rn∗ ⊗ Λ1 Rn∗ ⊗ Λn+2 Rn+3 , we were to take an element of K ⊗ Λn Rn+3 , where for the moment we let K = K(2,1n−p ) (Rn∗ ) = ker(Λn−p+1 Rn∗ ⊗ Λ1 Rn∗ → Λn−p+2 Rn∗ ), and apply our boundary map to it. In that case, it would end up in Λn−p+1 Rn∗ ⊗ Λn+1 Rn+3 , we would not have to include the extra term in order to “catch” the result of multiplying in ΛRn∗ , and our slimmed-down complex would start out looking like this: K(2,1n−p ) (Rn∗ ) ⊗ Λn+2 Rn+3 → Λn−p+1 Rn∗ ⊗ Λn+1 Rn+3 → Λp Rn+3 → Λp Rn .

84

Generalized Koszul complexes

We have observed that Λn−p+1 B ∗ may be described as K(1,1n−p ) (B ∗ ), so that the complex above may be rewritten K(2,1n−p ) (Rn∗ ) ⊗ Λn+2 Rn+3 → K(1,1n−p ) (Rn∗ ) ⊗ Λn+1 Rn+3 → Λp Rn+3 → Λp Rn . The natural question to ask is whether this slimming down can be continued all along our fat complexes. That is, can we do what is suggested above, and deﬁne a sleek complex associated to any map f : A → Rn and any integer p≤n: m

0 → K(m−n,1n−p ) (Rn∗ ) ⊗ ΛA → · · · → K(l,1n−p ) (Rn∗ ) ⊗ Λn+l A → · · · → K(2,1n−p ) (Rn∗ ) ⊗ Λn+2 A → K(1,1n−p ) (Rn∗ ) ⊗ Λn+1 A → Λp A → Λp Rn ? We notice that this “complex,” too, has dimension m−n+1, but it is considerably slimmer than the ones we have considered so far. We have placed quotation marks around the word complex because as yet we have not deﬁned the boundary maps and veriﬁed that they give us a complex. To deﬁne the boundary map, we will focus on the deﬁnition of K(l,1n−p ) (Rn∗ ) ⊗ Λn+l A → K(l−1,1n−p ) (Rn∗ ) ⊗ Λn+l−1 A. We have the commutative diagram → Dl−1 (Rn∗ ) ⊗ Λn−p Rn∗ ⊗ Λn+l−1 A Dl (Rn∗ ) ⊗ Λn−p Rn∗ ⊗ Λn+l A n−p+l n−p+l−1 ⊗1 ↓ ∂l−1 ⊗1 ↓ ∂l n∗ n−p+1 n∗ n+l n∗ Dl−1 (R ) ⊗ Λ R ⊗ Λ A → Dl−2 (R ) ⊗ Λn−p+1 Rn∗ ⊗ Λn+l−1 A where the vertical maps are the boundary maps of the complexes Dn−p+l Rn∗ and Dn−p+l−1 Rn∗ , each tensored with the indicated exterior powers of A. The horizontal maps involve the diagonalization of D(Rn∗ ), and the action of Rn∗ on ΛA. That is, the upper horizontal map is the composition Dl (Rn∗ ) ⊗ Λn−p Rn∗ ⊗ Λn+l A → Dl−1 (Rn∗ ) ⊗ Rn∗ ⊗ Λn−p Rn∗ ⊗ Λn+l A ∼ = Dl−1 (Rn∗ ) ⊗ Λn−p Rn∗ ⊗ Rn∗ ⊗ Λn+l A → Dl−1 (Rn∗ ) ⊗ Λn−p Rn∗ ⊗ Λn+l−1 A, and the lower one is similarly deﬁned. The commutativity of the diagram is easy to verify. Given that this diagram is commutative, it follows that the image of the left-hand vertical map is carried into the image of the right-hand vertical map. But these images are precisely the modules K(l,1n−p ) (Rn∗ ) ⊗ Λn+l A and K(l−1,1n−p ) (Rn∗ ) ⊗ Λn+l−1 A that we are after. To check that the composition of two successive such maps is zero is also very easy. With this discussion, we are now ready to deﬁne the families of complexes that we want to have at our disposal.

Families of complexes

85

III.3 Families of complexes For the rest of this chapter, F and G will denote free R-modules of ranks m and n, with m ≥ n. Let f : F → G be a map of free modules over the commutative ring R, and let q be a positive integer. Deﬁnition III.3.1 q 0 → Cm−n+1

We deﬁne the complex C(q; f ) as follows: → · · · → Ckq → · · · → Λn−q+s0 G∗ ⊗ Λs1 G∗ ⊗ Λn+|s| F si ≥1

→

Λn−q+s G∗ ⊗ Λn+s F → Λq F → Λq G,

s≥1

where C1q = Λq F and Ckq =

Λn−q+s0 G∗ ⊗ Λs1 G∗ ⊗ · · · ⊗ Λsk−2 G∗ ⊗ Λn+|s| F,

k ≥ 2,

si ≥1

|s| = si . The maps (except for Λq f : Λq F → Λq G) are the bar complex maps associated to the action of the algebra ΛG∗ on ΛF. As the signs of all our maps are quite crucial, we will make clear just what we mean by “boundary map” in this context. Namely, if a0 ⊗a1 ⊗· · ·⊗ak−2 ⊗x ∈ Ckq for k ≥ 3, then ∂(a0 ⊗ a1 ⊗ · · · ⊗ ak−2 ⊗ x) = a0 ⊗ a1 ⊗ · · · ⊗ ak−2 (x) +

k−3

(−1)k−i a0 ⊗ a1 ⊗ · · · ⊗ ai ∧ ai+1 ⊗ · · · ⊗ ak−2 ⊗ x.

i=0

Deﬁnition III.3.2

We deﬁne the complex T(q; f ) as follows:

q → · · · → Tkq → · · · → K(2,1n−q ) G∗ ⊗ Λn+2 F 0 → Tm−n+1

→ Λn−q+1 G∗ ⊗ Λn+1 F → Λq F → Λq G, where T1q = Λq F and Tkq = K(k−1,1n−q ) G∗ ⊗ Λn+k−1 F,

k ≥ 2.

86

Generalized Koszul complexes

The maps (except for Λq f : Λq F → Λq G and Λn−q+1 G∗ ⊗ Λn+1 F → Λq F ) are given as the following compositions of maps: K(k−1,1n−q ) G∗ ⊗ Λn+k−1 F → Λn−q+1 G∗ ⊗ Dk−2 G∗ ⊗ Λn+k−1 F → Λn−q+1 G∗ ⊗ Dk−3 G∗ ⊗ Λn+k−2 F. From the discussion of the last section, we know that the image of this composition lies in K(k−2,1n−q ) G∗ ⊗ Λn+k−2 F, so this deﬁnes our desired boundary map. We should point out, as we did in the earlier section, that the term Λn−q+1 G∗ ⊗ Λn+1 F is also of the general form: it is K(1,1n−q ) G∗ ⊗ Λn+1 F. In Chapter II, where we introduced the Koszul complex, we saw that the Koszul complex associated with a map g : F ⊕ R → R was the mapping cone of the Koszul complex associated to the map f : F → R mapped into itself by multiplication by the image of 1 under the map of R to R. The complexes we introduced above share that property with the Koszul complex, although we have to be a bit more careful in setting up our mapping cones in this case. To this end, we make two more deﬁnitions. Deﬁnition III.3.3

Let f : F → G be a map of free R-modules.

1. For every pair of integers (q, l), with 1 ≤ q ≤ n + 1 and l ≤ m, deﬁne the complex C(q, l; f ) as follows: (q,l)

(q,l)

0 → Cm−n+q−l → · · · → Ck → · · · → Λn−q+s1 G∗ ⊗ Λs2 G∗ ⊗ Λn−q+|s|+l F si ≥1

→

Λn−q+s G∗ ⊗ Λn−q+s+l F → Λl F,

s≥1

where (q,l)

Ck

=

Λn−q+s1 G∗ ⊗ Λs2 G∗ ⊗ · · · ⊗ Λsk ⊗ Λn−q+|s|+l F,

k ≥ 1,

si ≥1

and the maps are just those of the bar complex. 2. For every pair of integers (q, l), with 1 ≤ q ≤ n + 1 and l ≤ m, deﬁne the complex T(q, l; f ) as follows: (q,l)

(q,l)

0 → Tm−n+q−l → · · · → Tk

→ · · · → K(2,1n−q ) G∗ ⊗ Λn−q+2+l F → K(1,1n−q ) G∗ ⊗ Λn−q+1+l F → Λl F

where (q,l)

Tk

= K(k,1n−q ) G∗ ⊗ Λn−q+k+l F,

and the maps are essentially the ones we deﬁned for the complexes T(q; f ) above.

Families of complexes

87

Note that our deﬁnition does not preclude the possibility that l be negative, or that q be equal to n+1. In our immediate use of these complexes, these rather bizarre possibilities will not show up. In the next section, however, we will make use of this ﬂexibility when we consider various proofs of “generic acyclicity” (a term that will be deﬁned in that section) of our complexes. We now have the following proposition. Proposition III.3.4 Let F and G be free R-modules. Let f : F → G and γ : R → G be maps, and let g : F ⊕ R → G be the sum of the maps f and γ. Then for all q > 0, γ induces maps γ1 : C(q, q − 1; f ) → C(q; f ) and γ2 : T(q, q − 1; f ) → T(q; f ) such that the mapping cone of γ1 is C(q; g) and the mapping cone of γ2 is T(q; g). Proof

Let y0 = γ(1). We deﬁne γ1,0 = γ2,0 : Λq−1 F → Λq G by γ1,0 (x) = γ2,0 (x) = y0 ∧ Λq−1 f (x), (q,q−1)

and γ1,k : Ck

→ Ckq by

γ1,k (b1 ⊗ · · · ⊗ bk ⊗ x) = b1 ⊗ · · · ⊗ bk−1 ⊗ y0 (bk )(x). (q,q−1)

(q,q−1)

→ Tkq we use the fact that Tk is the image of To deﬁne γ2,k : Tk α n−q ∗ ∗ n+k−1 n−q+1 ∗ ∗ n+k−1 Λ G ⊗ Dk G ⊗ Λ F →Λ G ⊗ Dk−1 G ⊗ Λ F, while Tkq is the β

image of Λn−q G∗ ⊗ Dk−1 G∗ ⊗ Λn+k−1 F → Λn−q+1 G∗ ⊗ Dk−2 G∗ ⊗ Λn+k−1 F. We then note that we have a map, which we will denote by ∂γ : Dl G∗ → Dl−1 G∗ , deﬁned as the composition ∆⊗γ

Dl G∗ = Dl G∗ ⊗ R → Dl−1 G∗ ⊗ G∗ ⊗ G → Dl−1 G∗ ⊗ R = Dl−1 G∗ , 1⊗ω

where ∆ : Dl G∗ → Dl−1 G∗ ⊗G∗ is the indicated diagonal map, and ω : G∗ ⊗G → R is the map that sends an element b ⊗ a to a(b). Observe next that the following diagram commutes: Λn−q G∗ ⊗ Dk G∗ ⊗ Λn+k−1 F ↓α Λn−q+1 G∗ ⊗ Dk−1 G∗ ⊗ Λn+k−1 F

∂γ

→ Λn−q G∗ ⊗ Dk−1 G∗ ⊗ Λn+k−1 F ↓β

∂γ

→ Λn−q+1 G∗ ⊗ Dk−2 G∗ ⊗ Λn+k−1 F.

Then the map ∂γ = 1 ⊗ ∂γ ⊗ 1 carries the image of α to the image of β, and this is the map γ2,k that we are after. As in the case of the classical Koszul complex, the fact that the mapping cones of these maps are the appropriate complexes is not diﬃcult to prove; again one 2 simply uses the identiﬁcation of Λl (F ⊕ R) with Λl F ⊕ Λl−1 F.

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Notice that when G = R, so that n = 1 and q = 1, the complex C(1; f ) is just the Koszul complex (as is also T(1; f )), and the complex C(1, 0; f ) is also just the Koszul complex associated to the map f : F → R. III.3.1 The “homothety homotopy” When we introduced the Koszul complex associated to a map f : F → R, we showed that if a = f (x), for some x ∈ F, then multiplication by a on the Koszul complex is homotopic to zero. Since multiplication by an element of the ring is called a homothety, we can call the homotopy a “homothety homotopy.” What we will show here is that if f : F → G is a map as above, then for each λ ∈ Λn F and ξ ∈ Λn G∗ , the homothety ξ(λ) on the complex C(q; f ) is homotopic to zero. This not only shows that the homology of C(q; f ) is annihilated by ξ(λ), but that if A is any additive functor from the category of R-modules to itself which preserves homotheties, the element ξ(λ) ∈ R also annihilates the homology of A(C(q; f )) (since additive functors preserve chain homotopies). We take ξ ∈ Λn G∗ and λ ∈ Λn F, and we want to show that multiplication in the fat complex C(q; f ) by µ = ξ(λ) is homotopic to zero. We deﬁne σ 0 : Λq G → Λ q F by setting σ0 (y) = y(ξ)(λ). In order to show that ∂σ0 (y) = µy, we must prove some basic lemmas about the operation of ΛG∗ on ΛG in general. Lemma III.3.5

Let α ∈ Λl G∗ and u ∈ Λr G, v ∈ Λs G. Then α(u ∧ v) =

(−1)r|αi | αi (u) ∧ αi (v),

where we set ∆(α) = αi ⊗ αi , and |αi | denotes the degree of αi . (One says ∗ that the operation of ΛG on ΛG satisﬁes the measuring identity or that ΛG∗ measures ΛG.) Proof We will not give a detailed proof of this, but just an outline. For |α| = 1, this is easy: it is just the statement (which we used in Chapter II) that the boundary map of the Koszul complex is a derivation. We then proceed by induction on the degree of α, and assume that α = α1 ∧ α2 where |α1 | = 1. One then uses the fact that α2i ⊗ α2i ∆(α1 ∧ α2 ) = ∆(α1 )∆(α2 ) = (α1 ⊗ 1 + 1 ⊗ α1 ) + (−1)|α2i | α2i ⊗ α1 ∧ α2i , = α1 ∧ α2i ⊗ α2i

Families of complexes

89

together with the induction assumption that (v) . α(u ∧ v) = α1 (−1)r|α2i | α2i (u) ∧ α2i The rest is simply careful comparison of signs.

2

An immediate corollary of this is the following useful fact. Lemma III.3.6

Let u ∈ Λr G, v ∈ Λs G and α ∈ Λl G∗ . Then u(α)(v) =

(−1)|ui |(l−1) ui ∧ α(ui ∧ v).

Proof Again, we will not prove this; we just indicate that in this case, one can use induction on the degree of u. 2 Using the formula above, one sees immediately that ∂σ0 (y) = µy, so our homotopy is under way. That is, we have ∂σ0 (y) = (−1)|yi |(n−1) yi ∧ ξ(yi ∧ λ) = y ∧ ξ(λ) = µy, because most of the summands disappear in the above sum (since the λ is now to be considered as sitting inside Λn G, so that multiplication with yi is zero unless degree of yi is zero). To proceed with the deﬁnitionof the homotopy, we reﬁne our notation a bit. We have been writing ∆(x) = j xj ⊗ xj to indicate the total diagonal of x. However, we occasionally want to specify the degrees of the terms that occur in the sum, so to do that we will write ∆(x) = j l xjl ⊗ xjr−l to indicate the degree l of the term xjl , and the degree r − l of the term xjr−l if the element x is of degree r. We now deﬁne xjl (ξ) ⊗ xjq−l ∧ λ σ1 (x) = j

l
where x ∈ Λq F. In order to see that this works, we prove the following two lemmas: Lemma III.3.7 have j

For x ∈ Λq F, ξ and λ as before, and for each integer l, we

xjl (ξ)(xjq−l ∧ λ) =

q−l j

t=0

(−1)t

l+t xjq−l−t ∧ xjl+t (ξ)(λ). l

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Generalized Koszul complexes

Proof By the measuring identity, we see that xjl (ξ)(xjq−l ∧ λ) = ±ξit (xjq−l ) ∧ xjl (ξin−t )(λ) j

j

=

i,t

j

=

i,t

±xjq−l−t

t

j

=

±xjq−l−t ∧ xjt (ξit )xjl (ξin−t )(λ)

∧ xjl

xjt (ξit )ξin−t

(λ)

i

±xjq−l−t ∧ xjl (xjt (ξ))(λ)

t

j

l + t ± xjq−l−t ∧ xjl+t (ξ))(λ). = l t j 2 Lemma III.3.8 For x ∈ Λ F, and ξ and λ as before, we have xjl (ξ)(xjq−l ∧ λ) = x ∧ ξ(λ) − x(ξ) ∧ λ. q

j

Proof

l
As we saw above, for each l we have q−l l+t (−1)t xjl (ξ)(xjq−l ∧ λ) = xjq−l−t ∧ xjl+t (ξ)(λ). l t=0

So this says that β l β xjl (ξ)(xjq−l ∧ λ) = (−1) (−1) ∧ xjβ (ξ)(λ), x l jq−β l
β

l

and the conclusion follows (because we know what happens to the alternating sum of binomial coeﬃcients). 2 Now it is easy to see that σ1 gives us what we need. The next step is the “generic” one; that is, once we get this one, the others all are of the same type. We deﬁne β ⊗ xjn+s−q+l (ξ) ⊗ xjq−l ∧ λ σ2 (β ⊗ x) = j

l
for β ∈ Λn−q+s G∗ , x ∈ Λn+s F. There are two “tricks” to showing that this works: one is to recognize that β(xjn+s−q+l )(ξ) = β ∧ xjn+s−q+l (ξ) because ξ is of degree n (we are applying the corollary to the measuring formula). Then when we compute the boundary of l
Families of complexes

91

we can allow l = q − s, since the zero degree term in the middle cancels out in the boundary. But then, since the degree of x is greater than n (since s ≥ 1), the term x(ξ) = 0, so when we apply our lemma, we see that this deﬁnition of σ2 works. Now it is easy to see that for k ≥ 2, we may deﬁne β0 ⊗ · · · ⊗ βk−1 ⊗ xjn+|s|−q+l (ξ) σk+1 (β0 ⊗ · · · ⊗ βk−1 ⊗ x) = j

⊗

l
xjq−l

∧λ

for β0 ∈ Λn−q+s0 G∗ , βi ∈ Λsi G∗ , for i ≥ 1, and x ∈ Λn+|s| F (|s| = We therefore state (without proof) the following theorem.

i≥0 si ).

Theorem III.3.9 Let λ ∈ Λn F, ξ ∈ Λn G∗ , and let µ = ξ(λ). Deﬁne σ : C(q; f ) → C(q; f ) by σ0 (y) = y(ξ)(λ); σ1 (x) = xjl (ξ) ⊗ xjq−l ∧ λ; j

and σk+1 (β0 ⊗ · · · ⊗ βk−1 ⊗ x) =

l
j

β0 ⊗ · · · ⊗ βk−1 ⊗ xjn+|s|−q+l (ξ)

l
⊗ xjq−l ∧ λ for k ≥ 1. Then σ is a homothety homotopy for the homothety µ. III.3.2 Comparison of the fat and slim complexes As we would much prefer to work with the slim complexes, we would like to know that they, too, have homothety homotopies for every homothety µ of the previous subsection. In fact, just as a matter of curiosity, it is worth knowing just how the fat and slim complexes do compare. What we will show here is that for every q > 0, there are maps αq : T(q; f ) → C(q; f ), θq : C(q; f ) → T(q; f ) such that θq αq = 1, and αq θq is homotopic to the identity. Using these maps, we will be able to transfer the homotopy we have on the fat complexes to the slim ones. We will also have established the fact that these two families of complexes are homotopically equivalent, so that our cavalier “slimming down” that we did was justiﬁed. We will not present all the details of proof of these facts; the interested reader is referred to Reference [12] for the full story. We ﬁrst deﬁne the map αq : T(q; f ) → C(q; f ). In dimensions 0 and 1, of course, the map is the identity. For ease of notation, we will label these maps q q and α−1 respectively and, in general, we will denote by αkq the map that α−2 q q takes Tk+2 to Ck+2 .

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Generalized Koszul complexes

Deﬁnition III.3.10 (Deﬁnition of the map αq ) For q = −2 and −1 we deﬁne αkq to be the identity. For k ≥ 0, we deﬁne αkq as the composition q → Λn−q+1 G∗ ⊗ Dk G∗ ⊗ Λn+k+1 F → Tk+2

Λn−q+1 G∗ ⊗ D1 G∗ ⊗ · · · ⊗ D1 G∗ ⊗ Λn+k+1 F, % &# $ k

where the right arrow is the k-fold diagonalization of Dk G∗ . (We observe that q .) the latter term is a summand of Ck+2 It is a relatively straightforward calculation to see that αq , thus deﬁned, is a map of complexes. To deﬁne the maps θq : C(q; f ) → T(q; f ) requires a bit of extra preparation. In dimensions 0 and 1 it is of course deﬁned to be the identity and denoted by q q and θ−1 and as in the deﬁnition of the map αq , we will denote by θkq the θ−2 q q to Tk+2 . But now we introduce some notation to facilitate map that sends Ck+2 the deﬁnition. Notation Assume that a ﬁxed basis of G∗ is given, say y1 , . . . , yn . An element yj1 ∧ · · · ∧ yjl will be written either as j1 ∧ · · · ∧ jl or j1 · · · jl or simply as J. In short, the index on a basis element will be used to denote that element (as is the practice when working with tableaux), and products of elements will be denoted by products of their indices. For l = 1, we will usually write j or j1 instead of J. When working with products in the divided power algebra, we will use tableau notation in order to avoid confusion about whether juxtaposition means the usual product within that algebra, or the divided power when there are repeats. For example, we will (2) write, for yu1 yu2 , with u1 < u2 , the tableau [u1 u1 u2 ]. We will use freely the standard basis, consisting of standard tableaux, for “hook Weyl modules” (as described in Subsection III.1.2), and a typical basis q would be denoted by element of Tk+2   u1 · · · uk j1  j2    ⊗x  ..   . jn−q+1



  where x is an element of Λn+k+1 F, and  

j1 j2 .. .

u1

···

uk

    is a standard 

jn−q+1 tableau which stands for the image of the element j2 · · · jn−q+1 ⊗ [j1 u1 · · · uk ] ∈ Λn−q G∗ ⊗Dk+1 G∗ in Λn−q+1 G∗ ⊗Dk G∗ under the map which diagonalizes Dk+1 G∗ → G∗ ⊗ Dk G∗ , and then multiplies Λn−q G∗ into Λn−q+1 G∗ by using the G∗ factor.

Families of complexes

93

Recall that “standard tableau” means that the indices are strictly increasing in the column, and weakly increasing in the row. For reasons that will become apparent later, we will make one more (unusual) convention, which we will use in this section only, about our use of tableau notation in the case of rows: we will assume that the tableau is zero if the top row is not weakly increasing as written. (2) Thus, in the case of yu1 yu2 with u1 > u2 , we would have to write [u2 u1 u1 ] to represent it as a tableau. We are now in a position to deﬁne our maps θkq for all k ≥ −2. Deﬁnition III.3.11 (Deﬁnition of the map θq ) For k = −2 and −1, we have already said that the map is to be the identity. For k = 0, and Y = J ⊗ x, with J a basis element of Λn−q+s G∗ and x ∈ Λn+1 F, we deﬁne θ0q (Y ) = (−1)(s−1)(n−q+1) js · · · jn−q+s ⊗ j1 · · · js−1 (x). For q > 0, and Y = J ⊗ u1 ⊗ · · · ⊗ uk ⊗ x with all the ui basis elements of degree one, and J still of degree n − q + s, we deﬁne   js uk · · · u1  js+1    θk (Y ) = (−1)(s−1)(n−q+1)+(s−1)k  .  ⊗ j1 · · · js−1 (x).  ..  jn−q+s It is essential to remember here that the tableau is to be read as equal to zero if the row is not standard. Assume that the map θlq has been deﬁned on elements Y = J ⊗U1 ⊗· · ·⊗Ul ⊗x for Ui basis elements of arbitrary degree, and l < k, and that θkq has been deﬁned on elements Z = J ⊗U1 ⊗· · ·⊗Uk ⊗x with U1 , . . . , Ut basis elements of arbitrary degree (we make the convention that U0 = J), Ut+2 , . . . , Uk basis elements of degree 1, and Ut+1 basis element of degree st+1 ≤ r (r > 0). We now let Z = J ⊗ U1 ⊗ · · · ⊗ Ut+1 ⊗ · · · ⊗ Uk ⊗ x with the basis element Ut+1 = v ∧ W, v of degree 1, all of the basis elements Ut+2 , . . . , Uk are of degree 1, and degree(W ) = r. Deﬁne θkq (Z ) = θkq (B + (−1)(k−t−1)r E) where B = J ⊗ U1 ⊗ · · · ⊗ Ut v ⊗ W ⊗ Ut+2 ⊗ · · · ⊗ Uk ⊗ x, and E = J ⊗ U1 ⊗ · · · ⊗ Ut ⊗ v ⊗ Ut+2 ⊗ · · · ⊗ Uk ⊗ W (x). Notice that in position t + 1, the elements B and E are of degree less than or equal to r, while the terms of higher index are not aﬀected in degree. From this we see immediately (using a simple induction proof) that θkq (Z ) = 0 unless Ut+2 ≥ · · · ≥ Uk . While it is trivial to show that the map αq is a map of complexes, it is not trivial to prove that the same is true of θq . It is at this point that we seriously invoke the suggestion that the reader look at Reference [12] for details. In any event, it is quite straightforward to show that the composition θq αq is equal to the identity on the complex T(q; f ).

94

Generalized Koszul complexes

This fact is actually enough for us to transfer the homothety homotopy from the fat to the slim complexes. In fact, we have the following general lemma that will give us the result we want. Lemma III.3.12 Let X and Y be complexes, α : X → Y and θ : Y → X maps of complexes such that θα = idX . Let s : Y → Y be a homotopy on Y which makes the scalar µ homotopic to zero on Y. Then the map τ = θsα : X → X is a homotopy on X carrying µ to zero. Proof First we have to show that ∂X τ0 = µ. But ∂X τ0 = ∂X θ1 s0 α0 = θ0 ∂Y s0 α0 = θ0 µα0 = µθ0 α0 = µ. For i > 0, we have to show that ∂X τi + τi−1 ∂X = µ. But here again we have ∂X τi = ∂X θi+1 si αi = θi ∂Y si αi = θi (µ − si−1 ∂Y ) αi = µ − θi si−1 ∂Y αi while τi−1 ∂X = θi si−1 αi−1 ∂X = θi si−1 ∂Y αi , and this does it.

2

With this lemma, we now know that our slim complexes carry the desired homotopy. Although, as we said, we will not give a proof of the following fact, we state it for the sake of completeness. Theorem III.3.13 The composition of maps, αq θq is chain homotopic to the identity map on the complex C(q; f ). Thus, the complexes C(q; f ) and T(q; f ) are homotopically equivalent. III.4 Depth-sensitivity of T(q; f ) In this section we prove that the complexes T(q; f ), like the Koszul complex, give us a way of determining the depth of the ideal generated by the maximal minors (that is, the minors of order n = rankG). Recall that we are always assuming that m ≥ n, where m = rankF. The method of proof that we will use here is the one that was used in Reference [27], Proposition 2.1. Deﬁnition III.4.1 A sequence of additive functors {T i ; i ≥ 0} from an abelian category, A, to another, B, is called a homology functor if 1. For every exact sequence E : 0 → A → B → C → 0 in A, and every i ≥ 0, there is given a morphism ∂ i (E) : T i (C) → T i+1 (A) which is “natural”, that is,

Depth-sensitivity of T(q; f )

95

if E : 0 → A → B → C → 0 is another exact sequence, and ϕ : E → E is a map of complexes, then the diagram T i (C) ↓ T i (C )

∂ i (E)

→

∂ i (E )

→

T i+1 (A) ↓ T i+1 (A )

is commutative. 2. For each exact sequence E above, the sequence ∂ 0 (E)

0 → T 0 (A) → T 0 (B) → T 0 (C) → T 1 (A) → · · · ∂ i (E)

→ T i (C) → T i+1 (A) → · · · is exact. As one example, one that we will be dealing with almost immediately, we may consider the complex T(q; f ), and deﬁne the functors T i (A) = H i (HomR (T(q; f ), A)) . Another one, equally relevant, is to deﬁne the functors U i (A) = Hm−n+1−i ( T(q; f ) ⊗R A). Because of the fact that T(q; f ) is a free complex, tensoring with or taking Hom into a module, preserves exact sequences. Thus, to each exact sequence of R-modules, we get a long exact sequence of homology (or cohomology) modules. Hence these are examples of homology functors. Theorem III.4.2 Let I be an ideal in the commutative, noetherian ring R, and let {T i } be a homology functor from the category of ﬁnitely generated R-modules into itself which satisﬁes the following conditions: 1. T i (α) = α for every homothety α ∈ R; 2. SuppT i (A) ⊆ Supp R/I; 3. T 0 (A) = 0 if and only if Supp R/I ∩ Ass(A) = 0, where Ass(A) is the set of all associated primes of A. Then for each R-module, A, such that A/IA = 0, we have that depth(I; A) is equal to the smallest integer r for which T r (A) = 0. Furthermore, if T 0 (∗) = HomR (M, ∗) for some ﬁxed R-module M , then T d (A) = ExtdR (M, A), where d = depth(I; A). Proof We prove the ﬁrst part of the statement by induction on depth(I; A) (which we know to be a non-negative integer). Since A/IA = 0, we know that depth(I; A) = 0 if and only if every element of I is a zero divisor for A. But this means that depth(I; A) = 0 if and only if I is contained in the union of the associated primes of A. Thus depth(I; A) = 0 if

96

Generalized Koszul complexes

and only if Supp R/I ∩ Ass(A) = 0, and this is if and only if T 0 (A) = 0. Hence we have our result for depth(I; A) = 0. Let c(A) be the smallest integer r such that T r (A) = 0, and let us assume that depth(I; A) = d > 0. We want to show, as we have in the case depth(I; A) = 0, that d = c(A). Let α1 , . . . , αd be a maximal regular A-sequence contained in I, and consider the exact sequence 0 → A → A → A/α1 A → 0. We know that depth(I; A/α1 A) = d − 1 so, by induction, we have that c(A/α1 A) = d − 1. From the exactness of T i (α1 )

· · · → T i−1 (A/α1 A) → T i (A) → T i (A) → T i (A/α1 A) → · · · it follows that T i (α1 ) is a monomorphism for all i ≤ c(A/α1 A). But if T i (α1 ) is a monomorphism, then so is T i (α1h ) for all h. However, SuppT i (A) ⊆ Supp R/I, so that the radical of the annihilator of T i (A) contains the radical of I. Since α1 ∈ I, some power of α1 must annihilate T i (A), and since we are assuming that α1h = T i (α1h ), we cannot have that T (α1h ) is a monomorphism unless T i (A) = 0. From this we see that T i (A) = 0 for all i ≤ d − 1. To see that T d (A) = 0, we simply observe that 0 → T d−1 (A/α1 A) → T d (A), is exact, and that tells us what we want to know, that is, that c(A) = d. This concludes the proof of the ﬁrst statement of our theorem. Before we prove the second part, notice that the functors ExtiR (R/I, ∗) and ExtiR (R/radI, ∗) each satisfy the conditions of our theorem with respect to either of the ideals, I and radI. Thus we see immediately that depth(I; A) = depth(radI; A). Now for the proof of the second statement: we assume that there is a ﬁxed R-module, M, such that T 0 (∗) = HomR (M, ∗). If we take any prime ideal p, and use the fact that Supp HomR (M, R/p) = Supp T 0 (R/p) ⊆ Supp R/I, we see that Mp = 0 for all p not in Supp R/I, that is, Supp M ⊆ Supp R/I. Therefore, both {T i (∗)} and {ExtiR (M, ∗)} are homology functors satisfying the conditions of the theorem. Suppose that d = depth(I; A). If we let J = Ann(M )∩ Ann(T d (A)) ∩ I, and apply the fact that Supp R/I contains both Supp M and Supp T d (A), we see that radJ = radI. Therefore depth(J; A) = depth(I; A) = d, and we can ﬁnd a maximal A-regular sequence α1 , . . . , αd in J. Now, by the property of being homology functors, and from the fact that the αi annihilate T d (A) and ExtdR (M, A), we have T d (A) ∼ = T 0 (A/(α1 , . . . , αd )A) = HomR (M, A/(α1 , . . . , αd )A) ∼ = ExtdR (M, A), and the proof of the theorem is complete.

2

Depth-sensitivity of T(q; f )

97

In the examples we gave immediately after the deﬁnition of homology functor (Deﬁnition III.4.1), we see that if we choose I to be the ideal I(f ) generated by the minors of order n of the morphism f, then both satisfy the conditions (1) and (2) of our theorem with respect to I(f ). This is where we make use of the fact that we have homothety homotopies which show that the ideal I(f ) annihilates all of our homology in these complexes. In the case of the functors T i , we do see that T 0 (A) = HomR (M, A), where M = cokerf. It is easy to see that Supp M = SuppR/I(f ). For our homothety homotopy says that the 0-dimensional homology of our complex, namely M, is annihilated by all the n × n minors of f, so I(f ) ⊆ ann(M ). On the other hand, if a ∈ ann(M ), then an ∈ ann(coker(Λn f )) = I(f ). This says that a ∈ radI(f ), so we have I(f ) ⊆ ann(M ) ⊆ radI(f ), and establishes the asserted equality of supports. What about U 0 (A)? Can this also be expressed as Hom of some ﬁxed module into A? To see that the answer is yes, we ﬁrst consider the case m > n + 1, and recognize that U 0 (A) = ker(K(m−n,1n−q ) G∗ ⊗ Λm F ⊗ A → K(m−n−1,1n−q ) G∗ ⊗ Λm−1 F ⊗ A) if m > n + 1. Using the remark at the end of the subsection on hooks, we see that ∗ K(m−n,1n−q ) G∗ ⊗ Λm F ∼ = L(n−q+1,1m−n−1 ) G ⊗ Λ0 F and that

∗ K(m−n−1,1n−q ) G∗ ⊗ Λm−1 F ∼ = L(n−q+1,1m−n−2 ) G ⊗ Λ1 F .

Then using the fact that, for any ﬁnitely generated free module, H, and any ﬁnitely generated module, A, H ∗ ⊗R A ∼ = HomR (H, A), we see that U 0 (A) is equal to the kernel of the map HomR (L(n−q+1,1m−n−1 ) G ⊗ Λ0 F, A) → HomR (L(n−q+1,1m−n−2 ) G ⊗ Λ1 F, A). If we let N = coker L(n−q+1,1m−n−2 ) G ⊗ Λ1 F → L(n−q+1,1m−n−1 ) G ⊗ Λ0 F , then the kernel above is simply HomR (N, A). To see what map N is the cokernel of, consider the commutative diagram: Λt+1 G ⊗ Sν−2 G ⊗ F → Λt G ⊗ Sν−1 G ⊗ F → L(t,1ν−1 ) G ⊗ F → 0 ↓ ↓ ↓ ψt,ν Λt+1 G ⊗ Sν−1 G → Λ t G ⊗ Sν G → L(t,1ν ) G →0 ↓ ↓ ↓ ϕt,ν Λt+1 G ⊗ Sν−1 M → Λ t G ⊗ Sν M → Γt,ν →0 ↓ ↓ ↓ 0 0 0 where t = n − q + 1, ν = m − n − 1, M = coker(f ), and Γt,ν = coker(ϕt,ν ) = coker(ψt,ν ). Clearly, N = Γn−q+1,m−n−1 . The horizontal maps in the ﬁrst two rows are those of the graded Koszul complex on G, while the vertical maps

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between the ﬁrst and second rows are the operation of F on SG through the map f and, in the third column, we have the induced maps on the indicated modules. We are using the fact that if M = coker(f ), for any map f : C → D, then the sequence Sν−1 D ⊗ C → Sν D → Sν M → 0 is exact for all ν. Therefore we see that both T i and U i are homology functors satisfying the conditions of our theorem, or we will as soon as we show that SuppN = Supp M = Supp R/I(f ). Since M = 0 implies that N = 0, it suﬃces to show that M = 0 implies that N = 0. By localizing at primes, we may assume that R is local, and then by reducing modulo the maximal ideal, we may assume that R is a ﬁeld. In that case, all the modules are ﬁnite-dimensional vector spaces, so we may assume that G has a basis consisting of a basis of M together with a basis for the image of f. It is easy, then, to construct an element of Λt G ⊗ Sν M which is not in the kernel of ϕt−1,ν+1 . But, since the kernel of ϕt−1,ν+1 contains the image of ϕt,ν , this gives us an element not in the image of ϕt,ν , and we are done. When m = n, it is easy to see, using the same kinds of identiﬁcations used above, that U 0 (A) ∼ = HomR (Q, A), where Q = coker(Λn−q G∗ → Λn−q F ∗ ). Since m = n, it is easy to prove that SuppQ = Supp R/I(f ). When m = n + 1, similar identiﬁcations show that U 0 (A) = HomR (Q , A), where Q = coker(Λm−q F → Λm−q G). Again, it is easy to see that the support of Q is equal to that of R/I(f ). With these observations, we are able to apply our theorem to the functors T i and U i . In fact, we will state a number of results without proof, as the details of proof may be found in Reference [27]. (Although the proofs in Reference [27] relate to the complexes C(q; f ), our discussion above is suﬃcient to see that they apply equally to the complexes T(q; f ).) Theorem III.4.3 Given our map f : F → G and an R-module M such that M/(I(f ))M = 0, we have for each q with 1 ≤ q ≤ n, the following facts: 1. depth(I(f ); M ) = the smallest integer r for which H r (T(q; f ), M ) = 0, and furthermore, H d (T(q; f ), M ) = ExtdR (coker(Λq f ), M ) where d = depth(I(f ); M ). 2. m − n + 1 − depth(I(f ); M ) is equal to the largest integer r for which Hr (T(q; f ), M ) = 0. Furthermore  ExtdR (coker(Λm−q f ), M ) if m = n or n + 1; Hm−n+1−d (T(q; f ), M ) =  ExtdR (Γn−q+1,m−n−1 , M ) if m > n + 1. Corollary III.4.4 supM depth(I(f ); M ) ≤ m − n + 1, where M runs through all R-modules such that M/I(f )M = 0. In particular, depth(I(f ); R) ≤ m−n+1.

Another kind of multiplicity

99

We should note that there is a stronger result that can be proven, namely: supp dim Rp ≤ m − n + 1, where p runs through all minimal primes containing I(f ). The reader may ﬁnd this in Reference [27], theorem 3.5. Corollary III.4.5 Given our map f : F → G and an R-module M such that M/I(f )M = 0, the following statements are equivalent: 1. For some q, 1 ≤ q ≤ n, Hr (T(q; f ), M ) = 0 for all r = 0. 2. For some q, 1 ≤ q ≤ n, H r (T(q; f ), M ) = 0 for all r = m − n + 1. 3. For all q, 1 ≤ q ≤ n, Hr (T(q; f ), M ) = 0 for all r = 0 and H r (T(q; f ), M ) = 0 for all r = m − n + 1. 4. depth(I(f ); M ) = m − n + 1. In particular, if coker(f ) = 0, T(q; f ) is a free resolution of coker(Λq f ) for some q, 1 ≤ q ≤ n (or for all q, 1 ≤ q ≤ n), if and only if depth(I(f ); R) = m − n + 1. Corollary III.4.6 Let f : F → G be a map with coker(f ) = 0. If depth(I(f ); R) = m−n+1, then hdR (coker(Λq f )) = m−n+1 for all q, 1 ≤ q ≤ n. We ﬁnish this subsection with the generalized Cohen–Macaulay Theorem due to J. Eagon [37]. (We actually get the result not only for the ideal I(f ), but for coker(Λq f ) for all q, 1 ≤ q ≤ n.) Observe that if M is a Cohen–Macaulay module, then M is equidimensional, that is, dim Rp is constant (and equal to dim M ) for all p in Ass(M ). We also note that over a Cohen–Macaulay ring, equidimensionality is the same as unmixedness, since dim Rp + dim R/p = dim R for all prime ideals p. Lemma III.4.7 If M is a Cohen–Macaulay R-module, then ExtdR (N, M ) is equidimensional for any module N such that N ⊗R M = 0, where d is equal to depth(Ann(N ); M ). Proof See Reference [27], lemma 2.8.

2

Theorem III.4.8 Let f : F → G be our usual map and assume that depth(I(f ); R) = m − n + 1. If R is a Cohen–Macaulay ring, then coker(Λq f ) is unmixed for all q, 1 ≤ q ≤ n. III.5 Another kind of multiplicity In Section II.7 we restricted ourselves to ideals of deﬁnition of our noetherian local ring, R, although in Reference [9] the theory was developed for ideals, I, of R and modules, E, such that the length of E/IE is ﬁnite. The analogous restriction in this section would be to consider morphisms f : Rm → Rn such that the length of coker(f ) is ﬁnite, and then consider, for any R-module, E, the module coker(f ) ⊗R E. However, we shall adopt the more general setting of Reference [9] (as this is the point of view taken in Reference [27]), and consider

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morphisms f : Rm → Rn and modules, E, such that the length of coker(f ) ⊗R E is ﬁnite. Recall that if M is an R-module, we have associated to it the commutative algebra, S(M ), that is, the symmetric algebra of M over R. If M = Rm , we know that S(M ) = S(Rm ) is isomorphic to the polynomial ring in m variables: R[X1 , . . . , Xm ]. Given the module, M , we can form the graded module, S(M )⊗R M , over the graded ring, S(M ). What is more, we have the map of S(M ) ⊗R M into S(M ) given by multiplication by M , that is, since M = S1 (M ), we have for each ν the multiplication map of Sν (M ) ⊗R M into Sν+1 (M ). We will call this map τM , and the corresponding Koszul complex we will denote by K(τM ). When our module is Rm , we will write τm in place of τM , so that the Koszul complex will be denoted by K(τm ). If N is a graded S(M )-module, the complex K(τM ) ⊗S(M ) N will be denoted by K(τM ; N ). Suppose now that we have a map f : Rm → Rn . This induces an algebra map S(f ) : S(Rm ) → S(Rn ), and thus converts S(Rn ) into a graded S(Rm )-module (or algebra). Therefore, if E is an R-module, the graded S(Rn )-module, S(Rn )⊗R E, is a graded S(Rm )-module, and we have the complex K(τm ; S(Rn ) ⊗R E). It is this complex that we will want to study in more detail, in relation to a Hilbert–Samuel polynomial that we are about to introduce. Let us suppose that in addition to our map f : Rm → Rn , we have an R-module, E, with the property that length(coker(f ) ⊗R E) < ∞. Then it is straightforward to see that length(coker(Sν (f )) ⊗R E) < ∞ for all ν > 0 (since Supp coker(f ) = Supp coker(Sν (F )) for ν > 0). We can therefore deﬁne the function, Pf (E; ν) = length(coker(Sν (f ))⊗R E), which is a function from the positive integers to themselves. One of the main results we have about this function is the following (see Reference [27], theorem 3.1). Theorem III.5.1 Let f : Rm → Rn be a map, and E an R-module such that length(coker(f ) ⊗R E) < ∞. Then Pf (E; ν) is a polynomial function for suﬃciently large ν. Furthermore, ∆m Pf (E; ν) = (−1)m−q length(Hm−q (K(τm ; S(Rn ) ⊗R E))) q

for all suﬃciently large ν. Proof The proof of this result has the same ﬂavor as that of Theorem II.7.2. The important observation that gets the proof rolling is that we have the exact sequence of S(Rm )-modules 0 → S(Rm )E → S(Rn ) ⊗R E → coker(S(f )) ⊗R E → 0, and while coker(S(f )) ⊗R E is in general not ﬁnitely generated over S(Rm ), the module S(Rm )E is. Thus its Koszul homology is ﬁnitely generated, it is a ﬁnitely generated graded module over R (since the homology is killed by the

Another kind of multiplicity

101

image of τm ), and therefore all of its graded components are zero from a certain integer on. The rest of the proof is easy to follow. 2 Deﬁnition III.5.2

For any polynomial function, ϕ, set

u(ϕ) = (degϕ)! × (the leading coeﬃcient of ϕ). We now have a sequence of results from Reference [27], section 3, which enable us to deﬁne our generalized multiplicity and relate it to our Koszul complexes. The ﬁrst of these is Reference [27], theorems 3.3 and 3.4. Theorem III.5.3 Let f : Rm → Rn be a map, and E and R-module such that length(coker(f ) ⊗R E) < ∞. Then u(Pf (E; ν)) and degPf (E; ν) − n + 1 depend only on the R-modules E and coker(f ). If R is a local ring, then degPf (E; ν) = n − 1 + dimE. We also have the following corollary (corollary 3.6 of Reference [27]). Corollary III.5.4 Let R be a local ring, and f and E as above. Then m − n + 1 ≥ dimE, and hence m ≥ degPf (E; ν). Deﬁnition III.5.5 Given a module, M , of ﬁnite length over a local ring, R, choose an exact sequence (a presentation) Rm → Rn → M → 0. Then for each R-module, E, the product (dimR + n − 1)!× (the coeﬃcient of the term of degree n−1+dimR in the polynomial Pf (E; ν)) is a non-negative integer which depends only on M and E. We call this non-negative integer the multiplicity of M with respect to E, and denote it by eE (M ). In the case when E = R, we write eR (M ). When M = R/q with q an ideal of deﬁnition, we retrieve our older deﬁnition of multiplicity (except that we have it written now as eE (R/q) instead of eE (q)). Deﬁnition III.5.6 In view of Corollary III.5.4, we call a map f : Rm → Rn a parameter matrix for E if length(coker(f ) ⊗R E) < ∞ and m − n + 1 = dim E. The following proposition sums up where we are to this point (Reference [27], proposition 3.8). Proposition III.5.7 Let f : Rm → Rn be a parameter matrix for E, and let M = coker(f ). Then 1. eE (M ) = q (−1)m−q length(Hm−q (K(τm , (S(Rn ) ⊗R E)ν+q ))) for ν suﬃciently large. 2. eE (M ) ≥ 0 and eE (M ) = 0 if and only if dimE < dimR. 3. If 0 → E → E → E → 0 is an exact sequence, then eE (M ) = eE (M ) + eE (M ). In section 4 of Reference [27], there is a large amount of calculation all of which is directed at proving the following theorem (see Reference [27], theorem 4.2).

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Theorem III.5.8 Let R be a local ring, f : Rm → Rn a map of free R-modules, and E an R-module such that length(coker(f ) ⊗R E) < ∞. Then we have n−1 ∆m Pf (E; ν) = χH(Λq f, E) n−q where H(Λq f, E) stands for the Euler–Poincar´e characteristic of either the complex C(q; f ) (as in the paper, [27]), or T(q; f ). We remark that in Reference [27], the complexes T(q; f ) did not appear, as they were not yet known at the time of writing that paper. However, since we know that the two complexes are homotopic, their homologies are the same, as are their Euler–Poincar´e characteristics. The notation used in the statement of the theorem is used not only because it is the notation of the statement cited, but because it is “neutral,” that is, it can refer to either of the two complexes. We do not give a proof here of the above theorem; one proof can be found in the cited reference. Also, the works of D. Kirby and D. Rees [58, 59] give algebraic treatments of this topic, and the paper [60] gives a geometric approach to the subject.

IV STRUCTURE THEOREMS FOR FINITE FREE RESOLUTIONS

In the previous chapter, we saw a whole class of complexes associated with a given map, f , of ﬁnite free modules. We proved that all of them were sensitive to the depth of the ideal generated by the maximal minors of f , and gave a necessary and suﬃcient condition for them to be ﬁnite free resolutions of the cokernels of the exterior powers of f in terms of the depth of the ideal generated by the maximal minors of f . In this chapter, we deepen our understanding of what it means for a complex of free modules to be a resolution, that is, we confront the problem: what does it mean for a complex of free R-modules to be exact? Given that the underlying notion of homology, whether it be topological or algebraic, is the exact sequence, it seems reasonable to ask just what this notion means in everyday terms. Of course, when we say “everyday terms” we mean in terms of the data that we are given to hand: in the case of free resolutions, the matrices that represent the boundary maps of free complexes. This, after all, is what we would look at if we had a complex of vector spaces. Of course, there is a vast diﬀerence between modules which have ﬁnite free resolutions and those with inﬁnite ones, so it seems reasonable to start with the more familiar, that is, the ones whose resolutions are ﬁnite. Another reason that compels us to study ﬁnite resolutions is that the most familiar rings are regular: think of the power series rings and polynomial rings. (A commutative noetherian ring is called regular if its global dimension is ﬁnite.) And if one is to exploit the ﬁniteness of a resolution, one natural starting point is to make use of the “last” (left-most) matrix that occurs in it. So, from a general point of view, addressing the question of the structure of ﬁnite free resolutions suggests that the ﬁnal matrix of the resolution be a jumping-oﬀ point. In addition to this purely theoretical consideration, some problems, such as the Lifting Problem (which will be explained in this chapter), very much indicated that the last matrix of a ﬁnite free resolution was of signiﬁcance. We will see that the complexes studied in Chapter III help us to get some insight into the minors of the matrices that occur in a resolution. At certain crucial points, these complexes or their duals will play a role. We will also see that while our methods carry us some distance toward our goal, there is still a great deal more to be understood. But ﬁrst we have to get down to the basics

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of exactness, and then we will discuss several interesting applications (including another, better proof of the factoriality of a regular local ring), which in fact triggered the investigation. Unless otherwise stated, in this chapter we always assume that we are working over noetherian rings. IV.1 Some criteria for exactness Let R be a nontrivial commutative ring (with 1). Consider the following complex of ﬁnitely generated free R-modules fn

fk

f1

F : 0 → Fn → Fn−1 → · · · → Fk → Fk−1 → · · · → F1 → F0 . We discuss what conditions guarantee that F is exact, that is, that its homology modules (which exist only in positive dimension) are all trivial. Deﬁnition IV.1.1 Given a map ψ : E → F of ﬁnite free R-modules, deﬁne the rank of ψ, written rank(ψ), to be the largest integer k such that the map Λk ψ : Λk E → Λk F is not zero. If r denotes this rank, deﬁne I(ψ) to be the ideal Ir (ψ) introduced after Remark I.3.32, namely, the ideal of “r × r minors” of ψ. We see immediately that 0 ≤ r ≤ min{rankE, rankF } = s, since the map Λ0 ψ : Λ0 E → Λ0 F is the identity map on R, and Λs+1 ψ : Λs+1 E → Λs+1 F is clearly zero. We should also note that the ideal of 0 × 0 minors (i.e., I0 (ψ)) is always equal to (1) = R. Another special case to mention is that of ψ = 0. In this case, we have rank(ψ) = 0, and I(ψ) = R. We have already met an instance of I(ψ) in Section III.4; ψ was the map f : F → G with rankF = m ≥ n = rankG and rankf = n. Remark IV.1.2 In general, the ideal I(ψ) does not localize. That is, if ψ : F → G is a map of free modules over R, and we localize with respect to a multiplicative subset, S, we may consider I(ψ) ⊗ Rs and I(ψ ⊗ RS ). In general, these two ideals do not coincide. For instance, if e is a nontrivial idempotent of R, m a maximal ideal containing e, and ψ : R → R the map sending 1 to e, then I(ψ) = (e), (I(ψ))m = 0 but I(ψm ) = Rm . However, if I(ψ) contains a non-zero divisor, then we have I(ψ)RS = I(ψ ⊗ RS ) for every multiplicatively closed subset, S, of R. Deﬁnition IV.1.3 A projective R-module, P , is said to have deﬁned rank if the free Rm -module Pm has the same rank for every maximal ideal m. This common rank is deﬁned to be the rank of P . When R is a noetherian ring with no nontrivial idempotents, every projective module has deﬁned rank (cf., e.g. Reference [80],theorems 7.8 and 7.12). Example IV.1.4 Let e be a nontrivial idempotent of R. Then R/(e) is a projective R-module but its rank is not well deﬁned.

Some criteria for exactness

Proof

105

R/(e) is projective because the short exact sequence inc

0 → (e) → R → R/(e) → 0 splits, by means of the map R → (e) sending 1 to e. Suppose that m is a maximal ideal of R, and consider I(incm ). If e ∈ m, then 1 − e is not in m, and every element of (e) is annihilated by 1 − e. Thus / m, then (e) localized at m is im(incm ) = 0, and the rank of (R/(e))m is 1. If e ∈ Rm , so that the rank of (R/(e))m is 0. Since each of the elements e and 1−e must be contained in some maximal ideal, and since they cannot both be contained in the same one, we see that R/(e) does not have well-deﬁned rank. 2 Proposition IV.1.5 Given ψ : E → F as before, coker(ψ) is projective and has well-deﬁned rank if and only if I(ψ) = R. Proof We ﬁrst do the “if” part. Since I(ψ) = R contains a non-zero divisor, rank(ψ) does not change when localizing. Hence we assume that R is local and prove that coker(ψ) is free of rank equal to rankF − rank(ψ). If we call r the rank of ψ, I(ψ) = R implies that some r × r minor of ψ is invertible. Hence we may choose bases for E and F such that ψ has block matrix AB C D with A equal to the r × r identity matrix, Ir . Elementary row and column operations show that B and C may be assumed to be zero. But then D must be zero as well, otherwise the rank of the overall matrix would exceed r. It is now obvious that the cokernel of Ir 0 0 0 is free of rank equal to rankF − rank(ψ). ∼ Now we do the “only if” part. Since coker(ψ) is projective, we get F = 0 coker(ψ) ⊕ im(ψ), E ∼ = ker(ψ) ⊕ im(ψ), and ψ = idim(ψ) ⊕ ker(ψ) → coker(ψ) . Let m, n, t be the ranks of E, F , coker(ψ), respectively. If t = n, then ψm = 0 for every maximal ideal m, hence ψ = 0 and I(ψ) = R, as expected. If t < n, then ψm always has the matrix In−t 0 , 0 0 hence det In−t = 1 ∈ I(ψm ) and I(ψm ) = (I(ψ))m = Rm for every m. Again R = I(ψ), as required. 2 The above proposition reaﬃrms that the rank of R/(e) (of Example IV.1.4) ψ

is not well deﬁned since it has a presentation R → R → R/(e) → 0 such that

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I(ψ) = R (just take ψ to be multiplication by e and observe that the nontriviality of e means that I(ψ) = (e) is a proper ideal). We point out that the above proposition still holds under the assumptions that R is a noetherian ring with no nontrivial idempotents, and E and F are projective. That is, we can weaken the hypothesis that the modules be free, provided that we can still guarantee well-deﬁned rank. The following result generalizes a well-known property of vector spaces over a ﬁeld. Proposition IV.1.6

Given a complex ψ

ϕ

C: E→F →G of ﬁnite free R-modules such that I(ψ) = R = I(ϕ), C is exact if and only if rankF = rankψ + rankϕ. Proof As in the ﬁrst part of the last proof, we may assume that R is local. Thus a maximal minor of ψ (resp. of ϕ) is invertible, thanks to the fact that I(ψ) = R (resp. I(ϕ) = R). Hence we may choose bases for E, F and G such that ψ and ϕ have block matrices 00 Ir 0 and , respectively, 0 0 0 Is where Ir is the r × r identity matrix, Is is the s × s identity matrix, and r = rank(ψ), s = rank(ϕ). It is now obvious that C is exact if and only if r + s equals rankF . 2 Theorem IV.1.7

The complex fn

fk

f1

F : 0 → Fn → Fn−1 → · · · → Fk → Fk−1 → · · · → F1 → F0 given at the beginning of this section is exact if and only if (1) rankFk = rankfk+1 + rankfk (2) depthI(fk ) ≥ k for all k = 1, . . . , n. Notice that we make the convention that the ideal, (1), has inﬁnite depth. Proof We will start with the “only if” part. As usual, we want to use localization to prove the result. In order for I(fk ) to localize, we prove that I(fk ) contains a non-zero divisor. By inverting all non-zero divisors, we may assume that R is a semilocal ring (if every non-zero divisor is a unit, every maximal ideal is in the associator of the ring, and the associator of a ring is a ﬁnite set). Given any maximal ideal, m, codimRm = 0, since m is an associated prime of 0. By Theorem II.4.12, (coker(fk ))m has homological dimension zero, hence is free. It follows that the truncated complex fn

fk

0 → Fn → Fn−1 → · · · → Fk → Fk−1

Some criteria for exactness

107

is split exact after localization at m (split exact meaning exact with the further property that im(fi+1 ) is a direct summand of Fi for every i), that the rank of (coker(fk ))m is equal to (−1)i−k+1 rankFi i≥k−1

(a quantity independent of m), and that the projective R-module coker(fk ) has well-deﬁned rank. Therefore Proposition IV.1.5 applies and I(fk ) = R, as wished. Now we can prove (1): invert all non-zero divisors, getting I(fk ) = R for all k, and use Proposition IV.1.6. As for the proof of (2), take a prime ideal p ⊇ I(fk ) such that codim(Rp ) = depthI(fk ) (a prime associated to a maximal regular sequence of I(fk )). We claim that codim(Rp ) ≥ k. To see this, localize F at p: (coker(fk ))p is not free (by Proposition IV.1.5), hence hdR (coker(f1 ))p is at least k (we cannot shorten Fp ). Again by Theorem II.4.12, it follows that codim(Rp ) ≥ k, as claimed. This completes the proof of the “only if” part. Now we prove the “if” part. We may assume that R is local, with maximal ideal m (for exactness is a local property, and homology commutes with localization). Let d indicate the depth of m. For every k > d, condition (2) implies I(fk ) = R. . If In particular, coker(fd+1 ) is a free module by Proposition IV.1.5, say Fd+1 fd stands for the map Fd → Fd induced by fd , F can be viewed as the result of glueing together the following complexes: fd+1

C : 0 → Fn → · · · → Fd+1 → Fd → Fd → 0 and f

fd−1

d Fd−1 → Fd−2 → · · · → F0 , F : 0 → Fd →

both of which have to be proven to be exact. The exactness of C comes immediately from Proposition IV.1.6. The exactness of F (which satisﬁes the same conditions as F, but with length d equal to the depth of m) comes by induction on dim R. If dim R = 0, then d = 0, the complex is trivial and we are through. If dim R > 0, since localization preserves all assumptions, we know by induction that F turns out to be exact whenever localized at a non-maximal prime ideal of R. Hence H(F ), the homology of F , is all supported in m, m is an associated prime of H(F ), and depthH(F ) = 0. The conclusion now results from the so-called Acyclicity Lemma due to C. Peskine and L. Szpiro, which we quote below in a slightly weaker form than the original. 2 Proposition IV.1.8 (Acyclicity Lemma)

Let

E : 0 → En → En−1 → · · · → Ek → Ek−1 → · · · → E1 → E0 be a complex of ﬁnite free modules over the local ring R. If there is some k > 0 such that Hk E = 0 and Hk+t E = 0 for all t > 0, then depthHk E ≥ 1.

108

Proof

Structure theorems for ﬁnite free resolutions

Cf. Reference [69] (and also [23, lemma 3] and [41, lemma 20.11]). 2

We remark that both Proposition IV.1.6 and Theorem IV.1.7 above lend themselves to several generalizations (cf., e.g., [23]). In particular, the following holds ([23], corollary 2). Proposition IV.1.9 A:

Let M = 0 be an R-module, and let ϕn

ϕ1

0 → Fn → Fn−1 → · · · → F1 → F0

be a complex of ﬁnitely generated projective R-modules of well-deﬁned rank. Then A ⊗R M is exact if and only if for all k = 1, . . . , n (a) rank(ϕk+1 ⊗R M ) + rank(ϕk ⊗R M ) = rank Fk and (b) I(ϕk ⊗R M ) contains an M -sequence of length k or I(ϕk ⊗R M ) = R. We include this last result because it represents a curiosity in relation to an old question in homological algebra, namely, the “rigidity problem”. Roughly put, the rigidity question is this: If N and M are two modules of ﬁnite homological R dimension, and if TorR i (N, M ) = 0 for some given i, is it true that Torj (N, M ) = 0 for all j ≥ i? Positive results have been obtained on this problem when R is regular local, but it is not true in general. Possibly the most recent results can be found in Reference [54], where the authors examine the case when the ring is a hypersurface, that is, R = S/(f ), where S is a regular local ring. In any event, we mention this here because the above proposition seems to be made for studying the rigidity problem; the resolution, A, above could be the resolution of N (truncated at dimension i), in which case the Tor we are looking at is just the homology of A ⊗R M . However, none of the methods of attack on this problem has made use of this kind of result. There is an important corollary of Theorem IV.1.7, which is often considered as a second, independent, criterion of exactness for F. We have already remarked that exactness is a local property. Hence F is exact if and only if F ⊗R Rp is exact for every prime ideal p. The second criterion says that we may limit the set of prime ideals, p, for which exactness of F ⊗R Rp must be checked. Theorem IV.1.10 F is exact if and only if F ⊗R Rp is exact for all prime ideals p such that depth(pRp ) < n. Proof In order to prove the “if” part, it suﬃces to show that F satisﬁes conditions (1) and (2) of Theorem IV.1.7. In fact, if (2) holds, (1) follows automatically, since the ranks of the maps localize and (1) holds by assumption over every Rp . Hence we just prove (2), by contradiction. If depthI(fk ) < k for some k in {1, . . . , n}, then there exists a prime ideal p ⊇ I(fk ) associated to a maximal regular sequence contained in I(fk ). Hence rank((fk )p ) = rank(fk ) and depth(pRp ) = depthI(fk ) < k. It follows that I((fk )p ) = (I(fk ))p ⊆ pRp

Some criteria for exactness

109

and depthI((fk )p ) < k, contradicting what property (2) says about the exact 2 complex F ⊗R Rp . We end this section with some properties enjoyed not just by F, but by a larger family of exact complexes. Let G denote a (possibly inﬁnite) complex of ﬁnite free R-modules that is exact: gk

g1

· · · → Gk → Gk−1 → · · · → G1 → G0 . Suppose that depthI(gk ) ≥ 1 for every k ≥ 1. (By Theorem IV.1.7, this condition is automatically satisﬁed when F is ﬁnite.) Proposition IV.1.11

With the notation as above, we have:

(a) radI(gk ) ⊆ radI(gk+1 ) for all k ≥ 1. (b) If rank(g1 ) = rankG0 and depthI(g1 ) = k, then radI(g1 ) = · · · = radI(gk ). Proof (a) For x ∈ radI(gk ), let S = {xi }. Then S ∩I(gk ) = ∅ and I(gk )S = RS . Since by assumption I(gk ) contains a non-zero divisor, rank(gk ) = rank((gk )S ) and I((gk )S ) = (I(gk ))S . Hence coker(gk )S is projective, by Proposition IV.1.5. Since G exact implies GS exact, the projectivity of coker(gk )S implies that of coker(gk+1 )S . For the exactness of 0 → im(gk )S → (Gk−1 )S → coker(gk )S → 0 yields the projectivity of im(gk )S , and im(gk )S = coker(gk+1 )S because of the exactness of 0 → im(gk+1 )S → (Gk )S → coker(gk+1 )S → 0. Hence RS = I((gk+1 )S ) = (I(gk+1 ))S , again by Proposition IV.1.5, and x ∈ radI(gk+1 ). (b) Thanks to (a), it suﬃces to show that radI(gk ) ⊆ radI(g1 ). Assume for / radI(g1 ). By a contradiction that there exists some x ∈ radI(gk ) such that x ∈ localizing at S = {xi }, we may suppose that I(gk ) = R I(g1 ). As I(g1 ) is a proper ideal, coker(g1 ) = 0 (as we have seen in the discussion after Theorem III.4.2, I(ψ) and ann(coker(ψ)) have the same radical). Since for every ﬁnitely generated R-module M , hdR M ≥ depth(ann(M )), we get hdR (coker(g1 )) ≥ k, due to the hypothesis depthI(g1 ) = k.

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Structure theorems for ﬁnite free resolutions

Since I(gk ) = R, coker(gk ) is projective by Proposition IV.1.5. Hence im(gk ) = ker(gk−1 ) is projective, and coker(g1 ) has a projective resolution stopping at ker(gk−1 ). That is, hdR (coker(g1 )) < k, 2

a contradiction.

Remark IV.1.12 The hypothesis depthI(gk ) ≥ 1 for every k ≥ 1 is necessary. Consider for instance R = K[[X, Y ]]/(XY ), where K is any ﬁeld and X, Y are two indeterminates. Let X denote the image of X in R and take the complex X

R −→ R −→ R/(X) −→ 0. The kernel of multiplication by X is (Y ), where Y stands for the image of Y in R, for gX = hXY in K[[X, Y ]] implies g = hY . Hence we have an inﬁnite exact complex Y

X

Y

X

· · · −→ R −→ R −→ R −→ R −→ R/(X) −→ 0, where I(gk ) is either (X) or (Y ), so that the depth assumption fails to be veriﬁed. But (X) and (Y ) are primes, so radI(gk ) ⊆ radI(gk+1 ) fails too. IV.2 The ﬁrst structure theorem In this section we state and discuss the ﬁrst structure theorem. A sketch of its proof is in the next section. Theorem IV.2.1 (First Structure Theorem) fn

Let

fk

f1

F : 0 → Fn → Fn−1 → · · · → Fk → Fk−1 → · · · → F1 → F0 be an exact sequence of oriented free R-modules, and let rk denote rankfk . Then, (a) for all k ≥ 1, there exist unique maps ak : R → Λrk Fk−1 such that 1. an = Λrn fn 2. the following diagram is commutative Λ r k Fk a∗k+1 ↓ R

Λrk fk

−→ =

Λrk Fk−1 ↑ ak R

for each k < n (b) radI(fk ) = radI(ak ) for all k ≥ 2. Remark IV.2.2 (Oriented free modules) By deﬁnition, an oriented free module is a ﬁnite free module F , of rank r say, together with the choice of a generator of Λr F . As we have seen in Remark I.3.32, this amounts to ﬁxing (noncanonical) isomorphisms Λr−k F ∗ ∼ = Λk F , k = 1, . . . , r.

The ﬁrst structure theorem

111

Consider now the situation of Theorem IV.2.1. As F is exact, Theorem IV.1.7 says that rankFk = rk+1 + rk . As Fk is oriented, we have ﬁxed isomorphisms Λrk+1 Fk∗ ∼ = Λrk Fk . If k = n, that is, rn+1 = 0, the isomorphism is R ∼ = Λ r n Fn , and part 1. of (a) makes sense provided one thinks of an as the composite map Λrn fn R ∼ = Λrn Fn → Λrn Fn−1 . When k < n, Λrk+1 Fk∗ ∼ = Λrk Fk makes part 2. of ∗ (a) meaningful: one thinks of the dual map ak+1 : Λrk+1 Fk ∗ → R as the map Λrk Fk → R. Deﬁnition IV.2.3

The maps ak , k ≥ 1, are called multipliers.

We now present a series of applications of Theorem IV.2.1, the ﬁrst one being another (neater) proof of the fact that a regular local ring is factorial. Theorem IV.2.4 tion domain.

Let R be a regular local ring. Then R is a unique factoriza-

Proof By Proposition II.6.1, it suﬃces to show that hdR (R/(x, y)) ≤ 2 for all x and y in R. By Theorem II.5.3, we know that hdR (R/(x, y)) is ﬁnite. Hence R/(x, y) has a ﬁnite free resolution, say: fn

f2

(x,y)

0 → Fn → Fn−1 → · · · → F2 → R2 → R → R/(x, y) → 0. Now, Theorem IV.2.1 says that the following diagram commutes (x,y)

R2 −→ R ∗ ↑ a1 a2 ↓ R = R, hence the ideal ima∗2 can be generated by two elements, say I(a∗2 ) = (x , y ), and (x, y) = a1 (x , y ). That is, (x, y) is equal to (x , y ) times a non-zero divisor (every element of a regular local ring is a non-zero divisor). If we prove that hdR (x , y ) = 1, it will follow hdR (x, y) = 1, hence hdR (R/(x, y)) ≤ 2 as required. But Theorem IV.2.1 (b) says radI(f2 ) = radI(a2 ), which implies (recall Theorem IV.1.7) depthI(a2 ) = depthI(f2 ) ≥ 2. So depthI(a∗2 ) ≥ 2. It easily follows that x , y is a regular sequence, and hdR (x , y ) = 1 (the Koszul complex 2 is a resolution of R/(x , y ), of length 2). We now deal with some topics which in fact led to the ﬁrst structure theorem. Example IV.2.5 (Hilbert–Burch Theorem) Let a be an ideal which can be generated by n + 1 elements of R and such that hdR R/a = 2. Then we have (by Theorem IV.1.7): f2

f1

0 → Rn → Rn+1 → R → R/a → 0 and we ask whether we can describe a in terms of the maps f2 and f1 .

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Structure theorems for ﬁnite free resolutions

Theorem IV.2.1 says that the following diagram is commutative: f1

Rn+1 −→ R ↑ a1 ↓ a∗2 R = R. Moreover, Theorem IV.2.1 implicitly says that for all k ≥ 1, we have (∗)

I(fk ) = I(ak )I(ak+1 )

(this will be shown during the proof of the ﬁrst structure theorem). As depthI(fk ) ≥ 1 for all k ≥ 1 (Theorem IV.1.7), and depthI(fk ) ≥ 1 implies depthI(ak ) ≥ 1 for all k ≥ 2 (Theorem IV.2.1 (b)), we have from (∗) that depthI(a1 ) ≥ 1, that is, depth(a1 ) ≥ 1 and a1 is a non-zero divisor. Now, a2 = Λn f2 (Theorem IV.2.1 (a)); if a = (y1 , . . . , yn+1 ) and f2 = (cij ), 1 ≤ i ≤ n + 1, 1 ≤ j ≤ n, explicit computations show that yi = ±a1 ∆i , where ∆i is the maximal minor obtained by erasing the i-th row of (cij ). In other words, every ideal of homological dimension 1 looks determinantal. This result goes back to D. Hilbert [48, section 4], in a special case, and Reference [32] seems to contain the ﬁrst proof of the general case, although the same theorem cropped up independently, and with minor variations and motivations, at about the same time. The construction seen in the previous example is related to the “lifting problem” of Grothendieck, which was motivated by the Serre conjectures on the positivity of intersection multiplicity. This lifting problem is known to be not solvable in general (there are counterexamples due to M. Hochster [50]). Statement of the General Lifting Problem: Given a local ring S and a non-zero divisor, x, let R = S/(x). If M is an R-module, does there exist an . such that x is a non-zero divisor on M . and M ./xM .∼ S-module M = M? t t . If M is free of ﬁnite type, say M = R , clearly M = S does the job. Consider next any ﬁnitely generated M with a free resolution G, say: gk

g1

· · · → Gk → Gk−1 → · · · → G1 → G0 . One can construct a sequence g k : ··· → G . g1 . .k → G G k−1 → · · · → G1 → G0

just by choosing matrices for the maps gk and lifting these matrices. Suppose we is a complex. We then are lucky in our choices of the lifted matrices, so that G get an exact sequence of complexes: →G → G →0. 0→G x

Clearly Hi (G) = 0 for positive i (since G is a resolution of M ). Furthermore, = 0 because multiplication by x is onto, x belongs to for every i > 0, Hi (G)

The ﬁrst structure theorem

113

the radical, and Nakayama’s lemma applies. Hence the higher homology vanishes and the long exact homology sequence leaves us with the short exact sequence x → → H0 (G) → 0, H0 (G) 0 → H0 (G)

Since H0 (G) = M , setting implying that x is a non-zero divisor for H0 (G). . M = H0 (G) solves the lifting problem. The question then arises: when are we lucky enough to be able to choose our be a complex? lifted matrices in such a way that G .1 → G .0 , certainly a complex. If hdR (M ) = 1, G looks like 0 → G If hdR (M ) = 2, the construction given in Example IV.2.5 comes in. Take: f2

S n → S n+1 | | 0

→

f2

Rn → Rn+1

f1

→

R.

.i and set .i in f2 ; let yi = a1 ∆ Lift a1 to a1 and consider ∆ f1 = (y1 , . . . , y n+1 ). It automatically follows that f1 ◦ f2 = 0, and we are through. Beyond this, that is, if hdR (M ) > 2, anything can happen. is not always One would like to understand the arithmetic reasons why G a complex. A possible approach is to look for generic pairs (A, B), namely, a ring A and an exact sequence B of ﬁnitely generated free A-modules with the following property: given any ring A and any exact sequence B of free A modules with the same ranks of those of B, there exists a map A → A such that B = B ⊗A A . M. Hochster [51] has shown that the structure theorems of this chapter are enough to get a generic pair for resolutions of length 2. But the structure of resolutions of length 3 is far more complicated (cf., e.g. References [71] and [85]). Remark IV.2.6 Theorem IV.2.1 can sometimes provide conditions for the cokernel of a map to have ﬁnite homological dimension. We illustrate this in the following example. Let K be a ﬁeld and S the polynomial ring K[Xij ], 1 ≤ i ≤ n, 1 ≤ j ≤ m, with m ≥ n. Consider the generic map (Xij )

g : S m −→ S n . As shown in Section III.4, depthI(g) = m − n + 1 = hdR coker(g). Also let Ip+1 (g) denote the ideal generated by the (p + 1)-minors of the matrix (Xij ), and set R = S/Ip+1 (g). The generic map of rank p is the induced map (X ij )

f : Rm → Rn ,

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Structure theorems for ﬁnite free resolutions

where X ij denotes the image of Xij in R. (Notice that p must be less than n.) We are going to prove that coker(f ) cannot have ﬁnite homological dimension. If hdR coker(f ) were ﬁnite, then coker(f ) would have a ﬁnite free resolution (by Quillen-Suslin); indeed a resolution by free R-graded modules, because R inherits from S the structure of a graded ring (Ip+1 (g) is a graded ideal, g is a homogeneous map) and coker(f ) is a graded R-module. Theorem IV.2.1 would thus give the commutative diagram: Λp f

Λp Rm −→ Λp Rn ↓ a∗2 ↑ a1 R = R. In particular, every p-minor occurring in the ﬁrst p columns of (X ij ) would belong to the (proper) principal ideal of R generated by a∗2 (e1 ∧ · · · ∧ ep ), where {e1 , . . . , em } denotes the canonical basis of Rm . As the p-minors of g have degree p in S, while Ip+1 (g) is generated by elements of degree p+1, it would then follow that all p-minors in the ﬁrst p columns of (Xij ) would be contained in a proper principal ideal of S. However these same p-minors of (Xij ) would generate the ideal of p-minors of a generic n×p matrix, that is, an ideal of depth n−p+1 > 1: a contradiction. We end this section with an example showing that Theorem IV.2.1 may not hold for inﬁnite complexes. Example IV.2.7 Let K be a ﬁeld, X and Y indeterminates over K, and R = K[[X, Y ]]/(Y 2 − X 3 ) (cusp at the origin). Write X and Y for the images of X and Y in R. Then we have the following inﬁnite resolution: f3

f4

(X,Y )

f2

· · · → R2 → R2 → R2 → R → R/(X, Y ), 2 2 Y X Y X where every fk , k ≥ 2, has matrix . As det = 0, −X −Y −X −Y rank(fk ) = 1 for all k ≥ 2. If Theorem IV.2.1 held in this case, we should have commutative diagrams f2

with f2 =

Y −X

R2 −→ R2 ↓ a∗3 ↑ a2 R = R

2

X −Y

. As a1 = 1,

(a2 ◦ a∗3 )

(X,Y )

and

a∗2

R2 −→ R ↑ a1 , ↓ a∗2 R = R

X = (X, Y ) and a2 = . Hence Y

1 X = a2 (r) = ra2 (1) = r , 0 Y

Proof of the ﬁrst structure theorem

115

1 . Thus 0 2 Y 1 X = 0 −X −Y

where r stands for the constant a∗3

Y −X X Y would give r = , from which Y = rX, a contradiction. −X Y IV.3 Proof of the ﬁrst structure theorem IV.3.1 Part (a) We ﬁrst show how one deﬁnes an−1 starting from an (assume n ≥ 2). We have seen in Section III.4 that given f : F → G with rankF = m ≥ n = rankG, one can construct a family of depth-sensitive complexes T(q; f ). When q = n we have: T(n; f ) :0 → Dm−n G∗ ⊗ Λm F

ψm−n+1

→

Dm−n−1 G∗ ⊗ Λm−1 F →

ψk+1

· · · → Dk G∗ ⊗ Λn+k F → Dk−1 G∗ ⊗ Λn+k−1 F → ψ2 ψ1 · · · → G∗ ⊗ Λn+1 F → Λn F → Λn G ∼ = R,

where ψ1 = Λn f and, for every k > 0, ψk+1 is the composite map Dk G∗ ⊗ Λn+k F −→ Dk−1 G∗ ⊗ G∗ ⊗ Λn+k F −→ Dk−1 G∗ ⊗ Λn+k−1 F ∆⊗1

1⊗α

(α stands for the action of ΛG∗ on ΛF induced by f ∗ : G∗ → F ∗ ). Let us suppose that f coincides with the dual of the left-most non-zero map of our complex F, ∗ → Fn∗ . Then T(n; f ) looks like that is, f is fn∗ : Fn−1 ψ3

ψ2

Λrn f ∗

∗ ∗ ∗ → Fn ⊗ Λrn +1 Fn−1 → Λrn Fn−1 → n Λrn Fn∗ , · · · → D2 Fn ⊗ Λrn +2 Fn−1

and the following truncation of its dual complex Λrn fn

ψ∗

0 → Λrn Fn → Λrn Fn−1 →2 Fn∗ ⊗ Λrn +1 Fn−1 must be exact, because depth(fn ) ≥ n ≥ 2 (Theorem IV.1.7). Notice that ψ2∗ is just multiplication by the trace of fn , that is, the element of Fn∗ ⊗ Fn−1 corresponding to fn via the canonical isomorphism HomR (Fn , Fn−1 ) ∼ = Fn∗ ⊗ Fn−1 . Now we make a general observation. Given f

F → G → L → 0 exact, we have F ⊗ Λt−1 G → Λt G → Λt L → 0 exact,

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Structure theorems for ﬁnite free resolutions

where the left-most arrow is the composite map f ⊗1

m

F ⊗ Λt−1 G → G ⊗ Λt−1 G → Λt G. It follows that if the composite F → G → H is zero, then also the composite F ⊗ Λt−1 G

m◦(f ⊗1)

→

Λt G → Λt H

is zero, since the right-most map factors through Λt L: Λt G Λt L → Λt H. Thanks to the previous observation, it is immediate to see that if βn is the composite map f ∗ ⊗1

n ∗ ∗ ∗ ∗ Λrn−1 Fn−1 → Fn−1 ⊗ Λrn−1 −1 Fn−1 → Fn∗ ⊗ Λrn−1 −1 Fn−1 ,

∆

then the following composite ∗ Λrn−1 Fn−2

∗ Λrn−1 fn−1

→

βn

∗ ∗ Λrn−1 Fn−1 → Fn∗ ⊗ Λrn−1 −1 Fn−1

is zero. Some calculations then show that the following diagram is commutative: ∗ Λrn−1 fn−1

∗ Λrn−1 Fn−2

0→

R∼ =

→

→

Λrn Fn−1

→2

Λrn fn

→

Λ r n Fn

βn

∗ Λrn−1 Fn−1 i1

ψ∗

∗ Fn∗ ⊗ Λrn−1 −1 Fn−1 1 ⊗ i2

Fn∗ ⊗ Λrn +1 Fn−1 ,

where i1 and i2 are the appropriate isomorphisms given by the orientations (recall Remark IV.2.2). It follows that there exists a unique map ∗ → Λ r n Fn ∼ Λrn−1 Fn−2 = R,

whose dual (clearly verifying property 2) we deﬁne to be an−1 . Suppose now that we have ak+1 and ak ; we want to deﬁne ak−1 (assume k ≥ 2). As before, we see that the composite map ∗ Λrk−1 Fk−2

∗ Λrk−1 fk−1

−→

βk

∗ ∗ Λrk−1 Fk−1 −→ Fk∗ ⊗ Λrk−1 −1 Fk−1

is zero. Consider the sequence a

γ

k Λrk Fk−1 → Fk∗ ⊗ Λrk +1 Fk−1 , 0→R→

(IV.3.1)

where γ is multiplication by the trace of fk , that is, the element of Fk∗ ⊗ Fk−1 corresponding to fk via the canonical isomorphism HomR (Fk , Fk−1 ) ∼ = Fk∗ ⊗ Fk−1 .

Proof of the ﬁrst structure theorem

117

Again we have a commutative diagram ∗ Λrk−1 fk−1

∗ Λrk−1 Fk−2

0→

→ a

k →

R

∗ Λrk−1 Fk−1 Λrk Fk−1

βk

→ γ

→

∗ Fk∗ ⊗ Λrk−1 −1 Fk−1 Fk∗ ⊗ Λrk +1 Fk−1 ,

so that it suﬃces to show that the bottom row is exact, to be able to conclude as we did in the case of an−1 . In order to prove that (IV.3.1) is a complex, we ﬁrst show that the composite map Λrk fk

γ

Λrk Fk → Λrk Fk−1 → Fk∗ ⊗ Λrk +1 Fk−1

(IV.3.2) Fk∗

basis of dual to is zero. Let {x1 , . . . , xt } be a basis of Fk and {ξ1 , . . . , ξt } the {x1 , . . . , xt }. Then the trace element of fk can be written as i ξi ⊗ fk (xi ), and for every y1 ∧ · · · ∧ yrk ∈ Λrk Fk it turns out that (γ ◦ Λrk fk )(y1 ∧ · · · ∧ yrk ) = γ(fk (y1 ) ∧ · · · ∧ fk (yrk )) = ξi ⊗ fk (xi ) ∧ fk (y1 ) ∧ · · · ∧ fk (yrk ) i

=

ξi ⊗ (Λrk +1 fk )(xi ∧ y1 ∧ · · · ∧ yrk ),

i rk +1

fk = 0. which vanishes because Λ Next we prove that I(fk ) = I(ak )I(ak+1 ) (this is the equality announced and used in Example IV.2.5). As the inclusion ⊇ is straightforward, we just prove ⊆. The three formulas ±(j1 , . . . , jrk | i1 , . . . , irk )wj1 ∧ · · · ∧ wjrk (Λrk fk )(xi1 ∧ · · · ∧ xirk ) = (where {w1 , . . . , ws } stands for a basis of Fk−1 and the coeﬃcients occurring in the sum are k × k minors of fk ), ci1 ···irk xi1 ∧ · · · ∧ xirk ak+1 (1) = and ak (1) =

dj1 ···jrk wj1 ∧ · · · ∧ wjrk ,

inserted into the diagram (commutative by hypothesis) Λrk fk

Λrk Fk −→ Λrk Fk−1 a∗k+1 ↓ ↑ ak R = R,

(IV.3.3)

yield the following diagram xi1 ∧ · · · ∧ xirk ↓a∗k+1 ci1 ···irk and we are done.

Λrk fk

−→ a

k →

±(j1 , . . . , jrk | i1 , . . . , irk )wj1 ∧ · · · ∧ wjrk dj1 ···jrk wj1 ∧ · · · ∧ wjrk , ci1 ···irk

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Structure theorems for ﬁnite free resolutions

Since I(ak+1 ) ⊇ I(fk ) implies depthI(ak+1 ) ≥ 2, I(ak+1 ) contains a non-zero divisor. Take the zero composite map (IV.3.2), where Λrk fk factors as indicated by (IV.3.3). It follows that 0 = im(γ ◦ Λrk fk ) = I(ak+1 )im(γ ◦ ak ), and since I(ak+1 ) contains a non-zero divisor and im(γ ◦ ak ) is included in a free module, one necessarily has im(γ ◦ ak ) = 0, that is, the sequence (IV.3.1) is a complex. It remains to show that (IV.3.1) is exact. We resort to Theorem IV.1.10 and localize at primes of depth at most 1. Since k ≥ 2 and depthI(fk ) ≥ 2, I(fk ) blows up after localization (i.e., it becomes the whole ring) and we may assume R = I(fk ) = I(ak ) = I(ak+1 )

(IV.3.4)

(recall that I(fk ) = I(ak )I(ak+1 )). It is now obvious that ak is a monomorphism. To show exactness at Λrk Fk−1 , it suﬃces to prove that rank(ak ) + rank(γ) = rankΛrk Fk−1

(IV.3.5)

(by Theorem IV.1.7; the depth condition is satisﬁed because of (IV.3.4)). (IV.3.5) is equivalent to rank(γ) = rankΛrk Fk−1 − 1. To verify this latter equality, it is enough to show that (IV.3.2) is exact and use Theorem IV.1.7, since rank(Λrk fk ) = 1. But the exactness of (IV.3.2) is now obtained by noting that I(fk ) = R implies coker(fk ) projective (Proposition IV.1.5), so that Fk = ker(fk ) ⊕ im(fk ), Fk−1 = im(fk ) ⊕ coker(fk ) and 0

fk = idim(fk ) ⊕ {ker(fk ) → coker(fk )}. For the well-known property Λ(A⊕B) = ΛA⊗ΛB indicates that (IV.3.2) is made up of strands involving ker(fk ), im(fk ) and coker(fk ), strands whose exactness is easy to ascertain. This completes the proof of Part (a) of Theorem IV.2.1. IV.3.2 Part (b) Since the equality I(fk ) = I(ak )I(ak+1 ) proved above implies that I(fk ) ⊆ I(ak ), it suﬃces to show that I(ak ) ⊆ radI(fk ), for all k ≥ 2. Suppose the contrary, namely, that I(ak ) radI(fk ). Then there exists a minimal prime, p, associated to I(fk ) such that p I(ak ). By localizing at p, we may thus assume that R is local with maximal ideal m, that I(fk ) is primary to m and that I(ak ) = R. In particular, I(fk ) = I(ak+1 ) (again by the fact that I(fk ) = I(ak )I(ak+1 )). dj1 ···jrk wj1 ∧ · · · ∧ wjrk as in the proof of Part (a), If we write ak (1) = then (dj1 ···jrk ) = I(ak ) = R. Without loss of generality, we may suppose that

The second structure theorem

119

d1···rk = 1 and let F denote the free submodule of Fk−1 generated by w1 , . . . , wrk . We take the composite map fk

π

Fk → Fk−1 → F, where π stands for the projection, and look at the complex fk+1

fn

π◦fk

0 → Fn → Fn−1 → · · · → Fk+1 → Fk → F.

(IV.3.6)

By construction, rank(π ◦ fk ) = rankF = rk = rank(fk ) and I(π ◦ fk ) = I(fk ) = I(ak+1 ), so that using Theorem IV.1.7 we obtain that (IV.3.6) is exact. The exactness of (IV.3.6) shows that coker(π ◦ fk ), which is not equal to 0, has ﬁnite homological dimension. Since I(fk ) = R (so that neither coker(fk ) nor coker(π ◦ fk ) is projective), hdR coker(π ◦ fk ) = hdR coker(f1 ) − k + 1; hence k ≥ 2 implies hdR coker(π ◦ fk ) hdR coker(f1 ). As coker(π ◦ fk ) has ﬁnite homological dimension, Theorem II.4.12 applies and hdR coker(π ◦ fk ) codimR. On the other hand, since rank(π ◦ fk ) = rankF , the discussion after Theorem III.4.2 holds for π ◦ fk and I(π ◦ fk ) ⊆ ann(coker(π ◦ fk )). As I(π◦fk ) = I(fk ), this means that ann(coker(π◦fk )) contains an ideal primary to m, codim(coker(π ◦ fk )) = 0 and (again by Theorem II.4.12) hdR coker(π ◦ fk ) = codimR, a contradiction. This ﬁnishes the proof of Theorem IV.2.1. IV.4 The second structure theorem In Example IV.2.5 we considered an ideal a generated by n + 1 elements of R and such that hdR R/a = 2. We took the resolution f2

f1

0 → Rn → Rn+1 → R → R/a → 0 and were able to use the multipliers to describe f1 in terms of f2 (which had positive implications for the lifting problem). Suppose now that R is a local ring, that the ideal a is generated by three elements, and hdR R/a = 3. Then R/a has a resolution f3

f2

f1

0 → Rr → Rr+2 → R3 → R,

(IV.4.1)

where rank(f1 ) = 1, rank(f2 ) = 2 and r denotes rank(f3 ). It turns out that the multipliers are not enough to describe all of (IV.4.1) in terms of f3 . However

120

Structure theorems for ﬁnite free resolutions

there is something we can do in this case also, to get further information about these maps. Let C = (cij ) be the (r + 2) × r matrix of f3 . We will describe below an (r + 2) × 3 matrix, D = (dis ), whose entries, along with those of C, we use to describe the matrices of f2 and f1 . More precisely, given the (r + 2) × (r + 3) matrix (C|D) = (cij |dis ), the description of matrices for f2 and f1 , up to signs, and multiplied by non-zero divisors, goes as follows. Description of the 3 × (r + 2) matrix for f2 : The (s, t)-coordinate is the (r + 1) × (r + 1) minor of (C|D) obtained by erasing the row indexed by t and the columns indexed by the two elements of the set of indices{r + 1, r + 2, r + 3} which are not equal to r + s. Description of the 1×3 matrix for f1 : The (1, s)-coordinate is the (r + 2)× (r + 2) minor of (C|D) obtained by erasing the column indexed by r + s. The question is: How does one get the matrix D? We have seen in Section III.4 that given f : F → G with rankF = m ≥ n = rankG, one can construct a family of depth-sensitive complexes T(q; f ). Set q = 1 and f equal to the map f3∗ : (Rr+2 )∗ → (Rr )∗ . Then T(1; f3∗ ) looks like: ψ3

ψ2

f∗

3 T(1; f3∗ ) : 0 → Rr ⊗ Λr+2 (Rr+2 )∗ → Λr+1 (Rr+2 )∗ → (Rr+2 )∗ → (Rr )∗ ,

where ψ3 is the action of Rr on Λr+2 (Rr+2 )∗ induced by the map f3 , and ψ2 is given by the action of Λr (Rr )∗ on Λr+1 (Rr+2 )∗ (again induced by the map f3 ). By depth-sensitivity, the dual of T(1; f3∗ ) then provides the following exact sequence: f3

0 → Rr → Rr+2 → Λr+1 (Rr+2 ) → (Rr )∗ ⊗ Λr+2 (Rr+2 ),

(IV.4.2)

and we have the commutative diagram: 0 → Rr → Rr+2 → Λr+1 (Rr+2 ) → (Rr )∗ ⊗ Λr+2 (Rr+2 ) f∗

2 (R3 )∗ →

(Rr+2 )∗

f∗

3 →

(Rr )∗ .

Because of the exactness of (IV.4.2), it is possible to ﬁnd a (not necessarily unique) vertical map which yields the following commutative diagram: 0 → Rr → Rr+2 → Λr+1 (Rr+2 ) → (Rr )∗ ⊗ Λr+2 (Rr+2 ) ↑ f∗

2 (R3 )∗ →

(Rr+2 )∗

f∗

3 →

(Rr )∗ .

This vertical map is the map whose matrix we have called D. It is a relatively straightforward calculation to show that the descriptions of the matrices of f2 and f1 in terms of this D are valid. We take our cue from the foregoing discussion, and show how maps, similar to the map D above, may be constructed in general.

The second structure theorem

121

Let R be a commutative ring and fn

fk

f1

F : 0 → Fn → Fn−1 → · · · → Fk → Fk−1 → · · · → F1 → F0 , n ≥ 3, be an exact complex, with each Fk an oriented free R-module and rk = rank(fk ). The multipliers ak : R → Λrk Fk−1 of Theorem IV.2.1 give information about the rk × rk minors. But in (IV.4.1) above, we were looking for information about the submaximal minors: f3 had rank 2. Take therefore the composite map 1⊗a

m

n rn rn +1 a. Fn−1 , n : Fn−1 = Fn−1 ⊗ R → Fn−1 ⊗ Λ Fn−1 → Λ

and consider the following complex (where γ is multiplication by the trace of fn ): fn

a

γ

n 0 → Fn → Fn−1 → Λrn +1 Fn−1 → Fn∗ ⊗ Λrn +2 Fn−1 ,

which is exact for n ≥ 3 (n = depthI(fn )). Notice that a. n is precisely the map Rr+2 → Λr+1 (Rr+2 ) of (IV.4.2). Again make identiﬁcations and obtain fn

0 → Fn →

a

n −→

Fn−1 ∗ Λrn−1 −1 Fn−2

∗ Λrn−1 −1 fn−1

−→

Λrn +1 Fn−1

γ

→

Fn∗ ⊗ Λrn +2 Fn−1

∗ ∗ Λrn−1 −1 Fn−1 → Fn∗ ⊗ Λrn−1 −2 Fn−1 ,

∗ = 0 of Subsecwhere the bottom row is a complex (as in βn ◦ Λrn −1 fn−1 ∗ → Fn−1 whose tion IV.3.1). It follows that there exists a map Λrn−1 −1 Fn−2 dual we call bn−1 (b∗n−1 is like the vertical map (R3 )∗ → Rr+2 found above: just note that (R3 )∗ is isomorphic to Λ2 (R3 )). ∗ → Λrn−1 −1 Fn−2 to further maps bk : Fk∗ → How can we go from bn−1 : Fn−1 rk −1 Fk−1 that satisfy similar conditions? Λ Let us see these conditions ﬁrst. Suppose that we mimic case n − 1 : we take the composite map

a k+1 : Fk = Fk ⊗ R

1⊗ak+1

→

m

Fk ⊗ Λrk+1 Fk → Λrk+1 +1 Fk ,

and consider the following (where γ is multiplication by the trace of fk+1 ): fk+1

a k+1

γ

∗ Fk+1 → Fk → Λrk+1 +1 Fk → Fk+1 ⊗ Λrk+1 +2 Fk .

(IV.4.3)

If, and this is a big if, (IV.4.3) were exact, we could make the usual identiﬁcations and get b∗k as the vertical arrow induced by the commutative box of the following diagram: fk+1

Fk+1 →

Fk ↑

a k+1

γ

∗ −→ Λrk+1 +1 Fk → Fk+1 ⊗ Λrk+1 +2 Fk

Λrk −1 f ∗

∗ ∗ Λrk −1 Fk−1 −→ k Λrk −1 Fk∗ → Fk+1 ⊗ Λrk −2 Fk∗ .

122

Structure theorems for ﬁnite free resolutions

Hence the conditions would be: the diagrams ∗ a k+1

Λrk+1 +1 Fk∗ −→ Λrk −1 Fk

Fk∗ ↓ bk

Λrk −1 fk

−→ Λrk −1 Fk−1

are commutative (assume k ≥ 2). We thus want to show the exactness of (IV.4.3). Take: fn

fk+1

a k+1

γ

∗ ⊗ Λrk+1 +2 Fk . 0 → Fn → · · · → Fk −→ Λrk+1 +1 Fk → Fk+1

(IV.4.4)

If we prove that (IV.4.4) is exact, it will follow that (IV.4.3) is exact too (here is a case in which it is easier to deal with a longer sequence). In order to show the exactness of (IV.4.4), we can ﬁrst prove that it is a complex and then apply Theorem IV.1.7. That γ ◦ a k+1 = 0 and a k+1 ◦ fk+1 = 0 follows in the same way that im(γ ◦ ak+1 ) = 0 was proved in Subsection IV.3.1. The rank conditions can be checked by localizing at S = {xi }, where x is a non-zero divisor in I(fk+1 ), and proceeding as at the end of Subsection IV.3.1. As for the depth conditions, since k ≥ 2, ak+1 ) ≥ 2 we know from the properties depthI(ft ) is all right for all t ≥ k; depthI( of ak+1 ; depthI(γ) ≥ 1 follows from Lemma IV.4.1 below. Lemma IV.4.1

radI(γ) ⊇ radI(fk+1 ).

Proof Assume for a contradiction that there exists x ∈ / radI(γ) such that xt ∈ I(fk+1 ) for some t. Localize at S = {xi }. As S ∩ I(fk+1 ) = ∅ = S ∩ I(γ), I(fk+1 ) blows up and I(γ) does not. Hence a contradiction will follow, if one proves that I(fk+1 ) = R implies I(γ) = R. By Proposition IV.1.5, I(fk+1 ) = R implies that coker(fk+1 ) is projective; thus we have Fk+1 = ker(fk+1 ) ⊕ im(fk+1 ), Fk = im(fk+1 ) ⊕ coker(fk+1 ) and 0 fk+1 = idim(fk+1 ) ⊕ ker(fk+1 ) → coker(fk+1 ) . From the well known property: Λ(A ⊕ B) = ΛA ⊗ ΛB, it follows that the right-most map of (IV.4.4) is made up of strands involving ker(fk ), im(fk ) and coker(fk ), strands whose careful study shows coker(γ ∗ ) = (coker(fk+1 ))∗ ⊗ Λrk+1 (im(fk+1 ))∗ . Hence coker(γ ∗ ) is projective and Proposition IV.1.5 yields 2 I(γ ∗ ) = R. This whole preceding discussion amounts to a sketch of the proof of what is called the second structure theorem. Theorem IV.4.2 (Second Structure Theorem) fn

fk

Let f1

F : 0 → Fn → Fn−1 → · · · → Fk → Fk−1 → · · · → F1 → F0 ,

The second structure theorem

123

n ≥ 3, be an exact sequence of oriented free R-modules. Let rk denote rankfk , and let a .k : Fk−1 → Λrk +1 Fk−1 be the composite 1⊗a

m

Fk−1 = Fk−1 ⊗ R →k Fk−1 ⊗ Λrk Fk−1 → Λrk +1 Fk−1 , where ak is the multiplier of Theorem IV.2.1. Then for every k = 2, . . . , n − 1, there exists a map bk : Fk∗ → Λrk −1 Fk−1 (not unique) such that the following diagram commutes: ∗ a k+1

Λrk+1 +1 Fk∗ −→ Λ

rk −1

Fk

Λrk −1 fk

−→ Λ

Fk∗ ↓ bk rk −1

(IV.4.5)

Fk−1 .

Remark IV.4.3 A map bn : Fn∗ → Λrn −1 Fn−1 can be trivially added to the statement of the second structure theorem. Simply deﬁne it as the composite Λrn −1 fn F∗ ∼ = Λrn −1 Fn −→ Λrn −1 Fn−1 . n

Remark IV.4.4 Although Theorems IV.2.1 and IV.4.2 give such good information about the maximal and submaximal minors of every fk , it is very hard to guarantee the existence of a sequence F with prescribed maps ak and bk . We have just said in the previous remark that it is hard to recover F from the maps ak and bk . As a matter of fact, given a concrete resolution F, it may not be easy to ﬁnd its maps bk either. This motivates the following. Notice that the commutativity of the diagram in part (a) 2. in the statement of Theorem IV.2.1 amounts to ak (a∗k+1 (z(ε))) = Λrk fk (z)

(IV.4.6)

for all z ∈ Λrk Fk . Here ε ∈ Λrk+1 +rk Fk∗ denotes the orientation of Fk , that is, the chosen isomorphism in HomR (Λrk+1 +rk Fk , R) ∼ = Λrk+1 +rk Fk∗ (cf. Remark I.3.32), rk+1 ∗ Fk is the contraction of z on ε. and z(ε) ∈ Λ Notice that the commutativity of the diagram (IV.4.5) in the statement of Theorem IV.4.2 amounts to b∗k (x)(ak+1 (1)(ε)) = Λrk −1 fk∗ (x) ∗ . for all x ∈ Λrk −1 Fk−1 ∗ → Fk such that Notice also that it is possible to deﬁne maps c∗k : Λrk −1 Fk−1 ∗ fk (ck (x)) = x(ak (1)), because

fk−1 (x(ak (1))) = 0

∗ for all x ∈ Λrk −1 Fk−1 ,

(IV.4.7)

124

Structure theorems for ﬁnite free resolutions

so that one wonders whether the relatively easier maps ck can replace the maps bk . [Proof of (IV.4.7). It suﬃces to show y · fk−1 (x(ak (1))) = 0 for some non-zero divisor y ∈ R. Since we have already seen in the proof of Theorem IV.2.1, Part (a), that I(a∗k+1 ) contains a non-zero divisor, we may choose y to be a∗k+1 (z(ε)) for some suitable z ∈ Λrk Fk . We then have y · fk−1 (x(ak (1))) = fk−1 (x(ak (y))) = fk−1 (x(ak (a∗k+1 (z(ε))))), that is (recall (IV.4.6)), fk−1 (x(Λrk fk (z))), which is zero, since x(Λrk fk (z)) ∈ Fk−1 belongs in fact to im(fk ).] Proposition IV.4.5 For the maps bk of the second structure theorem, the following are equivalent: ∗ (1) b∗k (x)(ak+1 (1)(ε)) = Λrk −1 fk∗ (x) for all x ∈ Λrk −1 Fk−1 ∗ (2) fk (b∗k (x)) = x(ak (1)) for all x ∈ Λrk −1 Fk−1 .

Proposition IV.4.5 says that the maps ck are indeed another way of deﬁning the maps bk . It also coincides with case = 1 of the following stronger result. Theorem IV.4.6

Let fn

fk

f1

F : 0 → Fn → Fn−1 → · · · → Fk → Fk−1 → · · · → F1 → F0 be an exact sequence, let the maps ak be the multipliers of F, and let bk : Λ Fk∗ → Λrk − Fk−1 be a map. Then bk satisﬁes rk − ∗ fk (x) b∗ k (x)(ak+1 (1)(ε)) = Λ

∗ for all x ∈ Λrk − Fk−1

(IV.4.8)

if and only if it satisﬁes Λ fk (b∗ k (x)) = x(ak (1))

∗ for all x ∈ Λrk − Fk−1 .

(IV.4.9)

We do not give the proof, but refer the reader to Reference [20]. We point out that at one point it was hoped maps bk might help in the study of lower order minors, so it was encouraging to have description (IV.4.9), which is relatively easier than (IV.4.8) But it was soon discovered that maps bk do not always exist [17]. In a way, all of the foregoing chapter may be looked at as the ﬁrst steps of the “homological baby” coming to grips with the outside world of commutative algebra. When one gets right down to the core, it seems obvious, almost naive, to ask: What makes a complex exact? For vector spaces, the answer is so obvious that it was never even asked; for arbitrary commutative ground rings, this problem had to be confronted. While the structure theorems give a promising start

The second structure theorem

125

to understanding some basic questions, we see that they still leave a great many questions open. Even where the answers to some of our questions are negative, the counterexamples do not yield the insight we need in order to fully understand the phenomena that we want to study. To point to an even more basic, and hence irksome, question, consider the fact that a module, M , is completely determined by its presentation (which, for the sake of retaining our sanity, we will assume to be ﬁnite): F1 → F0 → M → 0. Presumably, then, it should be possible to read oﬀ all the information we might want to have about M from the presentation matrix. For example, what property does the matrix have to have in order that hdR M be ﬁnite? Notice that all the (limited) information we have obtained from the structure theorems comes from the assumption that the module is known to have ﬁnite homological dimension; using that, we have managed to drag information about the matrix at the tail of the resolution to the head (i.e., the presentation matrix). We know that the answer cannot be given solely in terms of the minors of the matrix, though, as the cokernel of the transpose matrix may have characteristics completely diﬀerent from the original. So, to be uncompromisingly candid, our baby steps do not begin to address some of the most elementary yet fundamental problems of commutative algebra. While this may be taken by some to be disheartening, to say the least, we can (and the authors do) take the view that there is a lot of growing yet to do, and perhaps the recipe resides somewhere in the willingness to accept that our accomplishments are as yet meager, but we should not lose sight of the fundamental questions.

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V EXACTNESS CRITERIA AT WORK

Since the time that the structure theorems of the last chapter appeared, there have been any number of applications and reﬁnements of these results. What we have tried to do is to select some area in which these theorems have played an essential role, and in which there are still lurking some fascinating, only partially explored areas of investigation. We have chosen to look at resolutions of Gorenstein ideals of depth three in a local ring, and their powers. (Recall [Deﬁnition II.4.10] that the depth of an ideal, also called its grade, is the length of a maximal R-sequence in I.) A Gorenstein ideal is a perfect ideal, I, of grade g, with the property ∼ that TorR g (R/I, k) = R, where k is the residue ﬁeld of the local ring, R. For example, any complete intersection (i.e. an ideal generated by an R-sequence) is a Gorenstein ideal, as the Koszul complex associated to any minimal set of its generators is a minimal resolution and the last term of that complex is precisely R. We will look at Gorenstein ideals of depth three for a number of reasons. First, in much the same spirit of the Hilbert–Burch Theorem (cf. Example IV.2.5), it is possible to describe such ideals in terms of the minors of a certain kind of matrix, namely an alternating one. We will show that all of these ideals are generated by the so-called Pfaﬃans of order n − 1 of an alternating matrix of odd order, n. Another reason is that for the ﬁrst time in this book, the algebra structure of such a resolution comes into play. It still is not completely clear just how large a role the algebra structures of ﬁnite free resolutions must play in their study, but these examples, as well as the vast amount of work on Poincar´e series, would indicate that such structure should not be ignored. The development in the ﬁrst section, where we deal exclusively with the Gorenstein ideal of grade three follows to a large extent the treatment in Reference [25]. In Section V.2, we build on the previous one and construct resolutions for all powers of the ideals in Section V.1. While this entails a bit of additional technique, the results exhibit a phenomenon which seems to occur with some frequency: a growth of homological dimension up to a critical point, and then a constant value from a certain point on. This class of ideals also provides us with the opportunity to illustrate the use of a counting method from representation theory to provide us with an idea of what the resolutions of such ideals may look like. An elaboration of the material of Section V.2 can be found in Reference [13].

128

Exactness criteria at work

V.1 Pfaﬃan ideals Before we begin with our Gorenstein ideals of depth three, we will dispose easily of those of lower depth, namely depths one and two. If I is Gorenstein of depth one, then it has a resolution of the type 0 → R → R → R/I → 0. Obviously, I is a principal ideal generated by a non-zero divisor. If I is Gorenstein of depth two, it has a resolution of the type 0 → R → R2 → R → R/I → 0. The condition that the last term of the minimal resolution be free of rank one (the condition that we placed on Tor) does not provide much wiggle room for the resolution. But now our Hilbert–Burch theorem tells us that, since the depth of I is two, I is generated by a regular sequence of length two, and our resolution is essentially the Koszul complex associated to those generators. Thus, we see that Gorenstein ideals of depth one or two must be generated by regular sequences. This is why the real investigation of Gorenstein ideals has to begin with depth three. Of course, one might be led to suspect that Gorenstein ideals are all complete intersections; what follows should disabuse the reader of that idea. V.1.1 Pfaﬃans Let R be a noetherian commutative ring, and F a ﬁnitely generated free R-module. A morphism f : F ∗ → F is called alternating if, with respect to some (and therefore every) basis and dual basis of F and F ∗ , the matrix W = (wij ) of f satisﬁes wij + wji = 0 whenever i = j, and wii = 0 for every i. Using the isomorphism HomR (F ∗ , F ) ∼ = (F ∗ )∗ ⊗ F ∼ = F ⊗ F, f corresponds to an element ϕ ∈ F ⊗F . In fact ϕ ∈ Λ2 F , where Λ2 F is embedded into F ⊗ F by means of the diagonal map. Since ϕ is homogeneous of degree 2 in ΛF , there exist divided powers ϕ(0) , ϕ(1) , ϕ(2) , . . . in kthe following sense. Let a ∈ ΛF be homogeneous of even degree, say a = i=1 ai , where each ai is a product of degree 1 elements; then, for every p ≥ 0, set a(p) = ai1 ∧ · · · ∧ aip . 1≤i1 <···
Pfaﬃan ideals

129

The sequence of elements a(0) , a(1) , a(2) , . . . (called a system of divided powers) satisﬁes the following properties: (1) a(0) = 1, a(1) = a, deg(a(p) ) = p deg(a) (p+q) (2) a(p) a(q) = p+q q a (3) (a(p) )(q) =

(pq)! (pq) . q!pq ! a

A fuller description of these divided powers can be found in Reference [25] (p. 482). Lemma V.1.1

For every a ∈ Λt F ∗ , we have: Λt f (a) = (−1)i ϕ(i) (a)(ϕ(t−i) ).

(V.1.1)

i≥0

Proof Use induction on t and the formula in Lemma III.3.2 (cf. Reference [25], pp. 458–460). 2 Corollary V.1.2

If rankF = 2k, then using the isomorphism HomR (Λ2k F ∗ , Λ2k F ) ∼ = Λ2k F ⊗ Λ2k F,

the element Λ2k f corresponds to ϕ(k) ⊗ ϕ(k) . Proof

Set t = 2k in (V.1.1). Then the only non-zero summand is (−1)k ϕ(k) (a) ϕ(k) ,

/ 0 since rankF = 2k. But ϕ(k) (a) = (−1)k ϕ(k) , a , where −, − stands for the action of (ΛF ∗ )∗ on ΛF ∗ and we identify ϕ(k) ∈ Λ2k F with its image under the canonical map Λ2k F → Λ2k (F ∗ )∗ → (Λ2k F ∗ )∗ . / 0 It follows that Λ2k f (a) = ϕ(k) , a ϕ(k) for every generator a of Λ2k F ∗ .

2

Corollary V.1.2 has the following meaning. Choose an orientation of F , that is, an isomorphism Λ2k F ∼ = R. Then Λ2k f corresponds to the determinant of the appropriate matrix of f , and such a determinant is the square of the element of R corresponding to ϕ(k) , called the pfaﬃan of f . We write Pf(f ), and also Pf(W ), if W is a matrix associated with f . If we take any alternating submatrix (of even order) W of W , we may consider Pf(W ). It is the analog of the minor of a determinant. The following lemma expresses a pfaﬃan of f of order 2m in terms of pfaﬃans of order 2m − 2 (rankF is unspeciﬁed). Lemma V.1.3

For every a ∈ Λ2m+1 F ∗ , we have: a(ϕ(m+1) ) = ϕ(m) (a)(ϕ).

130

Exactness criteria at work

Proof We use induction on m and the formula in Lemma III.3.1(cf. Reference [25], pp. 460–461). 2 The above implies, in particular, the explicit formula found in the following remark, and which will be used later in this chapter. Remark V.1.4 Let W = (wij ) be an alternating matrix of order m (m either even or odd). For every ﬁxed i0 , Pf(W ) =

m

(−1)i+i0 +1 γii0 wii0 Pf(W )ii0 ,

i=1

where γii0

  1 0 =  −1

if i < i0 if i = i0 if i > i0 ,

and Pf(W )h1 ,...,hr denotes the pfaﬃan of the alternating submatrix of W formed by deleting rows and columns indexed by h1 , . . . , hr . For every ﬁxed i0 and j0 such that i0 = j0 , m

(−1)i+i0 +1 γii0 wij0 Pf(W )ii0 = 0.

i=1

For every ﬁxed i0 , m

wii0 Ti (W ) = 0,

i=1

where Ti (W ) stands for (−1)i+1 Pf(W )i . Given again any alternating map f : F ∗ → F , for every m ≥ 1 we can consider the pfaﬃan ideal P f2m (f ) generated by all pfaﬃans of f of order 2m. Then an interesting corollary of Lemma V.1.3 is the following. Corollary V.1.5 m ≥ 1,

Let f : F ∗ → F be an alternating map. For each integer P f2m+2 (f ) ⊆ P f2m (f ).

Another corollary relates pfaﬃan ideals and determinantal ideals. Corollary V.1.6 m ≥ 1, we have:

Let f : F ∗ → F be an alternating map. For each integer

1. I2m (f ) ⊆ P f2m (f ) ⊆ radI2m (f ) 2. I2m−1 (f ) ⊆ P f2m (f ) 3. if rankf = 2m − 1, then I2m−1 (f ) is nilpotent; if rankF = 2m − 1, then det f = 0.

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131

Proof Recall formula (V.1.1). For every t, It (f ) is generated by the coordinates of Λt f (a) for various a, while the coordinates of ϕ(i) and ϕ(t−i) involve pfaﬃans of order 2i and 2(t−i), respectively. Hence if t is either 2m or 2m−1, every term on the right-hand side is contained in P f2n (f ) for some n ≥ m. By Corollary V.1.5, P f2n (f ) ⊆ P f2m (f ) and we get the second statement, as well as the ﬁrst inclusion of the ﬁrst statement. To ﬁnish the proof of statement 1, we simply remember that the square of each order 2m pfaﬃan of f is a minor of order 2m. Statements 1 and 2 imply the ﬁrst part of statement 3. As for the second part, again use formula (V.1.1): if rankF = 2m − 1, ϕ(i) = 0 for i ≥ m; but one of i and 2m − 1 − i is larger than or equal to m. 2 V.1.2 Resolution of a certain pfaﬃan ideal Let R be a noetherian commutative ring, and let n ≥ 3 be an odd integer; say, n = 2k + 1. Let F be a free R-module of rank n, and let f : F ∗ → F be an alternating map. Suppose that the pfaﬃan ideal P f2k (f ) has grade 3. We are going to build a sequence g∗

f

g

E : 0 → R → F∗ → F → R which is a resolution of R/P f2k (f ). As usual, let ϕ be the element of Λ2 F corresponding to f , so that, by Lemma V.1.1, f (a) = a(ϕ) for every a ∈ F ∗ . Let e be a generator of Λn F ∗ and deﬁne g : F → R by means of g(b) = ϕ(k) (e)(b). We claim that im(g) = P f2k (f ). For Lemma 3.3.2 implies that ϕ(k) (e)(b) = b(e)(ϕ(k) ), and as b ranges over a basis of F , b(e) ranges over a basis of Λ2k F . Hence im(g) is the ideal generated by the coordinates of ϕ(k) , that is, im(g) = P f2k (f ). We ﬁrst check that E is a complex. Since f ∗ = −f , it is enough to show gf = 0. Now, for a ∈ F ∗ , gf (a) = ϕ(k) (e)(f (a)) = ϕ(k) (e)(a(ϕ)) = (ϕ(k) (e) ∧ a)(ϕ) = −(a ∧ ϕ(k) (e))(ϕ) = −a(ϕ(k) (e)(ϕ)). But by Lemma V.1.3, ϕ(k) (e)(ϕ) = e(ϕ(k+1) ) = 0, since ϕ(k+1) ∈ Λn+1 F = 0. In order to show that E is exact, we use Theorem IV.1.7. We need to check that rankf + rankg = n, depthI(g) ≥ 3 and depthI(f ) ≥ 2 (note that I(g ∗ ) = I(g)).

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By statement 1 of Corollary V.1.6, I1 (g) = P f2k (f ) has the same radical as I2k (f ). By statement 3 of Corollary V.1.6, rankf = 2k, and I2k (f ) has grade 3 so rank g = 1 and I(g) = I1 (g) has grade 3 as well. This completes the proof that E is exact. Remark V.1.7 If, in addition, R is local, with maximal ideal m, and we assume im(f ) ⊆ mF , then P f2k (f ) = im(g) ⊆ mR

and

im(g ∗ ) ⊆ mF ∗ .

That is, E is a minimal free resolution of R/P f2k (f ). We are now ready to state the main theorem of this section. Theorem V.1.8

Let (R, m) be a noetherian local ring.

1. Let n ≥ 3 be an odd integer, and let F be a free R-module of rank n. Let f : F ∗ → F be an alternating map whose image is contained in mF . Suppose P fn−1 (f ) has grade 3. Then P fn−1 (f ) is a Gorenstein ideal, minimally generated by n elements. 2. Every Gorenstein ideal of grade 3 arises as in Part 1. Remark V.1.7 above indicates that we have completely proven Part 1 of the theorem (the property of being Gorenstein comes from having R as the left-most non-zero term in E). The proof of Part 2 is given in the rest of this section. V.1.3 Algebra structures on resolutions Let R be a commutative ring, and let d

d

d

P : · · · →3 P2 →2 P1 →1 P0 = R be a projective resolution of a cyclic R-module R/a, where a = im(d1 ). It is clear that P ⊗R P has the structure of a complex, with diﬀerential d(f ⊗ g) = df ⊗ g + (−1)deg f f ⊗ dg, and that any comparison map P ⊗R P → P covering the natural map R/a ⊗R R/a → R/a makes P into a homotopy-associative, homotopy-commutative diﬀerential graded algebra. (The property of being “homotopy-X” means the property of being X up to homotopy.) We claim that one can make P commutative, as well. Let the symmetric square S2 (P) be the quotient (P ⊗R P)/M, where M is the graded subcomplex of P ⊗R P generated by f ⊗ g − (−1)deg f deg g g ⊗ f | f, g homogeneous elements of P .

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133

Since d(M) ⊆ M, S2 (P) inherits the structure of a complex from P ⊗R P. Furthermore, each component of S2 (P) is a projective R-module, because for each k one has   S2 (P)k ∼ Pi ⊗R Pj  + Tk , = i+j=k, i<j

where

 0 Tk = Λ2 Pk/2  S2 Pk/2

if k is odd if k is of the form 4n + 2 if k is of the form 4n.

Since S2 (P) is isomorphic to P in degrees 0 and 1, the comparison theorem for projective complexes says that there exists a map of complexes ϕ : S2 (P) → P which extends the natural isomorphism R/a ⊗R R/a → R/a, and which is the identity on the subcomplex R ⊗R P ⊆ S2 (P). If we set f · g = ϕ(f ⊗ g), where f , g are in P, and f ⊗ g is the image, in S2 (P), of f ⊗ g, then the conditions the map satisﬁes are equivalent to the statement that · makes P into a (not necessarily associative) commutative, diﬀerential graded algebra, with the diﬀerential d, and with structure map R → P given by R = P0 . The uniqueness up to homotopy of comparison maps implies that the above algebra structure, induced by ϕ, is homotopy-associative. In fact, it is even associative if Pi = 0 for i ≥ 4 (recall that in this section, we are concerned with resolutions of length at most 3). Proposition V.1.9 With notation as above, if Pi = 0 for i ≥ 4, then P is a commutative, associative graded algebra. Proof

There is just one associativity to be checked: (f · g) · h = f · (g · h),

with f, g, h in P1 . Since d3 is a monomorphism, it is enough to show: d3 ((f · g) · h) = d3 (f · (g · h)). But both sides coincide with: d1 (f ) · g · h − f · d1 (g) · h + f · g · d1 (h), 2

for d is a derivation and d1 (f ), d1 (g), d1 (h) are in R.

We now specialize and suppose that R is a noetherian local ring, that a is a Gorenstein ideal of R of grade g, and that dg

d

d

F : 0 → Fg → Fg−1 → · · · →2 F1 →1 R

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Exactness criteria at work

is a minimal free resolution of R/a, equipped with a commutative multiplication F ⊗R F → F which we denote by a dot. Since a is Gorenstein, we can make the identiﬁcation Fg = R, so that, for each k ≤ g, the dot-multiplication map Fk ⊗R Fg−k → Fg = R ∗ induces a map sk : Fk → Fg−k .

Theorem V.1.10

For every k ≤ g, sk is an isomorphism.

Proof We claim that the following diagram is commutative up to sign: dg

F : 0 → Fg → Fg−1 → · · · → F2

↓ s2

↓ sg−1 d∗

1 F∗ : 0 → R → F1∗

d

→2 F1

d

→1 R

↓ s1 d∗ g−1

d∗ g

∗ ∗ → · · · → Fg−2 → Fg−1 → Fg∗ .

For, if a ∈ Fk and b ∈ Fg−k+1 , then a · b ∈ Fg+1 = 0, so 0 = dg+1 (a · b) = dk (a) · b + (−1)k a · dg−k+1 (b), or dk (a) · b = ±a · dg−k+1 (b). If, for α ∈ F∗ and b ∈ F, we write α, b ∈ R for the value of the linear functional α at b, we thus have sk−1 dk (a), b = dk (a) · b = ±a · dg−k+1 (b) = sk (a), dg−k+1 (b) / 0 = d∗g−k+1 (a), b . Hence sk−1 dk = ±d∗g−k+1 sk , as required. Now, since a is Gorenstein, both F and F∗ are minimal free resolutions of R/a and it follows that any map which extends an isomorphism in degree 0 must be an isomorphism. Thus each sk is an isomorphism. 2 V.1.4 Proof of Part 2 of Theorem V.1.8 Let a be a Gorenstein ideal of grade 3, minimally generated by n elements. Let d

d

d

F : 0 → F3 →3 F2 →2 F1 →1 R be a minimal free resolution of R/a, so that F1 ∼ = Rn . In the previous subsection we saw that F has the structure of a commutative, associative, diﬀerential graded algebra. By Theorem V.1.10, the multiplication of this algebra gives a perfect pairing F2 ⊗R F1 → F3 . If we identify F3 with R, such a pairing yields an identiﬁcation F2 = F1∗ . We ﬁrst show that, with respect to this identiﬁcation, d2 is alternating. Let {ei } and {εi } be dual bases for F1 and F2 = F1∗ , respectively,

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135

and write (wij ) for the matrix of d2 relative to these bases. Thus, writing −, − for the pairing between F2 and F1 , wij = εi , d2 (εj ) = εi · d2 (εj ) ∈ R = F3 , where · is the product for the algebra structure on F. We must show that wij = −wji and wii = 0. Applying d3 to the above equation, and using the formula for the diﬀerentiation of a product, we get d3 (wij ) = d2 (εi ) · d2 (εj ) + εi · d1 (d2 (εj )) = d2 (εi ) · d2 (εj ). Since · satisﬁes the graded commutative law and d2 (εi ) and d2 (εj ) have degree 1, we have d3 (wij ) = −d3 (wji ), d3 (wii ) = 0. Since d3 is a monomorphism, this shows that d2 is alternating. We next show that d2 satisﬁes the other conditions of Part 1 of Theorem V.1.8. By Theorem IV.1.7, rank d2 = n − 1 and In−1 (d2 ) has grade ≥ 2. By statement 3 of Corollary V.1.6, n − 1 is even, so n is odd. Since F is a minimal resolution, im(d2 ) ⊆ mF1 . To see that grade P fn−1 (d2 ) ≥ 3, we ﬁrst note that by statement 1 of Corollary V.1.6 , rad P fn−1 (d2 ) = rad In−1 (d2 ), so it is enough to show that grade In−1 (d2 ) ≥ 3. Since In−1 (d2 ) = I(d2 ), and I(d1 ) = I1 (d1 ) = a has grade 3 by assumption, we are done by Part (a) of Proposition IV.1.11. Having proved that d2 satisﬁes all the conditions of the map f in Part 1 of Theorem V.1.8, proceeding as in Subsection V.2.2, we can construct an exact sequence g∗

g

E : 0 → R → F1∗ →2 F1 → R. d

But under our identiﬁcations F3 = R and F2 = F1∗ , we have d3 = d∗1 . Since both d∗1 and g ∗ are kernels of d2 , there is a unit u ∈ R such that the following diagram is commutative: g∗

R −→ F1∗ ↓u . ∗ F R −→ 1 ∗ d1

Hence ug = d1 and P fn−1 (d2 ) = im(g) = im(ug) = imd1 = a, as required. This completes the proof of Theorem V.1.8. One should remark that, given a Gorenstein ideal, a, of grade 3 as above, minimally generated by n = 2k + 1 elements, we can rewrite the resolution g∗

g

E : 0 → R → F1∗ →2 F1 → R d

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Exactness criteria at work

as follows: g

f

g

E : 0 → Λn F1 → Λn−1 F1 → F1 → R, by means of the isomorphisms Λk F → Λn−k F ∗ such that a → a(e) [e = chosen orientation of Λn F ∗ ]. Then the pairing on E , which gives the duality expressed by Theorem V.1.10, is precisely the multiplication F1 ⊗R Λn−1 F1 → Λn F1 ,

R ⊗R Λn F1 → Λn F1

in the exterior algebra. Finally, we observe that there is a close analogy between E and the Koszul complex (which E must be, if n = 3). Namely, we note that the following diagram is commutative, where the ﬁrst row is a truncation of the Koszul complex associated with g : F1 → R, where γ : ΛF1 → ΛF1 is the map a → a ∧ ϕ(k−1) , and ϕ is the element of Λ2 F1 corresponding to f . Λ3 F1 → Λ2 F1 ↓γ

g

→ F1 → R

↓γ g

f

g

Λn F1 → Λn−1 F1 → F1 → R. V.2 Powers of pfaﬃan ideals Let R be a noetherian ring. Let n be a positive integer such that n = 2k + 1, k ≥ 1, let Zij , 1 ≤ i < j ≤ n, be 12 n(n − 1) independent indeterminates, and call X the n × n matrix (Xij ) such that   Zij if i < j Xij = −Zij if i > j  0 if i = j (X is called a generic alternating matrix). Finally, let S stand for the polynomial ring R[X] (=R[Zij ]). It is a fact that the ideal P f2k (X) has grade 3 (cf. e.g. Reference [55], corollary 2.5); hence P f2k (X) has a ﬁnite free S-resolution as described in Subsection V.1.2 above. In this section we describe a ﬁnite free S-resolution for each of the ideals [P f2k (X)]m , m ≥ 2. Interest in such a class of ideals was originally raised by the desire to compare its homological behavior with that of the family studied in Reference [24]. Let R[Uij , Tk ], 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ k ≤ m, be a polynomial ring in mn + m indeterminates, with m ≥ n. Let ϕ : Rm → Rn be the map with matrix (Uij ), and let a : R → Rm be the map with matrix (T1 , . . . , Tm ). Let J(ϕ, a) denote the ideal generated by the n × n minors of (Uij ) and by the entries of the product (T1 , . . . , Tm )(Uij ). It is proven in Reference [24] that J(ϕ, a) has grade m and is perfect, i.e., its homological dimension equals its grade. We will see below that, if n = 3 and m ≥ 3, [P f2k (X)]m is never perfect,

Powers of pfaﬃan ideals

137

for it always has grade 3 while its homological dimension depends on m (in a very nice way). V.2.1 Intrinsic description of the matrix X Let F0 be a free R-module of rank n = 2k+1, and let A be the symmetric algebra: A = S(Λ2 F0 ). Set F = A ⊗R F0 . We deﬁne a degree one A-map, f : F → F ∗ , in the following way. For every r ≥ 0, the map f , restricted to the component Ar ⊗R F0 , is the composite Ar ⊗R F0

−→

1⊗1⊗CF0

Ar ⊗R F0 ⊗R F0 ⊗R F0∗

−→ Ar ⊗R Λ2 F0 ⊗R F0∗ = Ar ⊗R A1 ⊗R F0∗ −→ Ar+1 ⊗R F0∗ , m⊗1

1⊗m⊗1

where m denotes multiplication in the appropriate algebras and CF0 stands for the element of F0 ⊗R F ∗ ∼ = HomR (F0 , F0 ) 0

corresponding to the identity on F0 . dual bases {e1 , . . . , en } and {ε1 , . . . , εn } for F0 and F0∗ , CF0 equals Choosing n j=1 ej ⊗ εj and the composite map: F0 −→ F0 ⊗R F0 ⊗R F0∗ −→ Λ2 F0 ⊗R F0∗ ,

m⊗1

1⊗CF0

sends each ei to j (ei ∧ ej ) ⊗ εj . Hence (ei ∧ ej ) is the matrix associated to f with respect to the bases of F and F ∗ induced by those of F0 and F0∗ (we denote the induced bases with the same symbols used for the original ones). By means of the identiﬁcation ei ∧ ej → Xij , the matrix (ei ∧ ej ) coincides with the generic alternating matrix X introduced at the beginning of this section, and A = S(Λ2 F0 ) is isomorphic to S = R[X]. From now on, we use S to indicate both R[X] and S(Λ2 F0 ), dropping the symbol A. The morphism, f , is called the generic alternating map. Asseen in Section 1, f corresponds to an element α ∈ Λ2 F ∗ , namely, we have α = i<j Xij εi ∧ εj . Moreover, the grade 3 ideal P f2k (X), which we denote by a, for short, has the by now familiar resolution (a resolution of S/a, i.e.): g∗

f

g

E : 0 → S → F → F ∗ → S. The morphism f is as above. As for g, choose an orientation e ∈ Λn F , that is, an identiﬁcation between Λ2k F and F ∗ . Then g can be identiﬁed with ω2k : Λ2k F → S,

a → β(α(k) ⊗ a),

where β stands for the natural pairing Λ2k F ∗ ⊗S Λ2k F → S given by the ΛF ∗ structure of ΛF . Concretely, g can be viewed as sending each εi to Ti (X) = (−1)i+1 Pf(X)i . Finally, g ∗ (1) = α(k) .

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Exactness criteria at work

V.2.2 Hooks again We already met in Subsection III.1.2 some “hook” GL(F )-representations, Lλ F , where λ stands for (λ1 , 1, . . . , 1) = (λ1 , 1t ), % &# $ t

or, more pictorially, λ1 boxes

# | | λ= t boxes | |

$% ···

&

,

.. .

with λ1 and t nonnegative integers such that λ1 ≥ 1 if t > 0. As an S-module, Lλ F is deﬁned as the cokernel of the composite δ : Λλ1 +1 F ⊗ St−1 F −→ Λλ1 F ⊗ F ⊗ St−1 F −→ Λλ1 F ⊗ St F, 1⊗m

∆⊗1

where ∆ (resp., m) is diagonalization (resp., multiplication) in the algebra ΛF (resp., SF ). Consider the basis of Λλ1 F ⊗St F induced by the basis {e1 , . . . , en } of F . Given an element e1 ∧ · · · ∧ eλ1 ⊗ ej1 · · · ejt of such a basis, denote its image in L(λ1 ,1t ) F by the “tableau” i1 j1 .. .

···

iλ1 .

jt As proved in Subsection III.1.2, the following statements hold: • A basis of L(λ1 ,1t ) F is given by all tableaux such that the indices in the ﬁrst

row are strictly increasing and those in the ﬁrst column are weakly increasing (“standard tableaux”). • A tableau which is not standard is equal to a Z-linear combination of standard tableaux (“straightening law”).

Powers of pfaﬃan ideals

139

Explicitly, the key step of the straightening law is: i1 j1 j2 .. .

···

iλ1 =

λ1

(−1)h−1

h=1

jt

j1 ih j2 .. .

i1

···

ih

···

iλ1

jt

(j1 < i1 < · · · < iλ1 ) where ih means ih omitted. Finally, suppose that F = F1 ⊕ F2 , with F1 generated by e1 , . . . , eh and F2 generated by eh+1 , . . . , en for some ﬁxed h. One has an isomorphism of free S-modules   L(λ1 ,1t ) F ∼ (L(µ1 ,1u ) F1 ⊗ St−u F2 ⊗ Λλ1 −µ1 F2 ) ⊕ L(λ1 ,1t ) F2 , = (µ1 ,1u )

where (µ1 , 1u ) ranges over all the hooks which are nested in (λ1 , 1t ). But as GL(F1 ) × GL(F2 )-modules, such an isomorphism usually holds only up to a ﬁltration. V.2.3 Some representation theory It will be shown in Chapter VI that L(λ1 ,1t ) F is just an instance of a wider family of GL(F )-representations, Lλ F , where λ = (λ1 , . . . , λq ) is a partition, that is, either (0) or a weakly decreasing sequence of positive integers. Lλ F is a free S-module and a GL(F )-representation. Moreover, the ΛF ∗ – module structure of ΛF allows the construction of a natural isomorphism Lλ F ⊗S Λn F ∗ ⊗S · · · ⊗S Λn F ∗ → Lλ∗ F ∗ , % &# $ q

where q is the number of non-zero parts of λ, and λ∗ is the partition (n − λq , n − λq−1 , . . .) (cf., e.g., Reference [4], proposition 2.4.2). Another key property of Lλ F is universality, namely, the construction of this module is essentially deﬁned over the integers and then carried over to any other ring. In particular, if we are in the situation of Subsection V.2.1 above, we have Lλ F = Lλ (F0 ⊗R S) ∼ = Lλ (F0 ) ⊗R S. As we shall see in a little while, this has interesting implications if we assume that our ground ring R is a ﬁeld of characteristic zero. For then every ﬁnite dimensional polynomial representation of GL(F0 ) is completely reducible, that is, it decomposes as a direct sum of irreducibles, and a complete set of irreducible representations of GL(F0 ) is {Lλ F0 | λ is a partition}.

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Exactness criteria at work

In particular, for every h ≥ 0, Sh (Λ2 F0 ) ∼ = ⊕Lλ F0 , where λ = (λ1 , . . . , λq ) ranges over all partitions having even entries and such that 2h = λ1 +· · ·+λq (cf., e.g. Reference [66], chapter 1; in particular, exercise 6 on p. 46). Also every GL(F0 )-module of type Lλ F0 ⊗R Lµ F0 is a direct sum of GL(F0 )irreducibles: Lλ F0 ⊗R Lµ F0 = ⊕γ(λ, µ; ν) Lν F0 , ν

and there is a combinatorial rule (known as the Littlewood-Richardson rule) describing the indicated multiplicities γ(λ, µ; ν) (cf., e.g. Reference [66], chapter 1; in particular, section 9). V.2.4 A counting argument This subsection contains some heuristic considerations leading to the construction of the resolutions of the powers am = [P f2k (X)]m . For simplicity, we restrict to the case m = 2. Having at hand the resolution E of a = P f2k (X), it is natural to assume that the ﬁrst map, ϑ1 , of a resolution of a2 must coincide with the second symmetric power of the map g : F ∗ → S having im(g) = a. For every r ≥ 0, ϑ1 is the composite Sr ⊗R S2 (Λ2k F0 )

−→

1⊗S2 ((ω2k )0 )

Sr ⊗R S2k (Λ2 F0 ) → S2k+r (Λ2 F0 ), m

where (ω2k )0 is the R-map Λ2k F0 → Sk (Λ2 F0 ) inducing ω2k over S. Since S2 (Λ2k F0 ) ∼ = S2 F0∗ = L(2k+1,1,1) F0∗ ∼ = L(2k,2k) F0 , ϑ1 is in fact a morphism Sr (Λ2k F0 ) ⊗R L(2k,2k) F0 −→ S2k+r (Λ2 F0 ). We study the latter morphism, when r ranges in N, in order to get a clue about ker(ϑ1 ). We do this by assuming that R is a characteristic zero ﬁeld, for the given modules are universal and we may hope that all the necessary information on our complex is available in the characteristic zero case. If R is assumed to be a ﬁeld of characteristic zero, then the irreducible components of Sr (Λ2k F0 ) ⊗R L(2k,2k) F0 either are mapped onto the corresponding irreducibles of S2k+r (Λ2 F0 ), or must occur in the kernel. For r = 0, we have S0 ⊗R L(2k,2k) F0 → S2k (Λ2 F0 ), which is just the inclusion of L(2k,2k) F0 into S2k (Λ2 F0 ) ∼ = ⊕Lλ F0 (each λ having even entries, adding up to 4k). Hence there is no contribution to ker(ϑ1 ). For r = 1, we have Λ2 F0 ⊗R L(2k,2k) F0 → S2k+1 (Λ2 F0 ). By the Littlewood– Richardson rule, one checks that the domain is isomorphic to L(2k+1,2k,1) F0 ⊕ L(2k,2k,2) F0

Powers of pfaﬃan ideals

141

(since F0 has rank k + 1). As S2k+1 (Λ2 F0 ) ∼ = ⊕Lλ F0 , each λ having even entries, adding up to 4k + 2, L(2k+1,2k,1) F0 must occur in the kernel. For r = 2, we thus have Λ2 F0 ⊗R L(2k+1,2k,1) F0 → S2 (Λ2 F0 ) ⊗R L(2k,2k) F0 → S2k+2 (Λ2 F0 ). Again by means of the Littlewood–Richardson rule, one checks that Λ2 F0 ⊗R L(2k+1,2k,1) F0 ∼ = L(2k+1,2k+1,2) F0 ⊕ L(2k+1,2k+1,1,1) F0 ⊕ L(2k+1,2k,3) F0 ⊕ L(2k+1,2k,2,1) F0 and that S2 (Λ2 F0 ) ⊗R L(2k,2k) F0 ∼ = L(2k+1,2k,3) F0 ⊕ L(2k,2k,4) F0 ⊕ L(2k+1,2k+1,1,1) F0 ⊕ L(2k+1,2k,2,1) F0 ⊕ L(2k+2,2k,2,2) F0 . Since S2k+2 (Λ2 F0 ) ∼ = ⊕Lλ F0 , each λ having even entries, adding up to 4k + 4, it follows that L(2k+1,2k+1,2) F0 must occur in the complex, in degree 3. For r = 3, however, we get 0 → Λ2 F0 ⊗R L(2k+1,2k+1,2) F0 → S2 (Λ2 F0 ) ⊗R L(2k+1,2k,1) F0 → S3 (Λ2 F0 ) ⊗R L(2k,2k) F0 → S2k+3 (Λ2 F0 ), that is, no new term is necessary in degree 4. Thus, assuming R to be a characteristic zero ﬁeld, one conjectures that a resolution of a2 may look like this: ϑ

ϑ

ϑ

0 → L(2k+1,2k+1,2) F →3 L(2k+1,2k,1) F →2 L(2k,2k) F →1 S. Of course some extra terms could be necessary in a characteristic-free setting. Yet we take the above as a reasonable candidate and start looking for possible deﬁnitions of the boundary morphisms, about which we have no hints. Note that our candidate can also be expressed in terms of F ∗ : ϑ

ϑ

ϑ

0 → L(2k−1) F ∗ →3 L(2k,1) F ∗ →2 L(2k+1,1,1) F ∗ →1 S. Λ2k−1 F ∗ S2 F ∗ Since the map f : F → F ∗ of E can easily be identiﬁed with u → p(2k+1,1) (αl1 ∧ u ⊗ αl1 ), L(2k) F ∗ → L(2k+1,1) F ∗ , l

142

Exactness criteria at work

where l αl1 ⊗ αl1 is the image of α given by the component Λ2 F ∗ → F ∗ ⊗S F ∗ of the diagonal map, and where p(2k+1,1) is the projection Λ2k+1 F ∗ ⊗S S1 F ∗ → L(2k+1,1) F ∗ , one conjectures that ϑ2 and ϑ3 are induced by Λ2k F ∗ ⊗S S1 F ∗ → Λ2k+1 F ∗ ⊗S S2 F ∗ , u ⊗ v → αl1 ∧ u ⊗ vαl1 l

and by Λ2k−1 F ∗ → Λ2k F ∗ ⊗S S1 F ∗ ,

u →

αl1 ∧ u ⊗ αl1 ,

l

respectively. One checks that ϑ2 and ϑ3 are actually well deﬁned, by invoking the following more general result, which holds for every ring R. Lemma V.2.1 Let ϕ be the map Λa F ∗ ⊗S Sb F ∗ → Λa+1 F ∗ ⊗S Sb+1 F ∗ , u⊗v → α ∧ u ⊗ vα . Then ϕ induces a morphism l1 l l1 ϕ : L(a,1b ) F ∗ → L(a+1,1b+1 ) F ∗ . Proof Recall the expression of a hook Schur module as a cokernel, and perform calculations applying the diagonal map to α = i<j Xij εi ∧εj (cf. Reference [13], p. 472). 2 It is not hard to see that our candidate is indeed a complex. That ϑ2 ◦ ϑ3 = 0 can be seen as follows. Let ϕ 2 and ϕ 3 be the maps inducing ϑ2 and ϑ3 , respectively. Then   3 (u)) = ϕ 2  Xij (εi ∧ u ⊗ εj − εj ∧ u ⊗ εi ) ϕ 2 (ϕ =

i<j

i<j

Xij

Xpq (εp ∧ εi ∧ u ⊗ εq εj − εp ∧ εj ∧ u ⊗ εq εi

p
− εq ∧ εi ∧ u ⊗ εp εj + εq ∧ εj ∧ u ⊗ εp εj ) and the double sum equals zero (think of what happens if one interchanges j ↔ q and i ↔ p). We also have that ϑ1 ◦ ϑ2 = 0 because if εi stands for ε1 ∧ · · · ∧ εi−1 ∧ εi+1 ∧ · · · ∧ εn , then Xpq ϑ1 (p(2k+1,1,1) (εp ∧ εi ⊗ εj εq )) ϑ1 (ϑ2 (p(2k,1) (εi ⊗ εj ))) = p=q

=

Xiq (−1)i−1 Tj (X)Tq (X)

q

1 i−1

= (−1)

Tj (X)

2 Xiq Tq (X)

q

and the sum in brackets is zero by the last formula of Remark V.1.4. This concludes our heuristic considerations.

Powers of pfaﬃan ideals

143

V.2.5 Description of the resolutions We begin the formal construction of some complexes Cm , m ≥ 2; these will be shown to be the required resolutions of the ideals am . Deﬁnition V.2.2 Let f : F → F ∗ be the generic alternating map of Subsection V.2.1, where rankF = n = 2k + 1 for some positive k. 1. If m ≥ 2k, then Cm is deﬁned to be the sequence ϕ

ϕ

0 → L(1m−n+2 ) F ∗ → L(2,1m−n+2 ) F ∗ → · · · ϕ

ϕ

ψ

· · · → L(n−1,1m−1 ) F ∗ → L(n,1m ) F ∗ → S. 2. If m = 2r, 1 ≤ r ≤ k − 1, then Cm is deﬁned to be the sequence ϕ

ϕ

0 → L(n−m) F ∗ → L(n−m+1,1) F ∗ → · · · ϕ

ϕ

ψ

· · · → L(n−1,1m−1 ) F ∗ → L(n,1m ) F ∗ → S. 3. If m = 2r + 1, 0 ≤ r ≤ k − 1, then Cm is deﬁned to be the sequence χ

ϕ

ϕ

0 → S → L(n−m) F ∗ → L(n−m+1,1) F ∗ → · · · ϕ

ϕ

ψ

· · · → L(n−1,1m−1 ) F ∗ → L(n,1m ) F ∗ → S. In all cases, ϕ stands for the map induced by the appropriate u ⊗ v → αl1 ∧ u ⊗ vαl1 , ϕ : Λa F ∗ ⊗S Sb F ∗ → Λa+1 F ∗ ⊗S Sb+1 F ∗ , l ∗

and ψ is the m-th symmetric power of the map g : F → S deﬁned in Subsection V.2.1 (explicitly, ψ(εi1 · · · εim ) = Ti1 (X) · · · Tim (X)). As for χ in the third case, it is deﬁned by χ(1) = α(k−r) . We point out that the morphisms ϕ of Cm are well deﬁned, thanks to the lemma of the previous subsection. When m = 1, Cm gives back E, the resolution of the pfaﬃan ideal a. In the special case n = 3, each Cm coincides with the well-known resolution of the m-th power of an ideal generated by a regular sequence (having three elements); one ﬁnds such a resolution in Reference [24]. It is interesting to notice that the hook Schur modules L(λ1 ,1t ) F ∗ were introduced in that paper for the ﬁrst time. Proposition V.2.3 Cm is a complex, for every m ≥ 1. Proof One shows that every composition ϕ ◦ ϕ is zero exactly in the same way we did for ϕ 2 ◦ ϕ 3 at the end of the previous subsection. One shows that every composition ψ ◦ ϕ is zero exactly in the same way we did for ϑ1 ◦ ϑ2 at the end of the previous subsection.

144

Exactness criteria at work

It remains to show that if m = 2r + 1, 0 ≤ r ≤ k − 1, then ϕ ◦ χ = 0, that is, ϕ(χ(1)) = ϕ(α(k−r) ) = 0. One writes down explicitly the divided power α(k−r) , applies ϕ, and checks vanishing, ﬁrst, by taking into account the key step of the straightening law recalled in Subsection V.2.2 and then by resorting to the formulas of Remark V.1.4. (Cf. Reference [13], pp. 474–477.) 2 Theorem V.2.4

For every m ≥ 1, the complex Cm is a resolution of S/am .

The proof of this theorem is discussed in the following subsection. Here we just make some comments. • The resolution Cm is minimal, that is, for every i ≥ 1, the image of the

i-th morphism of Cm is included in m(Cm )i−1 , where m denotes the ideal of S generated by the indeterminates occurring in the matrix X. • Cm is also generic (or universal), in the sense that it is deﬁned over Z[X] and then carried over to every other R[X], R noetherian. In particular, the Betti numbers of am (that is, the ranks of the modules (Cm )i ) are independent of the characteristic of R. • The existence of a generic minimal free resolution for S/am is remarkable, since it is known that S/P f2p (X), 2 ≤ p ≤ k − 1, may not admit such a resolution. More precisely, [62] contains a case in which the Betti numbers of S/P f2p (X) depend on the characteristic of R. Thus the ideals P f2p (X) behave pretty much in the same way as the determinantal ideals of a generic n1 × n2 matrix (cf. [48]). More on this in Section VII.5. • Cm shows that the homological dimension of S/am satisﬁes  if m ≥ 2k n hdS/am = m + 1 if m = 2r, 1 ≤ r ≤ k − 1  m + 2 if m = 2r + 1, 0 ≤ r ≤ k − 1 (notice that m + 2 = n, when m = 2(k − 1) + 1). Since, as we know, an ideal is called perfect if its depth and homological dimension coincide, it follows from grade am = grade a = 3 that am is perfect in every characteristic (“generically perfect”) when m = 1, 2 (no matter what the value of n) and whenever n = 3 (in this case, P f2k (X) is generated by a regular sequence). In all the other cases, am is not perfect. However, there is a regular pattern for hd S/am , namely, hd S/a2r+1 = hd S/a2r+2 = 2r + 3 if 0 ≤ r ≤ k − 2 hd S/am = 2k + 1

if m ≥ 2k − 1.

This regularity can be seen as a special case of a more general phenomenon related to “almost alternating maps,” cf. Reference [63], section 5. Another type of example can be found in References [5] and [6]. In those papers, the homological dimensions of certain generic modules associated to partitions are seen to depend on the size of the Durfee squares of those

Powers of pfaﬃan ideals

145

partitions. (A deﬁnition of this combinatorial invariant of a partition can be found in Subsection VII.3.1.) V.2.6 Proof of Theorem V.2.4 We proceed by induction on k = (n − 1)/2. When k = 1, that is, n = 3, we have already observed (immediately before Proposition V.2.3) that Cm resolves S/am . So we prove the statement for k ≥ 2, assuming it true for all values less than k. Since we already know that im ψ = am , it is enough to show that the positive homology vanishes. We make use of Theorem IV.1.10. Let p ∈ SpecS be such that  if m ≥ 2k  n m + 1 if m = 2r, 1 ≤ r ≤ k − 1 gradepSp < lengthCm =  m + 2 if m = 2r + 1, 0 ≤ r ≤ k − 1. The ideal, m, generated in S by the indeterminates occurring in X, has grade n(n − 1)/2 ≥ n. Thus grade mSp > grade pSp , no matter what m is, and one of the variables Xij is invertible over Sp , say X12 . But then the exactness of Cm ⊗S Sp follows, if we show that Cm is exact after localization at the powers of X12 . The idea is that if X12 can be assumed invertible, one can ﬁnd new dual bases such that the corresponding matrix associated with the generic alternating map is of type   0 1 0  −1 , 0 0 X X is a generic alternating matrix of order n = n − 2 = 2(k − 1) + 1, and the inductive hypothesis applies. More precisely, let R and S be the localizations at the powers of X12 of R[X12 , X13 , X14 , . . . , X1n , X23 , X24 , . . . , X2n ] and S, respectively, and keep denoting by F and F ∗ the corresponding objects over S . As in Reference [13], pp. 479–480 (also cf. [55]), there is an explicit way of choosing dual bases {ε1 , . . . , εn } and {e1 , . . . , en } for F ∗ and F , resp., such that f ⊗S S has the matrix above, the entries of the alternating submatrix X are algebraically independent over R , S = R [X ], and aS = a , where a is the ideal P f2(k−1) (X ) of S . Furthermore, ψ ⊗S S sends ε1 and ε2 to 0, and sends εi (3 ≤ i ≤ n) to X12 · Ti−2 (X ). As for the element of Λ2 F ∗ associated with f ⊗S S , it looks like ε1 ∧ ε2 +

1≤i<j≤n

(n = 2k − 1, as above).

Xij εi+2 ∧ εj+2

146

Exactness criteria at work

Accordingly, let us decompose F ∗ as H ∗ ⊕ G∗ , where H ∗ is generated by and G∗ by ε1 , ε2 (so that ε1 ∧ ε2 can be denoted by αG∗ and Xij εi+2 ∧ εj+2

ε3 , . . . , εn ,

1≤i<j≤n

by αH ∗ ). Recalling the end of Subsection V.2.2, we have (Cm ⊗S S )i

= L(a,1b ) (H ∗ ⊕ G∗ ) ∼ = [ ⊕ (L(µ1 ,1u ) H ∗ ⊗ Sb−u G∗ ⊗ Λa−µ1 F2 )] ⊕ L(a,1b ) G∗ , (µ1 ,1u )

where a = n+1−i, b = m+1−i, and some summands may be 0, if rank H ∗ = n is not large enough. We intend to use the decompositions of the modules L(a,1b ) (H ∗ ⊕ G∗ ) in order to ﬁlter the complex Cm ⊗S S . Thus the exactness of Cm ⊗S S will be reduced to the exactness of the factors of the ﬁltration. It is convenient to discuss separately the four cases: m ≥ 2k, m even m ≥ 2k, m odd m = 2r, 1≤r ≤k−1 m = 2r + 1, 0 ≤ r ≤ k − 1. Here we deal with the ﬁrst case only. The others are similar and can be found in Reference [13] (pp. 487–490). The H ∗ -content of a summand of L(a,1b ) (H ∗ ⊕ G∗ ) is deﬁned as the number µ1 + u, denoted by |H ∗ |. Hence L(a,1b ) (H ∗ ⊕ G∗ ) can be decomposed into the direct sum of two modules: the one, M0 (a, 1b ), comprising all summands with |H ∗ | even, and the other, M1 (a, 1b ), comprising all summands with |H ∗ | odd. If a morphism ϕ⊗S S is applied to L(a,1b ) (H ∗ ⊕G∗ ), then M0 (a, 1b ) is mapped to M0 (a+1, 1b+1 ) and M1 (a, 1b ) is mapped to M1 (a+1, 1b+1 ). If ψ⊗S S is applied to L(a,1b ) (H ∗ ⊕ G∗ ), it means that a = n and b = m, whence µ1 = n ; recalling the way ψ⊗S S acts on the elements of {ε1 , . . . , εn }, it follows that ψ⊗S S is zero on all summands of L(a,1b ) (H ∗ ⊕G∗ ), except for L(µ1 ,1m ) H ∗ ⊗Λ2 G∗ ⊆ M1 (n, 1m ) (here we use the fact that m is even). Therefore Cm ⊗S S is the direct sum of two subcomplexes, M0 and M1 ; M0 is given by the terms M0 (a, 1b ); M1 is given by the terms M1 (a, 1b ), together with S . One shows that both of them are exact, by ﬁltering them separately. As above, we will deal with only one case, that of M1 . The case of M0 is similar and can be found in Reference [13] (pp. 485–486). A ﬁltration {Xt } of M1 is described as follows (recall that m ≥ 2k is assumed to be even, say m = 2p). For every ﬁxed t ∈ {0, 1, . . . , p − 1}, in each L(a,1b ) (H ∗ ⊕ G∗ ) we assign to Xt all the summands having µ1 = n − j,

u = 2t − j

Powers of pfaﬃan ideals

147

for some t ≤ t and some non-negative integer j. To Xp we assign S and all the summands of L(a,1b ) (H ∗ ⊕ G∗ ) having µ1 = n − j and u = 2t − j for some t ≤ p and some non-negative integer j. It is easy to see that Xp = M1 (cf. Reference [13], proposition at the end of p. 481). Recalling the straightening law at the end of Subsection V.2.2, it is not hard to check that each Xt , 0 ≤ t ≤ p, is indeed a complex (cf. Reference [13], proposition on p. 482). We now describe the factors Xt /Xt−1 , 0 ≤ t ≤ p − 1 (X−1 is set equal to 0), and prove their exactness. The modules occurring in Xt /Xt−1 are given by µ1 = n − j and u = 2t − j, with j ranging between 0 and q, where 2t if 2t < n q= n − 1 if 2t > n . It follows that a − µ1 = 3 − (i − j) and b − u = m + 1 − 2t − (i − j). Since rank G∗ = 2 implies 0 ≤ a − µ1 ≤ 2, i − j can only be 1, 2, 3, whence b − u = m − 2t, m − 2t − 1, m − 2t − 2, respectively. Thus, no matter what j is, one ﬁnds terms of the following type: A(j) = L(n −j,12t−j ) H ∗ ⊗ Sm−2t−2 G∗ , B(j) = L(n −j,12t−j ) H ∗ ⊗ Sm−2t−1 G∗ ⊗ Λ1 G∗ , C(j) = L(n −j,12t−j ) H ∗ ⊗ Sm−2t G∗ ⊗ Λ2 G∗ . If one remembers the way αG∗ and αH ∗ operate, one checks that Xt /Xt−1 is the total complex of the following double complex, Dt , 0 ↑

0 ↑

0 ↑

(0)

0 → A(0) (0) ↑ ϕH ∗

ϕG∗

0 → A(1) (0) ↑ ϕH ∗ .. .

ϕG∗

→ B(0) (1) ↑ ϕH ∗ (0)

→ B(1) (1) ↑ ϕH ∗ .. .

(0)

(1)

ϕG∗

→ C(0) (2) ↑ ϕH ∗ (1)

ϕG∗

→ C(1) (2) ↑ ϕH ∗ .. .

(1)

↑ ϕH ∗ ϕ

(0) ∗

→0 →0

(2)

↑ ϕH ∗ ϕ

(1) ∗

↑ ϕH ∗

G G 0 → A(q − 1) → B(q − 1) → C(q − 1) → 0 (0) (1) (2) ↑ ϕH ∗ ↑ ϕH ∗ ↑ ϕH ∗

0 → A(q) ↑ 0

(0)

ϕG∗

→ B(q) ↑ 0

(1)

ϕG∗

→ C(q) ↑ 0

→0

148

Exactness criteria at work

In the diagram, for each w = 0, 1, 2, the map ϕH ∗ : L(µ1 ,1u ) H ∗ ⊗ Sv G∗ ⊗ Λw G∗ → L(µ1 +1,1u+1 ) H ∗ ⊗ Sv G∗ ⊗ Λw G∗ (w)

is deﬁned by p(µ1 ,1u ) (x ⊗ y) ⊗ z1 ⊗ z2 → p(µ1 +1,1u+1 ) ((αH ∗ )l1 ∧ x ⊗ y(αH ∗ )l1 ) ⊗ z1 ⊗ z2 . l

For each w = 0, 1, the map ϕG∗ : L(µ1 ,1u ) H ∗ ⊗ Sv G∗ ⊗ Λw G∗ → L(µ1 ,1u ) H ∗ ⊗ Sv+1 G∗ ⊗ Λw+1 G∗ (w)

is deﬁned by x ⊗ y ⊗ z → (−1)µ1

x ⊗ y(αG∗ )l1 ⊗ (αG∗ )l1 ∧ z.

l

We remark that each line of Dt is a complex essentially because ϕ ◦ ϕ = 0. The anticommutativity of the boxes is straightforward from the deﬁnitions, (w) particularly from the sign (−1)µ1 introduced in ϕG∗ . Now, each row of Dt is isomorphic to a short exact sequence 0 → Sh G∗ → Sh+1 G∗ ⊗ Λ1 G∗ → Sh+2 G∗ ⊗ Λ2 G∗ → 0, and the latter is isomorphic to the exact complex Λh+2 (G∗ ) of Subsection III.1.1, since Λ2 G∗ ∼ = S . But the exactness of the rows of Dt implies that the total complex of Dt , that is, Xt /Xt−1 , is exact as well, as anticipated. We ﬁnally show that Xp /Xp−1 is exact, so that the whole M1 is exact. The modules occurring in Xp /Xp−1 are S and those associated with µ1 = n − j and u = m − j, with j ranging between 0 and n − 1 (since m ≥ 2k gives 2p > n , and q = n − 1). Then a − µ1 = 3 − (i − j) and b − u = 1 − (i − j). Since 0 ≤ a − µ1 ≤ 2 and b − u ≥ 0, it follows that a − µ1 = 2 and b − u = 0. Hence (recall Λ2 G∗ ∼ = S ) Xp /Xp−1 is isomorphic to the complex ϕ

(2) ∗

ϕ

(2) ∗

H H L(2,1m−n +2 ) H ∗ → ··· Cm : 0 → L(1m−n +2 ) H ∗ →

ϕ

(2) ∗

ϕ

(2) ∗

H H L(n −1,1m−1 ) H ∗ → L(n ,1m ) H ∗ → S , ··· →

where ϕH ∗ is as before and L(n ,1m ) H ∗ → S is the only non-zero component of ψ ⊗S S , namely (recall Λn H ∗ ∼ = S ), the morphism deﬁned by (2)

(ε )b3 · · · (εn )bn → T3 (X )b3 · · · Tn (X )bn . % 3 &# $ b3 +···+bn =m

Cm

is a complex of the same type of Cm , relative to the generic alternating But matrix X of order 2(k − 1) + 1. Therefore it is exact by induction hypothesis. This concludes our discussion of the exactness of the complexes Cm .

VI WEYL AND SCHUR MODULES

In earlier chapters we introduced a very small dose of representation theory of the general linear group when we discussed, in particular, the hook shapes. In this chapter, we will deal more comprehensively with representations of that group, over an arbitrary commutative ring, R. Our discussion will help place the material in earlier chapters in proper perspective, and will move us forward to some interesting observations relating to determinantal ideals and other matters. In the classical theory, the fundamental shapes are the Ferrers diagrams corresponding to “partitions,” and the closely related “skew-partitions.” For our purposes, we will have to consider a slightly larger class of shapes, corresponding to the so-called almost skew-partitions. We should add that as of this writing, the class of shapes that is being studied is far broader than this. However, to study all of these would require quite a bit more of combinatorics than we propose to use here. We hope that the material we do cover, however, will help the interested reader to get into this other area if he so desires. We remind the reader that the details of the proofs of several of the theorems involving letter-place methods can be found in Appendix A following Chapter VII. VI.1 Shape matrices and tableaux We will start this section with the deﬁnition of a shape matrix, and then will talk about the partition or skew-partition, etc. associated with the matrix. However, the reader should know that historically the sequence was the other way around: the idea of partition and skew-partition, and Ferrers diagrams in general, arose ﬁrst; the idea of generalizing to shape matrices came afterwards. In practice, one usually describes the shape by the diagram. In the second subsection on tableaux, we will give a deﬁnition that generalizes the one used in Chapter III. There, too, we introduce a quasi order on tableaux that will be used in Appendix A. VI.1.1 Shape matrices Deﬁnition VI.1.1 A shape matrix is an inﬁnite integral of ﬁnite support, with all the aij equal to zero or 1. (To say it is to say that aij = 0 for only a ﬁnite number of indices i and (column) of the shape matrix A is the last row (column) in

matrix A = (aij ) has ﬁnite support j.) The last row which a non-zero

150

Weyl and Schur modules

term appears. Such a matrix is said to be row-convex (column-convex) if, in each row (column), there are no zeroes lying between ones. (All the shapes that we consider will be row- and column-convex.) The shape matrix B = (bij ) is a subshape of A (written B ⊆ A) if bij ≤ aij for every i and j. The shape matrix, A, is said to correspond to a partition if for all i, j, aij = 0 implies ai+1j = 0 and aij+1 = 0 implies aij = 0. It is said to correspond to a skew-partition or skew-shape if A = B − C, where B and C correspond to partitions, and C ⊆ B. It is said to be a bar shape if its only non-zero entries are in its last row, and it is row-convex. Finally, it is said to correspond to an almost skew-partition or almost skew-shape if A = B − C, where B corresponds to a skew-shape, and C is a bar subshape matrix of B the index of whose last row coincides with that of B, and whose ﬁrst non-zero entry in that row occurs in the same place as the ﬁrst non-zero entry of B. • Notice that, unless a given partition shape matrix is the zero matrix, a11 = 1.

We now illustrate each of these types of shape looks like this:  1 1 1  1 1 1   1 1 1  (P )  1 1 0   0 0 0  .. .. .. . . .

shape matrices. The typical partition 1 0 0 0 0 .. .

0 0 0 0 0 .. .

0··· 0··· 0··· 0··· 0··· .. .···

     ,   

and is often represented by the Ferrers diagram:

.

The typical skew-shape looks like  0  0   0  (S)  1   0  .. .

this: 0 0 1 1 0 .. .

1 1 1 0 0 .. .

1 1 0 0 0 .. .

1 0 0 0 0 .. .

and is often represented by the Ferrers diagram:

.

0··· 0··· 0··· 0··· 0··· .. .···

     ,   

Shape matrices and tableaux

A bar shape looks like:

(B)

        

0 0 0 0 0 .. .

0 0 0 1 0 .. .

and an almost skew-shape looks like:  0 0 0  0 0 1   0 1 1  (AS)  0 0 1   0 0 0  .. .. .. . . .

0 0 0 1 0 .. .

1 1 1 1 0 .. .

0 0 0 1 0 .. .

1 1 1 1 0 .. .

151

0··· 0··· 0··· 0··· 0··· .. .···

0 0 0 0 0 .. .

1 1 1 0 0 .. .

1 1 0 0 0 .. .

     ,   

0··· 0··· 0··· 0··· 0··· .. .···

     ,   

and would be represented by the Ferrers diagram:

.

Note that the diagram ignores the fact that the left-most column of the shape matrix consists completely of zeroes. Also notice that the empty diagram corresponds to the zero matrix. The shapes illustrated above are those that we will have most to do with, but clearly we can associate to any shape matrix, A = (aij ), a Ferrers diagram, or simply diagram: we just set up a grid equal to the eﬀective size of the matrix (say, s × t), and throw away the boxes whose entries are equal to zero. That is, the (i, j)th box lies in the diagram if and only if aij = 1. If A is the shape matrix, we will sometimes denote by (A) its corresponding diagram. The partition shape has a uniquely associated partition, namely the sequence λ = (λ1 , . . . , λs , . . .) where λi is the non-negative integer equal to the number of ones in row i. Clearly, the sequence is decreasing: λ1 ≥ λ2 ≥ · · · ≥ λs ≥ · · · . We say the length of λ is s if s is the smallest non-negative integer such that λs+t = 0 for every positive integer t. The skew-shape has two partitions uniquely associated to it, namely, λ = (λ1 , . . . , λs , . . .) and µ = (µ1 , . . . , µt , . . .), with µi ≤ λi for all i, and such that the length of µ is strictly less than that of λ. One then thinks of the shape as the result of removing from the shape of λ the subshape corresponding to µ. In fact, the notation most often used for a skew-shape is λ/µ. We say the length of λ/µ is the length of λ. If one removes the condition that the length of µ be

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strictly less than that of λ, then we have other pairs of partitions (λ , µ ) that will yield the same diagram. In that case, we still use the same notation, λ /µ (but the length of λ /µ stays equal to the length of the previous λ). Finally, we see that the almost skew-shape would be a skew-shape but for its last row, which, rather than projecting beyond (or ﬂush with) the penultimate row, does not make it out that far to the left. In short, it would be a skewpartition but for that inadequacy in the last row. In our examples above, the partition λ associated to (P ) is (4, 3, 3, 2); the pair of partitions associated to (S) are λ = (5, 4, 3, 2) and µ = (2, 2, 1). (For convenience we have eliminated the zeroes to the right in our notation.) For (AS), we might take the skew-partition to be (7, 7, 6, 5)/(3, 2, 1) with a bar having the entries (1, 1) in the fourth row, or we might take (7, 7, 6, 5)/(3, 2, 1, 1), with a bar having entries (0, 1) in the fourth row. Another way to denote an almost skew-shape, which closely parallels the notation for a skew-shape is to ﬁrst deﬁne an almost partition to be a sequence (µ1 , . . . , µn ) such that µ1 ≥ · · · ≥ µn−1 and 0 ≤ µn ≤ µ1 . We then can denote (not necessarily uniquely) an almost skew-shape by λ/µ, where λ is a partition of length n, and µ is an almost partition having exactly n terms and satisfying µi ≤ λi for all i. We can go one step further, and say that an almost partition, µ, is of type n − (i + 1) if i is the largest integer less than n such that µn ≤ µi , and we say that the type of the almost skew-shape λ/µ is equal to the type of µ. Clearly, the type is independent of the choice of λ and µ used to describe the almost skew-shape. The choice of the pair (λ, µ) can be made canonical if, in the case of type zero, we choose µn = 0, while for type greater than zero, we choose µn−1 = 0. We deﬁne the length of the almost skew-shape to be the length of the canonical partition, λ. With this terminology, we see that an almost skew-shape of type 0 is a skew-shape, and that for almost skew-shapes of length n, we can have types 0, 1, . . . , n− 2. In particular, almost skew-shapes of length 2 are necessarily skewshapes; for length 3, there are only skew-shapes and almost skew-shapes of type 1, and so on. We spoke of shape matrices as inﬁnite matrices in order not to have to specify last row or column when we talked about subshapes. However, we see that a shape matrix whose last row is row s and whose last column is column t, can be thought of as an s × t-matrix; when we draw diagrams or shapes, we will generally avoid the dots that we were forced to put into the illustrations above. If we have two shape matrices A and B, with B ⊆ A, we will assume that they are both s × t-matrices, simply by augmenting where necessary by zeroes.

Remark VI.1.2

Three immediate observations should be made here:

is also 1. If A is a partition (or skew-partition) matrix, then its transpose, A, a partition (or skew-partition) matrix.

Shape matrices and tableaux

153

2. If A is a partition matrix with associated partition λ, then we denote the by λ. partition associated to A 3. We see that if A is a skew-partition matrix with associated partitions λ and and µ has associated partitions λ µ, then the matrix A . The notation and terminology used above diﬀer slightly from those used elsewhere. For instance, in I. Macdonald’s book [66], the transpose is called the “conjugate,” and is denoted by A ; in Reference [3], it is called the “dual,” but denoted in the way it is above. Deﬁnition VI.1.3 We introduce some standard terminology for shapes, and partitions in particular: 1. The weight of a shape matrix A = (aij ) is aij , and is denoted by |A|.

2. If λ = (λ1 , . . . , λn ) and λ = (λ1 , . . . , λm ) are two partitions, we say λ ≥ λ if either λ = λ or if for some i, λj = λj for all j < i, and λi > λi . VI.1.2 Tableaux Deﬁnition VI.1.4 Let A be a shape matrix and S a set. A tableau, T , of shape A with values in S is a ﬁlling-in of the diagram (A) by elements of S. We denote the tableau by the ordered pair T = ((A); τ ), where τ is the ﬁlling-in of (A) by S. We could have said that τ is a function from (A) to S, but for the fact that we have not given a formal enough deﬁnition of “diagram” to do this. But if one regards the diagram as a collection of cells, then τ would be a function with domain (A). In most cases, we will simply refer to the tableau as T , with the set S an understood ordered basis of a ﬁnitely generated free module. This is how we saw tableaux used in earlier chapters. Occasionally we will use the term row tableau; this is simply a tableau the diagram of whose shape consists of one row. For further work with tableaux, it will be convenient to have a quasi order on tableaux with values in a totally ordered set. Suppose, then, that S is a totally ordered set, say S = {s1 , . . . , sn } with s1 < · · · < sn , and suppose T is a tableau with values in S. Deﬁne Tij to be the number of elements in {s1 . . . , si } that appear in at least one of the ﬁrst j rows of the diagram of T . Now suppose that T is another tableau. Deﬁnition VI.1.5 We say that T ≤ T if Tij ≥ Tij for every i and j. We say that T < T if T ≤ T and for some i, j we have Tij > Tij . To see that this is a quasi order and not an order, consider our set S with three elements: S = {s1 , s2 , s3 }, with s1 < s2 < s3 , and consider the diagram

154

Weyl and Schur modules

corresponding to the partition λ = (4, 2, 1). Then the two tableaux s1 T = s2 s3

s2 s3

s2

s3

s2 T = s3 s3

s3 s2

s2

s1

and

are such that T ≤ T and T ≤ T , but T and T are clearly not equal. However, the “≤” relation is both reﬂexive and transitive, as can easily be checked.

VI.2 Weyl and Schur modules associated to shape matrices To each ﬁnite free module, F, over a commutative ring, R, and each shape matrix we will associate two maps, a Weyl map and a Schur map, whose images will be called the Weyl and Schur modules of that shape. To do that, we ﬁrst look at some auxiliary ideas. If a = (a1 , . . . , al ) is a sequence of non-negative integers, let α = a1 + · · · + al . We deﬁne the maps δa : Dα F → Da1 F ⊗ · · · ⊗ Dal F and δa : Λα F → Λa1 F ⊗ al · · · ⊗ Λ F to be the diagonalization maps of the indicated divided and exterior powers of F into the indicated tensor products. We deﬁne the maps µa : Λa1 F ⊗ · · · ⊗ Λal F → Λα F and µa : Sa1 F ⊗ · · · ⊗ Sal F → Sα F to be the multiplication maps from the indicated tensor products of exterior and symmetric powers to the indicated exterior and symmetric powers. Deﬁnition VI.2.1 (Weyl and Schur maps) Let F be a free module over the commutative ring, R. For the s×t shape matrix A = (aij ), set ri = (ai1 , . . . , ait ), t s cj = (a1j , . . . , asj ), ρi = j=1 aij , γj = i=1 aij . The Weyl map associated to A, ωA , is the map ωA : Dρ1 F ⊗ · · · ⊗ Dρs F → Λγ1 F ⊗ · · · ⊗ Λγt F deﬁned as the composition ωA = µc1 ⊗ · · · ⊗ µct θW δr 1 ⊗ · · · ⊗ δr s where, since all the aij are equal to zero or one, we have identiﬁed Daij F with Λaij F for all i, j; the map θW is the isomorphism comprising all of these identiﬁcations together with rearrangement of the factors. Pictorially what we have

Modules associated to shape matrices

155

is the following:

  Da11 F ⊗ · · · ⊗ Da1t F         ⊗     θW . . −→ Dρ1 F ⊗ · · · ⊗ Dρs F → .       ⊗       Das1 F ⊗ · · · ⊗ Dast F   a Λ 11 F ⊗ · · · ⊗ Λa1t F         ⊗     .. → Λγ1 F ⊗ · · · ⊗ Λγt F. .       ⊗      as1  Λ F ⊗ · · · ⊗ Λast F

The Schur map associated to A, σA , is the map σA : Λρ1 F ⊗ · · · ⊗ Λρs F → Sγ1 F ⊗ · · · ⊗ Sγt F deﬁned as the composition σA = µc1 ⊗ · · · ⊗ µct θS δr1 ⊗ · · · ⊗ δrs where, since all the aij are zero or one, we have identiﬁed Λaij F with Saij F for all i, j; the map θS is the isomorphism comprising all of these identiﬁcations together with rearrangement of the factors. We can view the deﬁnition of the Schur map “pictorially” in the same way we did the Weyl map. Deﬁnition VI.2.2 (Weyl and Schur modules) Let F be a free Rmodule, and A a shape matrix. We deﬁne the Weyl module of F associated to A, denoted KA F , to be the image of ωA . We deﬁne the Schur module of F associated to A, denoted LA F, to be the image of σA . Remark VI.2.3 useful.

The following observations are easy to check and very

1. If A is a non-zero shape matrix with its initial column consisting only of zeros, and B is the shape matrix with that initial column removed, it is clear that the Weyl and Schur maps associated to A and B are the same. Hence, we will generally assume that our shape matrices have at least one entry in the ﬁrst column equal to one. 2. If A is a shape matrix, and B is the shape matrix obtained from A by a permutation of its rows (columns), then the associated Weyl and Schur modules of these matrices are isomorphic. With the deﬁnition of Weyl and Schur modules to hand, a natural question to consider is whether these modules are free over the ground ring and, if so, how can we describe a basis. For example, if we take a one-rowed partition λ, what

156

Weyl and Schur modules

are the Weyl and Schur modules associated to it? The Weyl module is the image of the map Dλ F →ωλ Λ1 F ⊗ · · · ⊗ Λ1 F , where ωλ is the diagonalization map. &# $ % λ

Clearly, then, the image is isomorphic to Dλ F itself, that is, Kλ F = Dλ F . In a similar way we can show that Lλ F = Λλ F . In both of these cases, the modules are clearly free R-modules, and we have a very concrete description of bases for them. Not only do we have explicit descriptions of their bases, we even have a description in terms of tableaux. In the case of Dλ F , a basis is parametrizable by the set of all one-rowed tableaux: xi1 xi2 · · · xiλ where {x1 , . . . , xm } is an ordered basis of the free module, F , and i1 ≤ · · · ≤ iλ . In the case of Λλ F , we have that a basis is parametrizable by all one-rowed tableaux: xi1 xi2 · · · xiλ where i1 < · · · < iλ . It will be proven later that if λ/µ is an almost skew-partition, then Kλ/µ and Lλ/µ are free, and bases for them can be parametrized by certain sets of tableaux of shape λ/µ satisfying combinatorial conditions. VI.3 Letter-place algebra We have run up, and will run up, against tensor products of divided powers, exterior powers and symmetric powers frequently. A tool that has proven helpful in dealing with these kinds of terms is the so-called letter-place algebra. In this section we will deﬁne this algebra, and develop some important combinatorial properties of it. Our treatment will be a little less general than that given in Reference [81]; the interested reader may go to that reference to see how multisigned alphabets are treated in a uniform and general way. We will deal with the divided powers case in some detail, and then we will quickly treat the cases of exterior powers and symmetric powers. Most of the proofs will be found in Appendix A. VI.3.1 Positive places and the divided power algebra Usually we are given a ﬁxed number, say n, of terms in the tensor product: Dk1 (F ) ⊗ · · · ⊗ Dkn (F ), where F is a free module. We intuitively look at such a product and know which is the ﬁrst factor, the second, and so on. The idea behind the letter-place approach is to clearly designate the places that the terms in the product are actually in. As an example of what we mean, suppose that x ∈ Dki (F ), and we want to write the element 1 ⊗ · · · ⊗ x ⊗ · · · ⊗ 1 in the tensor % &# $ i

product above. The letter-place algebra will allow us to write this element as (x|i(ki ) ). How this will help besides just shortening the amount we have to type and the space it takes to type it will become evident as we develop and use this approach. Just as with the symmetric and exterior algebras, we have that D(F ⊕ G) = D(F ) ⊗ D(G); it is, after all, the graded dual of the symmetric algebra. So,

Letter-place algebra

157

if we take D(F ⊗ Rn ) ∼ = D(F ⊕ · · · ⊕ F ), we see that D(F ⊗ Rn ) is equal to % &# $ n

D(F ) ⊗ · · · ⊗ D(F ). This is natural with respect to the action of GL(F ), but % &# $ n times

clearly not with respect to the action of GL(Rn ). In fact, we moved to the notation Rn rather than G to indicate that we have made a choice of basis in our free module, G. We can, though, use G in our preliminary discussion and, assuming that the rank of this free module is n, still see that “in some way”, D(F ⊗ G) ∼ = D(F ) ⊗ · · · ⊗ D(F ). Now we want to introduce convenient notation &# $ % n

to exhibit this isomorphism, as well as to get to the letter-place conventions. To this end, let us suppose that G has the (ordered) basis, {y1 , . . . , yn } with y1 < · · · < yn , and for any x ∈ D1 (F ), let us denote by (x|yi ) the element x ⊗ yi , (k) and by (x(k) |yi ) the element corresponding in D(F ) ⊗ · · · ⊗ D(F ) to (x|yi )(k) , % &# $ n

that is, to the element in that n-fold tensor product of D(F ) having x(k) in the ith factor. • The picture to keep in mind is: (x|yi ) is the element 1 ⊗ · · · ⊗ x ⊗ · · · ⊗ 1.

%

&#

$

i

Now k!(x ⊗ yi )(k) = (x ⊗ yi )k = (1 ⊗ · · · ⊗ x ⊗ · · · ⊗ 1)k &# $ % i

= 1 ⊗ · · · ⊗ x ⊗ · · · ⊗ 1 = k! (1 ⊗ · · · ⊗ x(k) ⊗ · · · ⊗ 1), % &# $ % &# $ k

i

i (k)

so that the above deﬁnition of (x(k) |yi ) makes sense. Finally, if l = l1 + · · · + ln , and x ∈ Dl (F ), we set (l ) (l ) (x|y1 1 · · · yn(ln ) ) = (x(l1 )|y1 1 ) · · · (x(ln )|yn(ln ) ), where x(l1 )⊗· · ·⊗x(ln ) indicates the image of the diagonal map into Dl1 (F )⊗ · · · ⊗ Dln (F ) applied to our element x. Remark VI.3.1 The identities and conventions that we adopt for our discussion are those that are clearly valid if one works over the ring of integers (as is the case illustrated above, where we have cancelled k! because there is no torsion over the integers). We will continue to do this in our treatment of the letterplace algebra and all other structures that are transportable from Z to arbitrary commutative base rings. A simple illustration, just to ﬁx our ideas, is this: Suppose x1 , x2 and x3 are in D1 (F ), and consider the element (2)

(2)

(x1 x2 x3 |y1 y2 y3 ) ∈ D2 (F ) ⊗ D1 (F ) ⊗ D1 (F ).

158

Weyl and Schur modules

Then this element is equal to (2)

(2)

(x1 x2 |y1 )(x2 |y2 )(x3 |y3 ) + (x1 x2 |y1 )(x3 |y2 )(x2 |y3 )+ (2)

(2)

(2)

(x1 x3 |y1 )(x2 |y2 )(x2 |y3 ) + (x2 |y1 )(x1 |y2 )(x3 |y3 )+ (2)

(2)

(2)

(x2 |y1 )(x3 |y2 )(x1 |y3 ) + (x2 x3 |y1 )(x1 |y2 )(x2 |y3 )+ (2)

(x2 x3 |y1 )(x2 |y2 )(x1 |y3 ). (a )

(a )

1 n (♠) We agree to set nthe symbol (w|y1 · · · yn ) equal to zero if the degree of w is not equal to i=1 ai . The element w is supposed to be a homogeneous element of D(F ). The letter-place notation we have been using above lends itself very naturally to writing tableaux. That is, suppose we wanted to write the product of the (2) (2) (2) above element, (x1 x2 x3 |y1 y2 y3 ) with, say, (x3 x1 |y1 y2 y3 ). As we saw above, each of these terms is a sum of a number of addends, so that the notation we have for each of these terms is already of considerable convenience. But now, instead of using juxtaposition to denote the product of these two terms, let us use “double tableau” notation, that is, let us write

(2)

x1 x2 x3 (2) x3 x1

(2)

y1 y1

y2 y2

y3 y3

for this product. Suppose that we choose an ordered basis for F , say {x1 , . . . , xm } with x1 < · · · < xm , and let us say that the elements xi above are among these basis elements. Then the double tableau above does not change value if we write it as: (DT )

(2)

x1 x2 x3 (2) x1 x3

(2)

y1 y1

y2 y2

y3 y3

.

We point this out to indicate that we may always assume that our tableaux are given in such a way that in each row, the elements are increasing. The terminology for this is that the tableaux are row-standard (a notion that we have already encountered in Chapter III). Recall that in Chapter III, we wrote out the rows of the tableau repeating letters instead of using divided powers. This helps to talk about the columns of a tableau; for instance, the tableau above has two rows and four columns (the number refers to the arrays in the letters as well as the places). Usually, we call the basis of F the letters, while the basis of G is called the places. A basic word of degree k is simply a basis element of Dk (F ), while a word of degree k is a linear combination of basic words of degree k. Usually

Letter-place algebra

we will write a word as w, and  w1  w2 (G)   ··· wn

159

we will write a general double tableau as  1(a11 ) 2(a21 ) 3(a31 ) · · · 1(a12 ) 2(a22 ) 3(a32 ) · · ·   ··· ··· ··· ···  1(a1n ) 2(a2n ) 3(a3n ) · · ·

where αi = (a1i + a2i + a3i + · · · ) ≥ αj for 1 ≤ i < j ≤ n, and we have written i for yi . We will continue to write i for yi as long as there is no danger of confusion. Also, in most cases, the words wi will be basic words, in which case (since they are basis elements of Dk (F )), they are increasing. Because of our convention (♠) above, we see that we may assume that the degree of the element wi is equal to our tableau is an element of Dk1 (F ) ⊗ · · · ⊗ Dkn (F ) when, for each αi . Note that j = 1, . . . , n, l ajl = kj . We will call a double tableau standard if the words wi are basic, the lengths of the rows are decreasing (from the top), it is row-standard, and also columnstandard in the sense that when we have used repeat notation instead of divided powers, the columns are strictly increasing from top to bottom. Our double tableau (DT ) above is not a standard double tableau; if we replace the element x1 in the second row by x2 , however, it will be standard. Clearly there is a set of double tableaux that are a basis for Dk1 (F ) ⊗ · · · ⊗ Dkn (F ), namely:   w1 1(k1 )  w2 2(k2 )   (W )   ··· ···  wn n(kn ) where the wi run through the basis elements of Dki (F ). But these tableaux are not in general standard. Even if it were the case that k1 ≥ · · · ≥ kn , so that the “place” side of the tableau were standard, the “word” side of the tableau would in general not be so. And if we had to reorder the rows so that they were decreasing in length, we would upset standardness in the column of places. What we do have is the following theorem. Theorem VI.3.2 The set of standard double tableaux having the ith place counted ki times is a basis for Dk1 (F ) ⊗ · · · ⊗ Dkn (F ). The proof breaks up into two parts: the double tableaux of type (G), with l ajl = kj , generate, and the number of such tableaux is equal to the number of tableaux of type (W ) above, for ﬁxed k1 , . . . , kn . The ﬁrst part is given in Appendix A, Section 1, and the second part in Appendix A, Section 2. VI.3.2 Negative places and the exterior algebra Now we take up the letter-place approach to the tensor product of a ﬁxed number of copies of ΛF for a ﬁxed free module, F . As in the previous discussion, we

160

Weyl and Schur modules

use the fact that Λ(F ⊗ Rn ) ∼ ⊕ · · · ⊕ F ), which is, in turn, isomorphic = Λ(F &# $ % n

to ΛF ⊗ · · · ⊗ ΛF . There are two natural ways to proceed with this discussion &# $ % n

from a letter-place point of view: we could make the letters of F be positive and the places of Rn negative, or vice versa. We will deal with the ﬁrst case in some detail, and indicate the necessary changes if we reverse sign. Take the basis of Rn to be {1, . . . , n}, but this time we will treat them as “negative” places (in fact, we have written them in bold face to distinguish them from the “positive” places of the previous subsection). To make the meaning of this clearer (if not altogether clear), we can think of the bases of our free modules as “alphabets” from which we make “words” by stringing them together (as we have been doing). But we can also think of the letters of our alphabet as being “signed,” that is, either positive or negative. In the preceding discussion of tensor products of divided powers, all of our letters and places were positive, so that we can assign the number 0 to all of them (to indicate that they are positive). However, in this case, we want to consider the basis elements of F as positive, while those of Rn as negative. So, we assign the value 0 to the basis elements of F , and we assign the value 1 to the basis elements 1, . . . , n to indicate that they are negative. In general, if you have signed alphabets A and B which are the bases of A and B, respectively, then the element a ⊗ b ∈ A ⊗ B is assigned the value |a ⊗ b| = |a| + |b| mod 2, where |x| stands for the sign of x. Of course, we will write the element a ⊗ b as (a|b) when we adopt the “letter-place” language as we did in the foregoing subsection. As before, then, we will write the element (x|i) to stand for the element x ⊗ i ∈ Λ(F ⊗ Rn ), where x is a basis element of F . We think of this, under the identiﬁcations made above, as the element 1 ⊗ · · · ⊗ x ⊗1 ⊗ · · · ⊗ 1 ∈ ΛF ⊗ · · · ⊗ ΛF . &# $ % % &# $ i

n

Since x has sign 0 and i has sign 1, the sign of (x|i) is 0 + 1 = 1. From the identiﬁcations we have made, we see that (x|i)(y|i) = −(y|i)(x|i). This, and the commutativity of multiplication in the case of divided powers is consistent with the sign convention: (a1 |b)(a2 |b) = (−1)|(a1 |b)||(a2 |b)| (a2 |b)(a1 |b). Our object is to work toward the same sort of double tableau notation for this tensor product that we had earlier. But before it was possible to take a positive place, i, say, and consider the element i(2) as in (xy|i(2) ). In this case, since a place i is negative, we see that i(2) = 0 (recall Remark VI.3.1), so we have to deﬁne what we mean by the element (w|p1 ∧ · · · ∧ pk ) where w is an element (word) of a basis of Dk F , and p1 , . . . , pk are distinct basis elements of Rn (so that p1 ∧ · · · ∧ pk is plus or minus a basis element of Λk Rn ). (k ) (k ) Suppose that w = a1 1 · · · al l , let k = k1 + · · · + kl and let b1 , . . . , bk be the sequence a1 , . . . , a1 , . . . , al , . . . , al . Let Sk1 ,...,kl denote the Young subgroup of the % &# $ % &# $ k1

kl

Letter-place algebra

161

symmetric group Sk consisting of those permutations that permute the ﬁrst k1 elements of 1, . . . , k among themselves, the next k2 elements among themselves, and so on. (This is a subgroup isomorphic to Sk1 ×· · ·×Skl consisting of k1 ! · · · kl ! elements.) Then we deﬁne (F I) (w|p1 ∧ · · · ∧ pk ) = (bσ(1) |p1 ) · · · (bσ(k) |pk ), σ

where σ runs through representatives of distinct cosets of Sk /Sk1 ,...,kl . In the summation above we have written the product in our exterior algebras as simple juxtaposition instead of using wedges. We do this to conserve a uniform notation for multiplication in the letter-place algebra, in which (as we will see later) letters and places may sometimes be positive and sometimes negative. A few simple examples will make this completely clear. • Consider (a(2) |p1 ∧ p2 ). We have

(a(2) |p1 ∧ p2 ) = (a|p1 )(a|p2 ). • Consider (a1 a2 |p1 ∧ p2 ). We have

(a1 a2 |p1 ∧ p2 ) = (a1 |p1 )(a2 |p2 ) + (a2 |p1 )(a1 |p2 ). • Consider (a(2) b(3) |p1 ∧ · · · ∧ p5 ). We have

(a(2) b(3) |p1 ∧ · · · ∧ p5 ) = (a|p1 )(a|p2 )(b|p3 )(b|p4 )(b|p5 ) + (a|p1 )(b|p2 )(a|p3 )(b|p4 )(b|p5 ) + (a|p1 )(b|p2 )(b|p3 )(a|p4 )(b|p5 ) + (a|p1 )(b|p2 )(b|p3 )(b|p4 )(a|p5 ) + (b|p1 )(a|p2 )(a|p3 )(b|p4 )(b|p5 ) + (b|p1 )(a|p2 )(b|p3 )(a|p4 )(b|p5 ) + (b|p1 )(a|p2 )(b|p3 )(b|p4 )(a|p5 ) + (b|p1 )(b|p2 )(a|p3 )(a|p4 )(b|p5 ) + (b|p1 )(b|p2 )(a|p3 )(b|p4 )(a|p5 ) + (b|p1 )(b|p2 )(b|p3 )(a|p4 )(a|p5 ), in other words, the 10 = 5!/2!3! terms that correspond to the ten distinct cosets of S5 /S2,3 . An important observation to make here, as we made before, when we use (k ) (k ) tableau notation as above (i.e.: (w|p1 ∧ · · · ∧ pk )), where w = a1 1 · · · al l , the word w is entered as our row-array b1 . . . bk . With this notation taken care of, we can consider double tableaux as we did earlier, but now the left side consists of words in the positive alphabet which is

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the basis of F , and the left side consists of words in the negative alphabet of places, the basis {1, . . . , n} of Rn . To see that our usual basis elements of Λk1 F ⊗ · · · ⊗ Λkn F can be expressed as double tableaux, consider the following example. • The element x2 ∧ x3 ∧ x5 ⊗ x1 ∧ x3 ⊗ x2 ∧ x4 ⊗ x3 ∧ x5 ∧ x6 ∈ Λ3 F ⊗ Λ2 F ⊗

Λ2 F ⊗ Λ3 F can be expressed as the double tableau:  (2)  x2 1 3  (3)   x3  1 2 4   (2)   x 1 4   5  x1 2     x4 3  x6 4

On the right hand side of the tableau we have omitted the wedge, and simply spread the basis elements out along the row. We used the divided power notation on the left hand of the column to simplify writing. Really, the top row of the tableau above should look like: (x2 x2 |1 3). In our situation, we see that if we interchange rows of the tableau, we must take sign into account. For example:  (2)  (2)   x2 x2 1 3 1 3  (3)  (3)    x3  x3  1 2 4 1 2 4   (2)  (2)    x    1 4 1 4  5  = −  x5   x1 2  x1 2        x4 3   x6 4  x6 4 x4 3 As in the previous case, we now have to deﬁne what we mean by a double standard tableau. We will call a double tableau standard if it is standard in the old sense on the left hand side of the vertical column, but on the right hand side, has the property that it is strictly increasing in the rows, and weakly increasing in the columns. Notice that this deﬁnition implies that the shape of the tableau is that of a partition. For instance, the tableau of the above example is very non-standard for all the possible reasons: the shape is not that of a partition (i.e., the lengths of the rows are not decreasing from top to bottom); on the left side the columns are not strictly increasing; nor are they on the right side. However, both sides of the tableau are row-standard: on the left side, the rows are weakly increasing, while on the right they are strictly increasing. These are the requisite conditions for row-standard when we are talking about positive and negative alphabets.

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An example of a standard double tableau would be the following:   (2) x3 x4 1 2 3   E :  x(2) , 1 3 5 3 x6 where we are again cheating a bit in that we are writing the left hand side in terms of divided powers rather than as spread-out sequences. Just to make sure that we understand what all the notation means, we remark that the tableau above is equal to: E = x3 ∧ x5 ⊗ x3 ⊗ x4 ∧ x5 ∧ x6 + x3 ∧ x5 ⊗ x4 ⊗ x3 ∧ x5 ∧ x6 + x4 ∧ x5 ⊗ x3 ⊗ x3 ∧ x5 ∧ x6 . In the same way the double standard tableaux generate the tensor product of divided powers, these double standard tableaux generate the tensor product of exterior powers. We have the following theorem. Theorem VI.3.3 The set of standard double tableaux having the ith place counted ki times is a basis for Λk1 F ⊗ · · · ⊗ Λkn F . A sketch of the proof of this theorem is in Section 3 of Appendix A. It should be fairly clear that the discussion above could just as well have been carried out if we had assumed that the alphabet for F were signed negatively, and that for the places signed positively. In that case, we would have simply written the basis elements of F in boldface, and those for Rn in ordinary typeface. There are one or two diﬀerences that we would have to remark in this case. One is that we would set (x1 |i)(x2 |i) = (x1 ∧ x2 |i(2) ). Another is that we would modify the fundamental identity (F I) earlier in this section as follows. If w were the word (k ) (k ) w = x1 ∧ · · · ∧ xk , and we had p = p1 1 · · · pl l with k = k1 + · · · + kl , then setting {q1 , . . . , qk } equal to the sequence {p1 , . . . , p1 , . . . , pl , . . . , pl }, we deﬁne % &# $ % &# $ k1

(F I)

(w|p) =

kl

(x1 |qσ(1) ) · · · (xk |qσ(k) ),

σ

where σ ranges over representatives of the cosets of the appropriate Young subgroup. However, if we prefer to keep the places in their original order (so that we could write the right-hand side of the term as a product of divided powers, say), then the order of the terms q1 , . . . , qk would be changed, and that would introduce a

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sign. For example, if we took (x1 ∧ x2 ∧ x3 |1(2) 2), we would get (x1 ∧ x2 ∧ x3 |1(2) 2) = (x1 |1)(x2 |1)(x3 |2) − (x1 |1)(x3 |1)(x2 |2) + (x2 |1)(x3 |1)(x1 |2). For “standardness” of double tableaux we would have strictly increasing rows in the letters, weakly increasing rows in the places; weakly increasing columns in the letters, strictly increasing columns in the places. The proof that these double standard tableaux form a basis is indicated in Section 3 of Appendix A. VI.3.3 The symmetric algebra (or negative letters and places) There is one last canonical algebra to consider, namely the tensor product of a ﬁxed number of copies of the symmetric algebra of F : S(F ) ⊗ · · · ⊗ S(F ). In &# $ % n

this case, we consider the basis elements of both F and Rn negative (but we will not bother to write these in boldface). Thus, an element (x|i) will have sign 0, and hence will always be positive (a not surprising development given that we are treating the symmetric algebra). Consequently, we write (x1 |i)(x2 |i) = (x2 |i)(x1 |i) and for w = l1 ∧ · · · ∧ lk , p = p1 ∧ · · · ∧ pk , we deﬁne (F I) (w|p) = (x1 |pσ(1) ) · · · (xk |pσ(k) ). σ

where now the σ runs over all permutations of {1, . . . , k}. With this identity, we get the usual basis theorem for the n-fold tensor product of the symmetric algebra: Theorem VI.3.4 The set of standard double tableaux having the ith place counted ki times is a basis for Sk1 F ⊗ · · · ⊗ Skn F . The reader is again referred to Section 3 of Appendix A for a brief indication of the proof. VI.3.4 Putting it all together We now want to put these various pieces together, and consider what happens when we have “letter alphabets” and “place alphabets” that contain both positive and negative elements. To use more descriptive notation, we will let L and P stand for the letter and place alphabets, respectively. Further, we will suppose that L = L+ "L− and P = P + "P − , where the plus and minus superscripts indicate the signs of the elements of these alphabets. If we now let L+ , L− , P + , P − stand for the free modules generated by these alphabets (or bases), we may consider what is called the letter-place superalgebra: S(L|P) = Λ(L+ ⊗ P − ) ⊗ Λ(L− ⊗ P + ) ⊗ D(L+ ⊗ P + ) ⊗ S(L− ⊗ P − ).

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The individual factors of the tensor product above have been described in detail; the product of two terms from diﬀerent components of the product is simply the tensor product of these terms, while the product (l1 |p)(l2 |p) = (−1)|(l1 |p)||(l2 |p)| (l2 |p)(l1 |p). VI.4 Place polarization maps and Capelli identities In Section VI.2 we deﬁned the Weyl and Schur maps, which entailed a good deal of diagonalization, identiﬁcation and multiplication from a tensor product of divided (exterior) powers to a tensor product of exterior (symmetric) powers. We now know that these tensor products of various powers can be expressed in letter-place terms, and we may ask if these complicated maps may be viewed in a diﬀerent way (hopefully, a simpler way) using the letter-place approach. The answer, as was no doubt anticipated, is yes, and the method will be that of place polarizations. In this section, we will consider two types of maps, both of which are called place polarizations: those from “positive places to positive places” and those “from positive places to negative places”. Deﬁnition VI.4.1 Let q ∈ P + , s ∈ P, s = q, and let (l|p) be a basis element of S(L|P). Deﬁne the place polarization, ∂s,q , to be the unique derivation on S(L|P) deﬁned by ∂s,q (l|p) = δq,p · (l|s), where δq,p is the Kronecker delta. When we say that this map is a derivation on S(L|P), we mean that it has the property ∂s,q {(l1 |p1 )(l2 |p2 )} = {∂s,q (l1 |p1 )}(l2 |p2 ) + (−1)|s||p1 | (l1 |p1 )∂s,q (l2 |p2 ). 2 = 0. A straightforward calculation shows that if s is a negative place, then ∂s,q 2 On the other hand, we can see easily that if s is positive, ∂s,q {(l1 |q)(l2 |q)} = 2{(l1 |s)(l2 |s)}, so that for q and s positive places, it makes sense to talk about (k) the higher divided powers of the place polarizations, ∂s,q , namely ∂s,q . In the case of the divided square just discussed, for instance, we see that the equation (2) may be interpreted as ∂s,q (l1 l2 |q (2) ) = (l1 l2 |s(2) ). In general, then, we have (k) (w|q (m) ) = (w|q (m−k) s(k) ), ∂s,q

where q and s are positive places. One fundamental identity, which is easy to prove, is the following. Fact 1 Let p, q, r be places with q and p positive, and consider the place polarizations ∂r,q , ∂q,p and ∂r,p . Then ∂r,p = ∂r,q ∂q,p − ∂q,p ∂r,q .

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In short: ∂r,p is the commutator of ∂q,p and ∂r,q . Proof It is clearly enough to consider these maps on a term of the form (w|p(a) q (b) r(c) . . .). If r is negative, then c can be 0 or 1, while if r is positive, there is no restriction on c. Let us suppose ﬁrst that r is positive. We have ∂r,p (w|p(a) q (b) r(c) . . .) = (w|p(a−1) q (b) rr(c) . . .) = (c + 1)(w|p(a−1) q (b) r(c+1) . . .). But we have ∂r,q ∂q,p (w|p(a) q (b) r(c) . . .) = (b + 1)∂r,q (w|p(a−1) q (b+1) r(c) . . .) = (b + 1)(c + 1)(w|p(a−1) q (b) r(c+1) . . .), and ∂q,p ∂r,q (w|p(a) q (b) r(c) . . .) = (c + 1)∂q,p (w|p(a) q (b−1) r(c+1) . . .) = b(c + 1)(w|p(a−1) q (b) r(c+1) . . .). Taking the diﬀerence, we get the desired result. If we assume that r is negative, and c = 1, we get zero when we apply the maps. On the other hand, if c = 0, the proof proceeds as it did for positive r. 2 We start with the case of positive-to-positive place polarizations. Assume that our places p, q and r are all positive. Then as we know, we can form the divided powers of all of the place polarizations involving these places, and ask if there are identities associated to these that generalize the basic identity proved above. Proposition VI.4.2 (Capelli Identities) Let p, q, r be places with p, q and r all positive, and consider the place polarizations ∂r,q , ∂q,p and ∂r,p . Then (a) (b) (b−k) (a−k) (k) (Cap) ∂r,q ∂q,p = ∂q,p ∂r,q ∂r,p ; k≥0

(Cap )

(b) (a) ∂q,p ∂r,q =

(a−k) (b−k) (k) (−1)k ∂r,q ∂q,p ∂r,p .

k≥0

Proof For a = b = 1, this is just the commutation rule (for both Cap and Cap ). One can now proceed, by direct calculation, or we can employ a standard trick and prove this by induction, ﬁrst on a for b = 1, and then for all a and b. The standard trick is to behave as though we were working over the ﬁeld of rational numbers (which allows us to divide by integers), and then assert that since our proposition works there, it works in general (see Remark VI.3.1). We will therefore proceed to prove our proposition by induction. As we pointed out, our proposition is true for a = b = 1, so we may let b = 1 and assume the

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167

identity true for a − 1. We then see that (a−1) (a−1) (a) ∂q,p = ∂r,q ∂r,q ∂q,p = ∂r,q {∂q,p ∂r,q + ∂r,p } a∂r,q (a−1) (a−1) ∂q,p ∂r,q + ∂r,q ∂r,p = ∂r,q (a−1) (a−2) (a−1) = {∂q,p ∂r,q ∂r,q + ∂r,q ∂r,p ∂r,q } + ∂r,q ∂r,p (a) (a−1) (a−1) = a∂q,p ∂r,q + (a − 1)∂r,q ∂r,p + ∂r,q ∂r,p (a) (a−1) = a{∂q,p ∂r,q + ∂r,q ∂r,p }.

Dividing through by a, we get our identity (Cap) for b = 1. We have used, (a) (b) without explicitly mentioning it in doing the calculation, that ∂r,p and ∂r,q commute, but this, we believe, is clear. Now, assuming it true for all a and for b − 1, a similar inductive argument proves the statement true for all a and b. 2 The proof of (Cap ) proceeds in the same way. We next turn to positive-to-negative place polarizations. In this case, we consider what happens if r is a negative place, with both p and q still positive. If we look at the above proof, and keep in mind that higher divided powers of positive-to-negative polarizations are zero, the identities (Cap) and (Cap ) make sense only when a = 1, and our proof in the case a = 1 above, is still valid. Hence we have the following proposition. Proposition VI.4.3 (Capelli Identities) Let p, q, r be places with p, q positive, and r negative. Consider the place polarizations ∂r,q , ∂q,p and ∂r,p . Then (Cap)+

(b) (b) (b−1) ∂r,q ∂q,p = ∂q,p ∂r,q + ∂q,p ∂r,p ,

(Cap )−

(b) (b) (b−1) ∂q,p ∂r,q = ∂r,q ∂q,p − ∂q,p ∂r,p .

VI.5 Weyl and Schur maps revisited Recall the set-up for the deﬁnitions of the Weyl and Schur maps. We let F be a ﬁnite free module over the commutative ring, R. For the n × m shape matrix n m A = (aij ), set pi = j=1 aij , γj = i=1 aij . The Weyl map associated to A, ωA , is a map ωA : Dp1 F ⊗ · · · ⊗ Dpn F → Λγ1 F ⊗ · · · ⊗ Λγm F that we deﬁned using many diagonalizations, identiﬁcations, and multiplications. Similarly, we deﬁned the Schur map σA : Λp1 F ⊗ · · · ⊗ Λpn F → Sγ1 F ⊗ · · · ⊗ Sγm F. We now maintain that these maps can be described using place polarizations; in particular, positive-to-negative place polarizations.

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For the Weyl map, we are going to consider the basis, L+ , of F as a positive letter alphabet (in the letter-place language), and our place alphabet P = P + " P − , where P + = {1, . . . , n} and P − = {1, . . . , m}. For the Schur map, we are going to regard the basis of F as a negatively signed letter alphabet, L− , and our place alphabet the same as the above. We next observe that S(L+ |P) = D(F ⊗ Rn ) ⊗ Λ(F ⊗ Rm ), which contains the subalgebras S(L+ |P + ) = D(F ⊗ Rn ) and S(L+ |P − ) = Λ(F ⊗ Rm ). Our discussion of the letter-place algebra tells us that D(F ⊗ Rn ) = DF ) ⊗ · · · ⊗ DF &# $ % n

while Λ(F ⊗ Rm ) = ΛF ⊗ · · · ⊗ ΛF . A similar discussion applies to the algebra &# $ % m

S(L− |P) = Λ(F ⊗ Rn ) ⊗ S(F ⊗ Rm ). What we will show is that our Weyl (or Schur) maps are compositions of place polarizations that take us from our desired domain to our desired target through S(L+ |P) (or S(L− |P)). Although we can carry out this project for arbitrary shapes, we will restrict ourselves to the class of shapes we have already discussed in Section VI.1, namely almost skew-shapes. Recall that an almost skew-shape can be represented as λ/µ where λ is a partition and µ is an almost partition. In order to conform to the notation used to describe the shape matrix, A, above, we will assume that our partition λ has length n, and that λ1 − µn = m if µ is a partition, and that λ1 − µn−1 = m if µ is not a partition. A quicker way to say this is that λ1 − min(µn , µn−1 ) = m. As we have noted before, we may as well set min(µn , µn−1 ) = 0. Using this notation for our shapes, we see that the numbers pi and γj above become: pi = λi − µi ;

˜j − µ γj = λ ˜j ,

for i = 1, . . . , n and j = 1, . . . , m, where the tilde denotes the transpose shape matrices of λ and µ. For each i = 1, . . . , n, let ∆i = ∂λi ,i · · · ∂µi +1,i . (Recall that we are assuming that min(µn , µn−1 ) = 0, so that m = λ1 .) Now we set ∆λ/µ = ∆n · · · ∆1 . We see that each ∆i is a composition of positive-to-negative place polarizations from the positive place, i, to the negative places µi + 1 to λi . Hence the map ∆λ/µ is a composition of such place polarizations from 1, . . . , n to 1, . . . , m. We see, therefore, that the image of ∆λ/µ is contained in that part of S(L+ |P) which contains no positive places, namely in Λ(F ⊗ Rm ) or, what is the same thing, it

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169

is a map ∆λ/µ : DF ⊗ · · · ⊗ DF → ΛF ⊗ · · · ⊗ ΛF . % % &# $ &# $ n

m

If we restrict it to Dp1 F ⊗ · · · ⊗ Dpn F , it is immediate to see that we end in Λγ1 F ⊗ · · · ⊗ Λγm F . It is laborious but straightforward to prove that this last map is the same as the Weyl map ωA for A = λ/µ; we will sketch a procedure for carrying out such an argument. We know that a basis for Dp1 F ⊗ · · · ⊗ Dpn F consists of double tableaux   w1 1(p1 )  w2 2(p2 )  . (W )   ··· ···  wn n(pn ) The result of applying ∆λ/µ to such a tableau  w1 µ1 + 1 · · ·  w2 µ2 + 1 · · ·   ··· ··· ··· wn µn + 1 · · ·

yields the tableau  λ1 λ2  . ···  λn

If one now reads this tableau as the element one obtains by diagonalizing wi over the negative places µi + i, . . . , λi and multiplying, one sees that this is precisely the deﬁnition of the map ωλ/µ . The discussion of the Schur map is identical to this one, with the proviso that we now consider the letters to be negative. However, we are still going from positive places to negative ones, in exactly the same way, so that while the domain and range of the Weyl and Schur maps are diﬀerent, the expression of them as composites of place polarizations is identical. VI.6 Some kernel elements of Weyl and Schur maps In this section, we will deﬁne some maps from the sum of tensor products of divided powers (exterior powers) to the domain of the Weyl (Schur) map, and show that the images are in the kernel of the Weyl (Schur) map. These maps are what were called in Reference [3] the “box map”; here we will see that they are expressible in terms of positive-to-positive place polarizations. Consider our almost skew-shape λ/µ : λ = (λ1 , . . . , λn ), µ = (µ1 , . . . , µn ). Remember that the shape is of type τ = n − (i + 1) if i is the largest integer diﬀerent from n such that µn ≤ µi . Thus, τ = 0 means that λ/µ is a skew-shape; τ > 0 means that the bottom row of the diagram of λ/µ is indented on the left from the penultimate row. We will introduce some more notation that we will use uniformly when we discuss these almost skew-shapes.

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Weyl and Schur modules

Notation (almost skew-shapes) We will set ti = µi − µi+1 for i = 1, . . . , n − 1. If τ = 0, this means that µn ≤ µn−1 and tn−1 = µn−1 − µn = µn−1 . If τ > 0, this means that µn−1 − µn = −µn < 0; moreover there is an i = n − 1 − τ such that µi+1 < µn ≤ µi , and we set s = µn − µi+1 . Finally, we denote our shape λ/µ by the notation (p1 , . . . , pn ; t1 , . . . , tn−1 ). With this notation, we see that the diagram of an almost skew-shape of type τ = n − (i + 1) > 0 looks like this: t1 .. .

.. .

ti .. .

.. .

.. .

tn−2

.. .

p1 p2 .. . pi pi+1 .. . pn−2 pn−1 pn

tn−2 + · · · + ti+1 + s

with 0 < s ≤ ti . Of course, tn−2 + · · · + ti+1 + s = µn − µn−1 = −tn−1 > 0. We will now restrict ourselves to the Weyl case until the end of this section, where we indicate how the results apply to the Schur case as well. Assume that our shape (p1 , . . . , pn ; t1 , . . . , tn−1 ) is a skew-shape, that is, assume that tn−1 ≥ 0. For each i = 1, . . . , n − 1, and for each ki > 0, we consider the module Dp1 ⊗ · · · ⊗ Dpi +ti +ki ⊗ Dpi+1 −ti −ki ⊗ · · · ⊗ Dpn and the (positive-to-positive) place polarization (t +k )

i i : Dp1 ⊗ · · · ⊗ Dpi +ti +ki ⊗ Dpi+1 −ti −ki ⊗ · · · ⊗ Dpn → Dp1 ⊗ · · · ⊗ Dpn . ∂i+1,i

Here, and from now on in most cases, we omit the underlying free module, F , from our notation. Deﬁne λ/µ,i to be the map λ/µ,i : Dp1 ⊗ · · · ⊗ Dpi +ti +ki ⊗ Dpi+1 −ti −ki ⊗ · · · ⊗ Dpn → Dp1 ⊗ · · · ⊗ Dpn , ki >0 (t +k )

i i which, on each summand, is equal to ∂i+1,i . Now deﬁne Rel(λ/µ) = Dp1 ⊗ · · · ⊗ Dpi +ti +ki ⊗ Dpi+1 −ti −ki ⊗ · · · ⊗ Dpn ,

i

ki

where the sum is taken over i = 1, . . . , n − 1, and all positive ki . And now deﬁne λ/µ : Rel(λ/µ) → Dp1 ⊗ · · · ⊗ Dpn to be the map which, for each i, is the map λ/µ,i .

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171

In short, λ/µ is the sum of many, many place polarizations. We will often write Rel(p1 , . . . , pn ; t1 , . . . , tn−1 ) for Rel(λ/µ) when we want to make the data for the shape more explicit. The reason for this elaborate notation is that we will eventually show that the image of the map λ/µ is the kernel of the Weyl map ∆λ/µ = ωλ/µ . For an almost skew-shape of type τ > 0, the kernel of the Weyl map will be given by relations of the kind above, plus τ additional kinds of terms. It is evident from the deﬁnition of the map λ/µ above that the relations on the Weyl map for a skew-shape involve shuﬄing between consecutive pairs of rows of the shape. The additional terms that we must consider for the almost skew-shape of type τ > 0 involve shuﬄing between the last row and those rows beyond which it does not protrude (to the left), as well as the lowest row beyond which it does protrude. In our diagram of the almost skew-shape of type τ > 0, this means that we have to shuﬄe the last row with the rows from n − 2 up through the ith. This makes n − (i + 1) = τ rows, and hence τ kinds of terms to describe these shuﬄes. We now formally describe these additional terms. For j = i + 1, . . . , n − 2, deﬁne tj Dp1 ⊗ · · · ⊗ Dpj +kj ⊗ Dpj+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn −kj #λ/µ,j : kj =1

→ Dp1 ⊗ · · · ⊗ Dpn (k )

to be the map which on each component is the place polarization ∂n,jj , and for i = n − (τ + 1), deﬁne s #λ/µ,i : Dp1 ⊗ · · · ⊗ Dpi+ti−s+k ⊗ Dpi+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn−(ti−s)−k k=1

→ Dp1 ⊗ · · · ⊗ Dpn to be, again, the map which on each component is the place polarization (t −s+k) ∂n,ii . We next deﬁne, for an almost skew-shape, λ/µ of type τ > 0, the overall relations Rel(λ/µ) = Rel(p1 , . . . , pn ; t1 , . . . , tn−1 ) by: Rel(p1 , . . . , pn ; t1 , . . . , tn−2 , 0) n−2

tj

Dp1 ⊗ · · · ⊗ Dpj +kj ⊗ Dpj+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn −kj

j=i+1 kj =1 s

Dp1 ⊗ · · · ⊗ Dpi +ti −s+k ⊗ Dpi+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn −(ti −s)−k ,

k=1

and the map λ/µ : Rel(p1 , . . . , pn ; t1 , . . . , tn−1 ) → Dp1 ⊗ · · · ⊗ Dpn

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Weyl and Schur modules

in the by now obvious way. We point out that Rel(p1 , . . . , pn ; t1 , . . . , tn−2 , 0) are the relations of the skewshape obtained from the almost skew-shape by sliding the last row ﬂush with the penultimate one. An easy way to think about the additional terms that we have to add to the relations is this. Think of removing all the rows between the bottom one and the jth. In that case, the bottom row is still pushed in from the jth one, and so push it ﬂush with it. The relations we write down are the terms that we would get from the relations on that skew-shape, except that we interpolate the rows that we removed. The last exceptional term, that is, the ith one (where τ = n − i − 1), can also be thought of in the same way: we remove the intervening rows, and observe that the last row is now protruding ti −s units beyond the jth. Thus, the relations we write down are precisely those we would have from that skew-shape after reinserting the excised rows. Of course, one might ask why the summations in these terms do not go beyond tj for the ﬁrst type of term, or beyond s for the second. The fact is that one could remove the upper limit of summation, as we shall see, but what we are trying for is the smallest number of terms that we know we can get away with. To see that the upper limit on the sums is not essential, we observe that an immediate corollary of the Capelli identity (Cap ) is the following: (v) (v) (v) (v−l) (l) (l) = (−1)v ∂q,p ∂r,q − (−1)l ∂r,p ∂r,q ∂q,p . ∂r,p l>0

Thus, the image of a polarization of high order from one place to another far down the line is already in the image of one of the same order from, say, adjacent places, plus the images, by lower order polarizations, between the original places. This is what permits us to limit the index of summation. The main focus of this section is the proof of the following essential result. Theorem VI.6.1

Let λ/µ be any almost skew-shape. Then the composition λ/µ

∆λ/µ

Rel(λ/µ) −→ Dp1 ⊗ · · · ⊗ Dpn −→ Λγ1 ⊗ · · · ⊗ Λγm is zero. That is, the image of λ/µ is contained in the kernel of the Weyl map. Proof The proof depends heavily on the Capelli identity (Cap) involving positive-to-negative polarizations. First assume that our shape is a skew-shape, that is, that τ = 0. We want to show that on each summand Dp1 ⊗ · · · ⊗ Dpi +ti +ki ⊗ Dpi+1 −ti −ki ⊗ · · · ⊗ Dn , (t +k )

i i is zero. Since ∆λ/µ is a composition of compositions of the map ∆λ/µ ∂i+1,i positive-to-negative place polarizations (recall that ∆λ/µ = ∆n · · · ∆1 ), we can (ti +ki ) commute ∂i+1,i past all the maps ∆1 , . . . , ∆i , and thus we have to study what

Some kernel elements of Weyl and Schur maps

173

(t +k )

i i the composition ∆n · · · ∆i+1 ∂i+1,i ∆i · · · ∆1 looks like. We further dissect this

(t +k )

i i composition, and focus on the parts surrounding ∂i+1,i , namely

(t +k )

i i ∂λi+1 ,i+1 · · · ∂µi+1 +1,i+1 ∂i+1,i ∂λi ,i · · · ∂µi +1,i .

Using recursion and the identity (Cap), one sees that (t +k )

i i ∂λi+1 ,i+1 · · · ∂µi+1 +1,i+1 ∂i+1,i

pi+1

=

(t +k −α)

i i ∂i+1,i

α=0

∂λi+1 ,i+1 · · · ∂ˆβ1 ,i+1 · · · ∂ˆβα ,i+1 · · · ∂µi+1 +1,i+1 ∂β1 ,i · · · ∂βα ,i

βα <···<β1

where µi+1 < βα < · · · < β1 ≤ λi+1 , and the hat over the ∂ means that term is to be omitted. Next we observe that if α < ti , then each term ∂λi+1 ,i+1 · · · ∂ˆβ1 ,i+1 · · · ∂ˆβα ,i+1 · · · ∂µi+1 +1,i+1 ∂β1 ,i · · · ∂βα ,i has at least pi+1 − ti polarizations from i + 1 to some negative place. But since our original element has the place i + 1 only pi+1 − ti − ki times, our element is carried to zero when α < ti . But if α ≥ ti , then the terms ∂β1 ,i · · · ∂βα ,i composed with ∆i polarize the positive place i to α + pi ≥ ti + pi negative places between µi+1 + 1 and λi . Since λi − (µi+1 + 1) = λi − µi + µi − µi+1 − 1 = pi + ti − 1, the composition of these polarizations must take i to the same negative place twice, which results in the zero map. Now we assume that τ > 0, so that we must consider the extra summands that comprise Rel(λ/µ). These are of two types: D1 ⊗ · · · ⊗ Dpj +kj ⊗ Dpj+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn −kj and D1 ⊗ · · · ⊗ Dpi +ti −s+k ⊗ Dpi+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn −(ti −s)−k , To see what happens to terms of the ﬁrst type under the composition, we again (k ) see that the map λ/µ applied to them is simply ∂n,jj . We therefore just have (k )

to look at ∆n ∂n,jj ∆n−1 . If we apply (Cap) as we did above, it is even easier to see that the composition is zero, since the place j is sent to the negative places µj + 1, . . . , λj , and µn > µj . For the term D1 ⊗ · · · ⊗ Dpi +ti −s+k ⊗ Dpi+1 ⊗ · · · ⊗ Dpn−1 ⊗ Dpn −(ti −s)−k , the same argument that was used above yields the result. 2 ¯ λ/µ to be the cokernel of λ/µ . Then the Corollary VI.6.2 Let us deﬁne K ¯ λ/µ → Kλ/µ . identity map on Dp1 ⊗ · · · ⊗ Dpn induces a map θλ/µ : K Proof This follows immediately from the result above.

2

All of the above discussion carries over to the Schur map and Schur modules, simply by replacing divided powers by exterior powers and exterior powers by

174

Weyl and Schur modules

symmetric powers. Or, if one wishes, one can simply replace the positive letter alphabet by its negative counterpart. All the maps that we deﬁne are in terms of the place alphabets, and these have not changed. VI.7 Tableaux, straightening, and the straight basis theorem The last theorem is a step toward giving us a presentation of our Weyl (Schur) modules: since the image of ∆λ/µ is the Weyl module, Kλ/µ , it suggests that perhaps the sequence Rel(λ/µ) → Dp1 ⊗ · · · ⊗ Dpn → Kλ/µ → 0 is exact. At least we know it is a complex. In this section, we will prove a basis theorem for our Weyl (Schur) modules, from which the exactness of the above sequence will follow. VI.7.1 Tableaux for Weyl and Schur modules The Weyl module corresponding to the shape, , is the image of D4 ⊗ D3 under the map ∆2 ∆1 , where ∆1 = ∂5,1 ∂4,1 ∂3,1 ∂2,1 ; ∆2 = ∂3,2 ∂2,2 ∂1,2 . Suppose {xi } is a basis for our free module, F (unspeciﬁed rank at this point), (2) and suppose we take the basis element of D4 ⊗ D3 : x2 x3 x4 ⊗ x1 x2 x4 . In our double tableau notation for D4 ⊗ D3 , this would be written x2 x2 x3 x4 1(4) , x1 x2 x4 2(3) and its image under ∆λ/µ would be x2 x2 x3 x4 x1 x2 x4

2 1

3 4 2 3

5

.

What we will do is write this element as x1

x2 x2

x2 x4

x3

x4

,

namely as a tableau. This may cause some initial confusion as the element we are representing by this tableau is in reality a sum of basis elements in Λ1 ⊗ Λ2 ⊗ Λ2 ⊗ Λ1 ⊗ Λ1 rather than simply a ﬁlling of a diagram. To be more meticulous,

Straightening and the straight basis theorem

175

we should really introduce some term such as Weyl-tableau to indicate that it is more than just a ﬁlled diagram. However, it will be clear from the context of our discussions, when we are using the term “tableau” in this extended sense, and when we are using it in the strictly combinatorial or typographic sense. This notation is not only the standard one used for these modules, but it is also extremely eﬃcient. All of the above carries over mutatis mutandis for Schur modules: the divided powers are replaced by exterior powers, and the exterior powers are replaced by symmetric powers. In addition, the positive letters are replaced by negative letters. The next deﬁnitions of various kinds of standardness and straightness of tableaux apply to tableaux of positive or negative letters; we will therefore introduce a notation that will apply to both cases simultaneously. Notation (signed inequalities) If A is a multi-signed alphabet, we say that a <+ b if a < b or a and b are positive, and a = b. We say that a <− b if a < b or a and b are negative, and a = b. In less formal language, a <+ b means, for example, that if a and b are both positive, then a ≤ b. Otherwise, a < b. With this notation we proceed with some deﬁnitions. Deﬁnition VI.7.1 We say that a tableau (of any shape) is row-standard if in each row it is <+ -increasing; we say it is column-standard if in each column it is <− -increasing. We say it is standard if it is both row- and column-standard. Deﬁnition VI.7.2 In a row-standard tableau, two elements aik , ajk (with i < j) in the same column are said to form (or be) an inversion if they violate column-standardness. That is, if aik is not <− ajk . The inversion is said to be unﬂippable if there is an element in the tableau, bik−1 , immediately to the left of aik , such that ajk <− bik−1 . Otherwise the inversion is called ﬂippable. The row-standard tableau is said to be straight if every inversion is unﬂippable. Clearly, since a standard tableau has no inversions, a standard tableau is necessarily straight. We do have a partial converse. Proposition VI.7.3 If λ/µ is a skew-shape and T is a row-standard tableau of that shape, then T is straight if and only if it is standard. Proof It suﬃces to show that if the tableau is not standard, then it is not straight. If T is not standard, it has an inversion, say aik is not <− ajk with i < j. Let us suppose that this inversion is the left-most one that exists in the tableau. If aik has no element immediately to the left of it, then there is no way the inversion is unﬂippable, so T is not straight. If we do have an element immediately to the left of aik , say bik−1 , then bik−1 lies above an element cjk−1 ,

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Weyl and Schur modules

since our shape is a skew-shape. But since the k th column is the ﬁrst one in which we have an inversion, the column to the left of it is <− -increasing. Hence bik−1 <− cjk−1 <+ ajk (the last inequality holding due to the row-standardness of T ). Thus the inversion is ﬂippable, and T is not straight. 2 From now on, we will focus our attention on Weyl modules; the appropriate changes for Schur modules will mostly be left to the reader. In the Weyl case, <+ -increasing means weakly increasing, while <− -increasing means strictly increasing. VI.7.2 Straightening tableaux What we want to do now is show that our Weyl modules are generated by the straight tableaux. There is a procedure that is called “straightening”, that is used to do this. Before we get into the general details, we give two examples of straightening. Example VI.7.4

Let us take as our ﬁrst example the tableau we have just x2 x2 x3 x4 . It fails to be standard because of an looked at, namely, x1 x2 x4 (2)

inversion in the second column. This element is the image of x2 x3 x4 ⊗x1 x2 x4 ∈ (3) D4 ⊗ D3 . However, let us look at x1 x2 x3 x4 ⊗ x4 ∈ D6 ⊗ D1 ⊂ Rel((5, 3)/(1, 0)), and at its image under (5,3)/(1,0) which, in this instance, means applying the (2) map ∂2,1 . We get (2)

(3)

(3)

(2)

x2 x3 x4 ⊗ x1 x2 x4 + x2 x4 ⊗ x1 x3 x4 + 2x2 x3 ⊗ x1 x4 + (2)

(2)

x1 x2 x3 x4 ⊗ x2 x4 + x1 x2 x4 ⊗ x2 x3 x4 + (2)

(2)

(3)

(2)

2x1 x2 x3 ⊗ x2 x4 + 2x1 x2 ⊗ x3 x4 . Notice that the ﬁrst summand is the term we started with. If we apply the Weyl map to this sum, we get zero by the theorem of the last section. This means that our original tableau is equal to the negative of the sum of the tableaux we obtain by writing the images of all the remaining summands as tableaux. One sees easily that all of these tableaux are standard, so that our original tableau is a linear combination of standard tableaux. Example VI.7.5

Another example to consider is the following tableau corx3 x3 x3 responding to an almost skew-shape of type 1: x2 x4 x4 . Here we x4 have an inversion that is ﬂippable (in the second column, second and third rows). In this case, we swoop up the third with the second row, and consider (3) (3) the element x3 ⊗ x2 x4 ⊗ 1 ∈ D3 ⊗ D4 ⊗ D0 ⊂ Rel((4, 3, 2)/1, 0, 1)). This

Straightening and the straight basis theorem

(3)

(2)

(3)

177

(3)

maps (under ∂3,2 ) to the sum: x3 ⊗ x2 x4 + x3 ⊗ x4 ⊗ x2 , one of whose summands is our original term. If we arrange these in tableaux, we have our x3 x3 x3 original tableau plus another, namely x4 x4 x4 , which is not straight x2 because of a ﬂippable inversion in the second column, ﬁrst and third row. We will see soon that this tableau is “better” than the one we started with, but we can go one step further to actually express our original tableau as a sum of straight ones. For now we can swoop up the third and ﬁrst rows together, getting (3) (3) −x2 x3 ⊗ x4 ⊗ 1 ∈ D4 ⊗ D3 ⊗ D0 ⊂ Rel((4, 3, 2)/(1, 0, 1)), and the image of (3) (3) (2) (3) this element under ∂3,1 is −x3 ⊗ x4 ⊗ x2 − x2 x3 ⊗ x4 ⊗ x3 . This shows that x2 x3 x3 our original tableau is equal to the straight tableau: x4 x4 x4 . x3 The main task is to formalize the argument underlying the procedure illustrated above. Let us assume, then, we have a tableau, T , which is not straight, which means it has some ﬂippable inversions. Of those, choose the ones furthest to the left, and of those, choose the one: aik ≥ ajk ; i < j, for which i is maximal and j − i is minimal. (Remember that in the case of skew-shapes, we would automatically get that j = i + 1, but not so here.) Since this inversion is ﬂippable, we either have no element aik−1 , or we do and aik−1 ≤ ajk . Suppose ﬁrst that aik−1 does not exist. Then if i < l < j, we must have aik < alk , for otherwise we should have a ﬂippable inversion, and the minimality of j −i says that this can not happen. But then alk > ajk , and by the maximality assumption on i, this must be an unﬂippable inversion. This means that alk−1 exists, and alk−1 > ajk . But our tableau is row-standard (we are always assuming that), so since ajk−1 ≤ ajk (assuming ajk−1 exists), we have the inversion alk−1 > ajk−1 . By the assumption on k, this inversion must be unﬂippable, so we have alk−2 > ajk−1 , and so on. Following this pattern, we see that the only way to avoid ﬂippable inversions between rows l and j is for row l to protrude further to the left than row j. This has to mean that j = n, and we have, for all l with i < l < j, that µl < µn ≤ µi which means, in particular, that the almost skew-shape is of type n − i − 1. This was under the assumption that aik−1 does not exist, and that j > i + 1. So now suppose that aik−1 does exist, but that we still have j > i + 1. If for some l with i < l < j we had aik ≥ alk , we would have aik−1 > alk , since the minimality of j − i forces this to be an unﬂippable inversion. But again, as above (since our tableau is row-standard), this forces us to conclude that µl > µi , and since l < j ≤ n, this is impossible. So we see that if j − i > 1, we must have, for all l with i < l < j, that aik < alk , which now implies that for all

178

Weyl and Schur modules

such l, alk > ajk , hence we have an inversion. Since (by maximality of i), this inversion must be unﬂippable, we run through the usual argument to conclude that µj > µl . Again we conclude that j = n, and the fact that our shape is of type ≥ n − i − 1. Let us observe too that if we have i < l < l < j, then we must have alk < al k . Because if not, then (by maximality of i), the inversion has to be unﬂippable, and by the usual leftward descent argument, we would conclude that µl < µl , which can not be since l < j ≤ n. So ﬁnally, let us assume that j − i = 1. In that case, we have no extra information except that if aik−1 exists, then aik−1 ≤ ai+1k . Furthermore, we have no conclusion to draw about the type of the shape. We will, for the sake of simplicity of writing, refer to this ﬂippable inversion of our non-straight tableau as the extreme ﬂippable inversion of T . There are a few more observations we should make about these extreme ﬂippable inversions. As we have seen, we have aik−1 ≤ ajk if aik−1 exists. Let α be the minimum value such that ajα exists and either aiα−1 does not exist, or aiα−1 ≤ ajα . If row i does not protrude further to the left than row j or, what is the same, µj ≤ µi , then α = µj + 1. If, on the other hand, µj > µi (in which case we must have j = n), then we see that µn + 1 ≤ α ≤ k. This number α will come into play when we describe our straightening algorithm. We should note that if ajα−1 exists, then ajα−1 = ajα since we have ajα−1 < aiα−2 ≤ aiα−1 ≤ ajα . Our second example above suggests that we ask what happens if aik−1 = aik . We saw in that example that we scooped up all of the second row along with the third row in order to avoid inconvenient coeﬃcients when we diagonalized. We have to take this situation into consideration; it is a rather special case, but merits special attention. If we have aik−1 = aik , then we have aik−1 = aik = ajk (since aik−1 ≤ ajk ). If ajk−1 exists, then we have aik−1 ≥ ajk−1 , so aik−2 must exist and be greater than ajk−1 . Thus, in this case, our α = k, and we also see that we must have j = n. (If ajk−1 does not exist, we certainly have α = k and j = n.) This analysis of the type of conﬁguration that can yield a non-straight tableau is what informs our straightening procedure. But before we actually describe this procedure, and maintain that applying it actually makes our tableaux “better”, we have to say what “better” means. Deﬁnition VI.7.6 Let T be a tableau. We deﬁne the column word of T , denoted uT , to be the word we obtain by writing down the elements of the columns of T starting from the bottom of the left-most column of T , working up that column, returning to the bottom of the next column, etc. We deﬁne the modiﬁed column word of T , denoted wT , to be the word obtained from T reading the columns in decreasing order. Finally, we deﬁne the reversed column word of T , denoted wT , to be the word obtained from T reading the columns in increasing order.

Straightening and the straight basis theorem

179

Just to be sure that there is no confusion about this deﬁnition, let us look at x3 x3 x3 . The column word is uT = x4 x2 x4 x3 x4 x3 x3 ; the the tableau, x4 x4 x4 x2 modiﬁed column word is wT = x4 x4 x3 x2 x4 x3 x3 , and the reversed column word is wT = x4 x2 x3 x4 x3 x4 x4 . Notice that the tableau above appeared in Example VI.7.5, but there it was being treated as an element of Kλ/µ . Here, however, we are regarding the tableau purely combinatorially. Deﬁnition VI.7.7 Given two tableaux, T and T , corresponding to the same diagram, we say that T < T if uT < uT in the lexicographic ordering of words. In Examples VI.7.4 and VI.7.5 above, the tableaux that were produced via straightening were all less than the original tableaux we started with. We are now set to show that the straight tableaux generate our Weyl module or, what is the same, that every tableau can be written as a linear combination (with integer coeﬃcients) of straight tableaux. This is what we mean by the straightening procedure referred to earlier. We proceed as we have done before: we take a tableau, T , assume that it is not straight, and then show that it is a linear combination of tableaux, T , which are all smaller than T in the order relation just deﬁned. Since we are dealing with ﬁnite sets, this will do the job. It will then remain to show that the straight tableaux are linearly independent, and hence form a basis for the Weyl module. We will always assume that we start with row-standard tableaux; when we read a tableau as an element of the Weyl module, if it were not row-standard, making it so would not change the element. Let T be a (row-standard) non-straight tableau, and assume that its extreme ﬂippable inversion occurs in adjacent rows, i and i+1 < n, where n is the number of rows in the tableau. Focusing our attention just on those two rows, we have the situation: T =

b1

t ···

at+1 bt+1

··· ···

ak−1 bk−1

ak bk

··· · · · bq

aq

,

with the inversion ak ≥ bk . (We are taking liberties here, calling just the two rows T , but we think the meaning is clear. We have also changed the names of the elements in the interests of clarity.) We have drawn the diagram as though the cell immediately to the left of ak exists; of course that need not be the case. But with our assumptions, t ≥ 0 and we cannot have ak−1 = ak . We now let w ∈ Dk−1−t be the word at+1 · · · ak−1 (meaning that if there are repeats they are considered as divided powers rather than simple powers), we let v ∈ Dq+1+ be the word b1 · · · bk · · · bk+ ak · · · aq , where bk = · · · = bk+ = bk++1 ( may be zero), and u ∈ Dq −k− the word consisting of bk++1 · · · bq .

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Weyl and Schur modules

From the Capelli identities, we see that the map (k−1−t) (k+) ∂3,2

: Dk−1−t ⊗ Dq+1+ ⊗ Dq −k− −→ D0 ⊗ Dq−t ⊗ Dq

∂2,1

applied to our element w ⊗ v ⊗ u gives us the element k−1−t

(k+−h)

(−1)h ∂3,2

(k−1−t−h) (h) ∂3,1 (w

(∂2,1

⊗ v ⊗ u)),

h=0

and we observe that (k−1−t−h) (h) ∂3,1 (w

∂2,1

⊗ v ⊗ u) ∈ D0 ⊗ Dq−t+k+−h ⊗ Dq −k−+h ,

with q − t + k + − h ≥ q + + 1 > q. We thus see immediately that (k−1−t) (k+) ∂3,2 (w ⊗ v ⊗ u) is in the kernel of the Weyl map. ∂2,1 To see how this helps us, we must look at just what this term looks like. First of all, we see that this term is a sum of tableaux, one of which is T itself. The other tableaux that occur will all have w in the top row, u in the bottom row, and the elements of v distributed around the top and bottom rows, as indicated (k+) by the polarization map ∂3,2 . To obtain a tableau other than T , we must either have some term among the b1 , . . . , bk replaced by terms from among ak , . . . , aq , or all of those elements as they stand, but again some of the ai terms must come down to the right of the bk . In either case, when one writes the column word for such a tableau, one sees easily that it is greater in the lexicographic order than the one for our original tableau, T . Since the sum of these tableaux is in the kernel of the Weyl map, we see that T (considered as an element of the Weyl module) is equal to the negative of the sum of the other tableaux, all of whose column words are greater than that of T . We now examine what happens when the extreme ﬂippable inversion involves the last row of the tableau. In this case, the two rows may or may not be adjacent, but we will draw the picture as though they were, since all of our discussion (and maps) involves just these two rows. We also may assume that the shape is not a skew-shape but is an almost skew-shape of positive type, so that our top row of the two projects to the left strictly beyond the bottom one. This means that we are in the situation pictured thus: T =

a1

··· s

as+1 bs+1

··· ···

ak−1 bk−1

ak bk

··· · · · bq

aq

,

with, again, the ﬂippable inversion ak ≥ bk and ak−1 ≤ bk . (Because we are assuming that s > 0, we know that ak−1 exists.) Let α be the index of the column as described in our previous discussion, that is, such that either aα−1 does not exist, or aα−1 ≤ bα . We clearly have s + 1 ≤ α ≤ k. Because there are no ﬂippable inversions to the left of the kth column, we see that we have aj < bj for α ≤ j ≤ k − 1. On the other hand, we have aj > bj for s + 1 ≤ j < α. Recall that from our earlier discussion, bα−1 = bα (if bα−1 exists).

Straightening and the straight basis theorem

181

We now have to be a bit more careful about our deﬁnition of the word w: let w be the word consisting of the elements in the top row to the left of ak−δ , where δ is such that ak−δ−1 < ak−δ = · · · = ak . We let v be the word bα · · · bk+ ak−δ · · · aq , that is, we do not include the terms bs+1 · · · bα−1 . Finally, we let u be the word bs+1 · · · bα−1 bk++1 · · · bq . We have w ⊗ v ⊗ u ∈ Dk−δ−1 ⊗ Dq−α+2+δ+ ⊗ (k−δ−1) (k++1−α) ∂3,2 (w ⊗ v ⊗ u). Again Dq −s+α−(k++1) . Now let us consider ∂2,1 using the Capelli identities, we see that (k−δ−1) (k++1−α) ∂3,2

∂2,1

=

k+−α

(k++1−α−h) (k−δ−1−h) (h) ∂2,1 ∂3,1

(−1)h ∂3,2

h=0 (α−−2−δ) (k−α++1) ∂3,1 .

+ (−1)k+−α+1 ∂2,1

The important thing to notice here is that the terms under the summation sign are all in the kernel of the Weyl map (since they all involve positive powers of ∂32 ), so that under the Weyl map, the sum (k−1−δ) (k++1−α) ∂3,2

∂2,1

(α−−2−δ) (k−α++1) ∂3,1

− (−1)k−α++1 ∂2,1

applied to w ⊗ v ⊗ u gives zero. If one analyzes the tableaux that show up under this sum, one sees that in addition to our tableau, T , one gets tableaux with w in the top row and the terms of v spread between the top and bottom rows (as in the previous situation), and other tableaux in which the word w gets spread around the top and bottom rows. Because of the inequalities we observed at the beginning of this discussion, we see that all the tableaux that appear, other than T , have column words greater in the lexicographic ordering than that of T . Hence, we have again “straightened” our original tableau, and we have shown that the straight tableaux generate the Weyl module for an almost skew-shape. VI.7.3 Taylor-made tableaux, or a straight-ﬁlling algorithm Before we address the problem of proving that the set of straight tableaux is linearly independent, we describe an algorithm, which we will call the Taylor algorithm, that will produce straight tableaux of a given shape from certain reverse column words. The algorithm was developed by B. Taylor, and, as with most of these constructions involving straight tableaux, applies to the larger class of row-convex shapes. We, however, will describe this procedure just for our almost skew-shapes. Let us denote by D the diagram of an almost skew-shape, with columns c1 , . . . , cm . We make a few observations. Fact 2 If T is a straight tableau of diagram D, then its reverse column word has the property that its subwords corresponding to each column are strictly increasing. Consequently, if we arrange the elements of the reverse column word as elements of Λγ1 ⊗ · · · ⊗ Λγm (where γi is the number of boxes in ci ), we get a basis element of this tensor product of exterior powers.

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Weyl and Schur modules

Proof This is the same as saying that in each column, there are no repeats. Suppose for some i < j we had aik = ajk . Then straightness implies that aik−1 > 2 ajk and this is clearly impossible. Fact 3 If T is a straight tableau, and w = x11 ∧· · ·∧x1γ1 ⊗· · ·⊗xm1 ∧· · ·∧xmγm is its reverse column word (written as a basis element of Λγ1 ⊗ · · · ⊗ Λγm ), then in each column cj , the element xjl appears in the ﬁrst box (from the top) not occupied by xj1 . . . , xjl−1 which either has no neighbor to its immediate left, or has one whose value is less than or equal to xjl . Proof Certainly this is the case if l = γj . If l = γj , and if the assertion were not true, then some xjs , with s > l, must be in that box. But xjs > xjl , so we have an inversion. However, there is either no element to the left of the box occupied by xjs , or there is and its value is less than or equal to xjl . Therefore, this is a ﬂippable inversion, contradicting the straightness of our tableau. 2 We stress again that all of what we have said makes sense if we think of the tableau simply as a ﬁlled-in diagram. Whether the ﬁlled-in diagram is to be regarded as an element of the Weyl module or not, depends upon the context in which we are using it. What we propose to do now is start with a basis element of Λγ1 ⊗· · ·⊗Λγm , and associate to it, when possible, a ﬁlling of the diagram which is straight and whose reverse column word is the given basis element we started with. When B. Taylor introduced this algorithm[83], he called it the straight-ﬁlling algorithm, and the resulting ﬁlled diagram a straight ﬁlling. In fact, we will see that if this procedure does not produce a straight ﬁlling, then there is no straight tableau with that given reverse column word. Algorithm (Taylor Algorithm) Let D be the diagram of an almost skew-shape with columns c1 , . . . , cm , and let w = x11 ∧ · · · ∧ x1γ1 ⊗ · · · ⊗ xm1 ∧ · · · ∧ xmγm be a basis element of Λγ1 ⊗ · · · ⊗ Λγm . Arrange x11 , . . . , x1γ1 in increasing order in column c1 . Next, place x21 in the ﬁrst box of c2 which either has no neighbor to its immediate left, or has such a neighbor whose entry is less than or equal to x21 . If there is no such box in c2 , the output of the algorithm is “no straight ﬁlling”. If there is such a box, ﬁll it with x21 . Assuming that we have placed x21 , . . . , x2j , we place x2j+1 in the ﬁrst empty box of c2 which either has no neighbor to its immediate left, or has such a neighbor whose entry is less than or equal to x2j+1 . Again, if there is no such box, our output is “no straight ﬁlling”; if there is, we ﬁll it with x2j+1 . We continue in this way with the remaining columns, obtaining an output of “no straight ﬁlling”, or a ﬁlling which we shall call T (w). Let us look at an example or two. As our shape, we will take one we have used earlier, namely (4, 3, 2)/(1, 0, 1). This has three rows and four columns, with γ1 = 1, γ2 = 3, γ3 = 2, γ4 = 1. The word w1 = x4 ⊗ x1 ∧ x2 ∧ x4 ⊗ x4 ∧ x5 ⊗ x6

Straightening and the straight basis theorem

produces the ﬁlling x4

x1 x4 x2

x4 x5

x6

183

, while the word w2 = x4 ⊗ x1 ∧ x2 ∧ x5 ⊗

x1 x4 x4 x5 x6 . On the other hand, words x2 such as x4 ⊗ x1 ∧ x2 ∧ x4 ⊗ x5 ∧ x6 ⊗ x4 or x5 ⊗ x1 ∧ x2 ∧ x4 ⊗ x4 ∧ x6 ⊗ x4 produce “no straight ﬁlling.” We point out a few things about this algorithm. First of all, if we start with a straight tableau, T , with reverse column word w (written as a basis element of our tensor product of exterior powers), then T (w ) will clearly have the reverse column word w , if T (w ) exists. But from our second fact above, T (w ) clearly does exist and equals T . We therefore see that if two straight tableaux have the same reverse column word (and hence the same modiﬁed column word), then they are equal. Hence the straight tableaux are precisely those whose reverse column words produce a successful outcome of the straight-ﬁlling algorithm. x4 ∧ x6 ⊗ x4 produces the ﬁlling x4

VI.7.4 Proof of linear independence of straight tableaux Remember that the Weyl map is a map ωλ/µ : Dp1 ⊗ · · · ⊗ Dpn → Λγ1 ⊗ · · · ⊗ Λγm associated to the almost skew-shape λ/µ. In the preceding subsection, we have spoken about the basis elements of Λγ1 ⊗ · · · ⊗ Λγm in the “classical” (i.e., nonletter-place) sense, and we have discovered that among those basis elements is a subset, namely, that subset consisting of basis elements which produce a straight tableau of shape λ/µ under the Taylor algorithm. This subbasis generates a free submodule of Λγ1 ⊗ · · · ⊗ Λγm which we shall denote by Λ(λ/µ). By the same token, our classical basis of Dp1 ⊗ · · · ⊗ Dpn contains a subset consisting of those elements which yield a straight tableau when they are used to ﬁll the diagram of λ/µ. These form a basis of a free submodule of Dp1 ⊗ · · · ⊗ Dpn which we will denote by D(λ/µ), and from the fact that the straight tableaux generate the Weyl module, Kλ/µ , we know that the image of the Weyl map is equal to the image of the composite map: ωλ/µ

inc

D(λ/µ) −→ Dp1 ⊗ · · · ⊗ Dpn −→ Λγ1 ⊗ · · · ⊗ Λγm , where the map labeled “inc” is the inclusion map. In addition to this inclusion map, we also have the projection map proj:Λγ1 ⊗ · · · ⊗ Λγm → Λ(λ/µ). If we can show that the composite map: inc

ωλ/µ

proj

D(λ/µ) −→ Dp1 ⊗ · · · ⊗ Dpn −→ Λγ1 ⊗ · · · ⊗ Λγm −→ Λ(λ/µ) is an isomorphism, then it will automatically follow that the straight tableaux are linearly independent, and therefore are a basis for Kλ/µ .

184

Weyl and Schur modules

Since D(λ/µ) and Λ(λ/µ) are both free modules with selected bases, we will show that the composite map above is an isomorphism by showing that the matrix of this map, with respect to a suitable total order of the bases, is upper triangular with ones on the diagonal. However, to make the picture of what we are about to do clear, we ﬁrst give an example. We will take our favorite almost skew-shape: λ/µ = (4, 3, 2)/(1, 0, 1) whose diagram is D =

(2)

. Consider the element x1 x4 x6 ⊗ x4 x5 ⊗ x2 ∈ D3 ⊗

x1 x4 x6 x4 x5 , x2 which is straight, so this element is a basis element of D(λ/µ). For convenience of writing, let us agree to replace xi by its index, i. Also, instead of writing wedges for the product in the exterior algebra, let us simply juxtapose, and instead of writing the tensor product symbol, we will simply write a slash. Using this shorthand, we see that in Λ1 ⊗ Λ3 ⊗ Λ2 ⊗ Λ1 , T is the sum of elements:

D3 ⊗ D1 ; under our Weyl map it goes to the tableau T = x4

4/142/45/6 + 4/142/65/4 + 4/152/64/4 + 4/452/14/6 + 4/452/64/1 + 5/142/64/4 + 4/642/15/4 + 4/642/45/1 + 4/652/14/4 + 5/642/14/4. Of course, in calculating out all the terms that occur as summands of T , many are zero, as any repeat in a column is automatically zero in the exterior powers. What we are left with, then, is a sum of terms which, but for a sign, are basis elements of Λ1 ⊗ Λ3 ⊗ Λ2 ⊗ Λ1 . If we rewrite these terms so that the sequences between slashes are increasing, we can ask which of them yields a straight ﬁlling under the Taylor algorithm. With very little diﬃculty one sees that there are only two: the “original” ﬁlling, and the one that corresponds to 4/125/46/4, 1 4 4 . whose associated ﬁlling is 4 5 6 2 We see, then, that the composite map applied to our element yields 4/142/45/6 + 4/152/64/4. Let us notice that the second ﬁlling was constructed from the term we denoted 4/152/64/4, and the reversal of order in the third column was due to the Taylor algorithm. If we write T and T for the ﬁllings corresponding to the two elements above, and uT , uT , their corresponding column words, we see immediately that uT < uT in the lexicographic order; thus T > T . If, now, we let T be the straight ﬁlling of w (T ) (which is pictured immediately above), then uT < uT , so T > T . This suggests the order we should put on the basis elements of Λ(λ/µ) and D(λ/µ). Namely, we order the basis elements of Λ(λ/µ) according to the lexicographic order on the column words of their straight ﬁllings, and we order the

Straightening and the straight basis theorem

185

basis elements of D(λ/µ) according to lexicographic order of the column words of their straight ﬁllings. With this, we are able to prove the statements which follow. Proposition VI.7.8 an isomorphism.

The composite map proj ωλ/µ inc : D(λ/µ) → Λ(λ/µ) is

Proof The main argument in this proof is to show that with the ordering of the basis elements that we have deﬁned, the matrix of the map is upper triangular with ones on the diagonal. To this end, consider a basis element X = X1 ⊗ · · · ⊗ Xn ∈ D(λ/µ), and use it to ﬁll the diagram of λ/µ. Its image under the Weyl map is then the tableau that we get using that ﬁlling (see example above). This tableau is straight, since the hypothesis that X is in D(λ/µ) tells us precisely that, and the value of this tableau in Λγ1 ⊗· · ·⊗Λγm is a sum of elements which are obtained by ﬁrst diagonalizing each Xi into Λ1 ⊗ · · · Λ1 , and then multiplying the entries in each column. % &# $ pi

To clarify this procedure, visualize the diagram as ﬁlled with the rearranged terms; thus we obtain rows whose entries are permutations of the original row, and the “columns” to which we refer are the columns of the diagram. (Again, we recommend that you look at the example above). Because the multiplication is occurring in an exterior algebra, any repeats will yield zero, so the summands that remain will be those in which no repeats appear in any of the columns. There are two facts that are fairly clear: the original ﬁlling is among the summands (the identity permutation on the rows is among the permutations of the rows that we are considering); the column word of any of the ﬁllings that occur, other than the original one, is greater than that of the original one. This last can be proven easily by observing that if there is any change in the ﬁrst column, it must be due to the substitution of one of the original entries by a larger one in its row. A similar argument, column by column, proves the assertion. The next thing we must do is apply the projection map to the surviving summands; this now gives us the sum of those summands whose reverse column word yields a straight ﬁlling. But it is easy to see that if T is any ﬁlling whose reverse column word yields the straight ﬁlling, T , then the column word of T is greater than that of T in the lexicographic order. For any change in any column will be the result of moving a larger element in a column below a smaller one in that same column (and one can now argue column by column as above). As a result we see that the matrix of this map is indeed upper triangular, with ones on the diagonal, and the proposition is proven. 2 We now have the straight basis theorem, as well as the presentation theorem for almost skew-shapes. Theorem VI.7.9 Let F be a free R-module, and λ/µ an almost skew-shape. The following statements are true:

186

Weyl and Schur modules

1. The Weyl module, Kλ/µ (F ), is a free R-module with basis consisting of the straight tableaux in a basis of F . ¯ λ/µ (F ) → Kλ/µ (F ) is an isomorphism. 2. The map θλ/µ : K 3. The functor Kλ/µ (F ), considered as a functor of F , is universally free. Proof Our proposition above tells us that the straight tableaux are linearly independent; in fact they are in one-one correspondence with the basis elements of D(λ/µ) and of Λ(λ/µ). Since we already know that they generate Kλ/µ , they form a basis of the Weyl module. This disposes of 1. The map θλ/µ is clearly a surjection, so that the cosets of the basis elements ¯ λ/µ . But it is clear that these elements are also linearly of D(λ/µ) generate K ¯ λ/µ . Hence the map θλ/µ must be an independent, so they form a basis of K isomorphism. This takes care of 2. ¯ λ/µ is the cokernel of a natural map between two universFinally, since K ally free functors, it must be universally free. But from 2, we conclude that Kλ/µ is also universally free. (For a reminder of what universal freeness is, see Subsection III.1.2.) 2 VI.7.5 Modiﬁcations for Schur modules We have focused our attention in the preceding sections on Weyl modules; the same results obtain for Schur modules, but some minor modiﬁcations have to be made in the deﬁnitions and proofs. We will brieﬂy discuss these changes, but will not go into extensive detail. First notice what row- and column-standard and straight mean for tableaux of negative letters. Row-standard means strictly increasing in rows; columnstandard means weakly increasing in columns. Straight means the following: • In a row-standard tableau, two elements aik , ajk (with i < j) in the same column are said to form (or be) an inversion if they violate column-standardness. That is, if aik > ajk . The inversion is said to be unﬂippable if there is an element in the tableau, bik−1 , immediately to the left of aik , such that bik−1 ≥ ajk . Otherwise the inversion is called ﬂippable. The row-standard tableau is said to be straight if every inversion is unﬂippable.

In a row-standard tableau of negative letters, the straightening is a bit easier than is the case for positive letters, as there can be no repeated elements in any given row. Hence, whereas in the straightening algorithm for positive letters we had to scoop up elements in a row that repeated, here our straightening procedure will not require that. Except for that, our straightening of tableaux goes through as before, and we have the fact that the straight tableaux generate the Schur module. To prove linear independence of the straight tableaux, we must modify the Taylor algorithm. Since the Schur map goes from tensor products of exterior powers to tensor products of symmetric powers, and a basis of the tensor product of symmetric

Weyl–Schur complexes

187

powers is a tensor product of monomials, we have to show how to associate to such a tensor product of monomials a straight tableau (or no tableau at all). But clearly all we have to do is what we did with the positive letters, except for insisting that if an element is put into a box which has a box to its immediate left, then it has to be strictly greater than the occupant of that box. Otherwise, the proofs go through mutatis mutandis. VI.7.6 Duality As anticipated in Remark III.1.5 for the special case of hooks, there is a duality between some Weyl and Schur modules. It is well known, [3], that if A is the is its transpose (again the shape matrix shape matrix of a skew-shape, λ/µ, and A ∗ ∼ of a skew-shape), then KA (F ) = (LA(F ))∗ . The proof depends in part on the fact that for such shapes, the modules in question are universally free. Now, if A is the shape matrix of an almost skew-shape of positive type, its transpose is no longer of the same kind. Therefore if we want an isomorphism like the one stated, we would at least have to saturate the class of almost skew-shapes with respect to transposition, and develop all of the preceding material for that larger class of shapes. Since all of the shapes in that class would be row-convex, and the straightening techniques and algorithms used in this chapter apply to row-convex shapes, one could probably arrive at such a duality statement. VI.8 Weyl–Schur complexes In this section, we simultaneously extend the notions of Weyl and Schur modules to a complex which we choose to call the Weyl–Schur (or W–S) complex. We make this choice even though it was just called the Schur complex when it ﬁrst appeared [3]. If λ/µ is an almost skew-shape, the W–S complex associated with it will be denoted by Kλ/µ (ϕ), to be consistent with the notation we have been using throughout this chapter. Sometimes the notation, Lλ/µ (ϕ), is used as well. We will consider a given map ϕ : G → F , where F and G are free modules, in place of just a free module, F . We will deﬁne D(ϕ) to be the algebra D(G)⊗Λ(F ) endowed with an endomorphism, ∂ϕ : D(ϕ) → D(ϕ), deﬁned as the composition of the maps: id⊗µ

∆⊗id

D(G) ⊗ Λ(F ) −→ D(G) ⊗ D1 (G) ⊗ Λ(F ) −→ D(G) ⊗ Λ(F ), where ∆ is the diagonal map of D(G) → D(G)⊗D1 (G), and µ is the composition: ϕ⊗id

m

D1 (G) ⊗ Λ(F ) −→ Λ1 (F ) ⊗ Λ(F ) −→ Λ(F ). In this second composition, m stands for multiplication in the exterior algebra. We set Dk (ϕ) = Dα (G) ⊗ Λk−α (F ), 0≤α

188

Weyl and Schur modules

and note that our map ∂ϕ gives us a complex 0 → Dk (G) → Dk−1 (G) ⊗ Λ1 (F ) → · · · → D1 (G) ⊗ Λk−1 (F ) → Λk (F ) → 0. The reader will recognize this to be one of the complexes we considered in Chapter III. We can actually say a bit more, namely that this endomorphism of D(ϕ) makes D(ϕ) into a diﬀerential graded algebra; that is, with the graded algebra structure of the tensor product of two algebras, the map ∂ϕ is a derivation. This means that it satisﬁes the identity: ∂ϕ ((a1 ⊗ b1 )(a2 ⊗ b2 )) = ∂ϕ (a1 ⊗ b1 )(a2 ⊗ b2 ) + (−1)b1 (a1 ⊗ b1 )∂ϕ (a2 ⊗ b2 ), where (−1)b1 means (−1) raised to the degree of the element b1 . The proof of this is straightforward; it merely uses the fact that the diagonal map preserves multiplication. In a similar way, we can deﬁne Λ(ϕ) to be the algebra Λ(G) ⊗ S(F ) endowed with an endomorphism, ∂ϕ : Λ(ϕ) → Λ(ϕ), deﬁned as the composition of the maps: ∆ ⊗id

id⊗µ

Λ(G) ⊗ S(F ) −→ Λ(G) ⊗ Λ1 (G) ⊗ S(F ) −→ Λ(G) ⊗ S(F ), where ∆ is the diagonal map of Λ(G) → Λ(G)⊗Λ1 (G), and µ is the composition: ϕ⊗id

m

Λ1 (G) ⊗ S(F ) −→ S1 (F ) ⊗ S(F ) −→ S(F ). In this composition, too, m stands for multiplication in the symmetric algebra. As above, we deﬁne Λk (ϕ) = 0≤α Λα (G) ⊗ Sk−α (F ), and note that our map ∂ϕ gives us a complex 0 → Λk (G) → Λk−1 (G) ⊗ S1 (F ) → · · · → Λ1 (G) ⊗ Sk−1 (F ) → Sk (F ) → 0. This in turn makes Λ(ϕ) into a diﬀerential graded algebra. If we want to apply letter-place methods in this situation as we did earlier, we ﬁrst establish the following proposition. Proposition VI.8.1 Let ϕ1 : G1 → F1 and ϕ2 : G2 → F2 be two maps of free modules. Then ϕ1 ⊕ ϕ2 : G1 ⊕ G2 → F1 ⊕ F2 is a map of free modules, and D(ϕ1 ⊕ ϕ2 ) ∼ = D(ϕ1 ) ⊗ D(ϕ2 ). The corresponding statement for Λ(ϕ1 ⊕ ϕ2 ) is also true. Proof We will simply show that as graded modules they agree; the detailed veriﬁcation of the algebra and complex isomorphisms is straightforward, but tedious. We have Dα (G1 ⊕ G2 ) ⊗ Λk−α (F1 ⊕ F2 ) (∗) Dk (ϕ1 ⊕ ϕ2 ) = α

=

α,β,γ

Dβ (G1 ) ⊗ Dα−β (G2 ) ⊗ Λγ (F1 ) ⊗ Λk−α−γ (F2 ),

Weyl–Schur complexes

while (∗∗) (D(ϕ1 ) ⊗ D(ϕ2 ))k =

189

Dα (ϕ1 ) ⊗ Dk−α (ϕ2 )

α

=

Dβ (G1 ) ⊗ Λα−β (F1 ) ⊗ Dγ (G2 ) ⊗ Λk−α−γ (F2 ).

α,β,γ

If we set γ = α − β, we see that the last sum in (∗∗) rearranges into the last sum in (∗). 2 Corollary VI.8.2

If we denote by ϕ ⊗ Rn the direct sum ϕ ⊕ · · · ⊕ ϕ, we have % &# $ n

D(ϕ ⊗ R ) = D(ϕ) ⊗ · · · ⊗ D(ϕ) . % &# $ n

n

This clearly allows us to think of a basis of Rn as, say, positive places, and to develop the letter-place context for this situation. In fact, we will think of a basis of F , say F, as an ordered set of negative letters, and a basis of G, which we will write G, as an ordered set of positive letters. For convenience (it is easy to show that our choice makes no essential diﬀerence to our discussion), we shall assume that every element of G precedes all the elements of F, and so we have a total order on G " F. We will use the customary double tableau notation, but since the “letters” are basis elements of D(G) ⊗ Λ(F ), we should explain what we mean by an expression such as (w ⊗ u|1(k1 ) 2(k2 ) · · · n(kn ) ), where w ∈ Dk (G), u ∈ Λl (F ) and k + l = k1 + · · · + kn . To be consistent with our previous notation for letter-place identities, we set (w(s1 ) ⊗ u(t1 )|1(k1 ) ) · · · (w(sn ) ⊗ u(tn )|n(kn ) ), (w ⊗ u|1(k1 ) 2(k2 ) · · · n(kn ) ) = where the sum ranges over all pairs si , ti such that si + ti = ki . We have written w(si ), u(ti ) to indicate the components of the n-fold diagonalization of w and u of degrees si , ti . Again, an example will serve to make this much clearer. Example VI.8.3

Consider (x(2) ⊗ y1 ∧ y2 |12(2) 3). This is equal to

(x ⊗ 1|1)(x ⊗ y1 |2(2) )(1 ⊗ y2 |3) − (x ⊗ 1|1)(x ⊗ y2 |2(2) )(1 ⊗ y1 |3) + (x ⊗ 1|1) (1 ⊗ y1 ∧ y2 |2(2) )(x ⊗ 1|3) + (1 ⊗ y1 |1)(x(2) ⊗ 1|2(2) )(1 ⊗ y2 |3) − (1 ⊗ y2 |1) (x(2) ⊗ 1|2(2) )(1 ⊗ y1 |3) + (1 ⊗ y1 |1)(x ⊗ y2 |2(2) )(x ⊗ 1|3) − (1 ⊗ y2 |1) (x ⊗ y1 |2(2) )(x ⊗ 1|3). We have been very meticulous about writing, say, 1 ⊗ y1 instead of just y1 . In the future, we will generally drop the 1 and the tensor product sign when the meaning is evident.

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Weyl and Schur modules

Once we have this notation at our disposal, we are set to use our old double tableau notation. If A stands for the multi-signed alphabet G " F, the notation and deﬁnitions of Subsection VI.7.1 apply to our situation. At this point, it must be fairly clear to the reader that the same “straightening” arguments we have used before will work here to show that the double standard tableaux generate the n-fold tensor product of D(ϕ). We refer the reader to Section 4 of Appendix A to see how to prove linear independence in much the same way as before. An illustrative example will also be found there. As a result we have the theorem: Theorem VI.8.4 Let ϕ : G → F be a morphism of free R-modules. Then the double standard tableaux discussed above form a basis for D(ϕ ⊗ Rn ). The same result obtains for Λ(ϕ ⊗ Rm ). Clearly, one can now proceed with place polarizations as in Section VI.4. Because the polarizations operate only on the places, all the structure attached to the letters is preserved by them. In particular, the complex structure that we have imposed on our algebra is preserved. As a result, given an almost skew-shape, λ/µ, (or any shape for that matter), we can deﬁne the map ∆λ/µ (ϕ) : Dp1 (ϕ) ⊗ · · · ⊗ Dpn (ϕ) → Λγ1 (ϕ) ⊗ · · · ⊗ Λγm (ϕ) as we did the Weyl and Schur maps, polarizing the positive places (basis of Rn ) to the negative places (basis of Rm ) according to the recipe dictated by the diagram of λ/µ. (The pi and γj are read from the diagram of λ/µ in the usual way.) This, by our observation above, is a map of complexes, which we call the Weyl–Schur map (or W–S map); as a result, its image is also a complex. Deﬁnition VI.8.5 The image of the map ∆λ/µ is the Weyl–Schur (W–S) complex, Kλ/µ (ϕ), associated to the map ϕ and the shape λ/µ. Remark VI.8.6 It is important to point out here that, since the map ∆λ/µ does not depend on the map ϕ at all, the chains of the complex Kλ/µ (ϕ) depend only on the shape λ/µ and the modules G and F . With this deﬁnition, we can proceed to deﬁne the complex Rel(λ/µ) as we did for Weyl and Schur modules, as well as the map of complexes, λ/µ , whose image ¯ λ/µ (ϕ), the cokernel of the is in the kernel of ∆λ/µ (ϕ). We denote again, by K ¯ λ/µ (ϕ) map λ/µ and by θλ/µ the naturally induced map of complexes from K to Kλ/µ (ϕ). Given our signed inequalities, namely, <+ and <− , our deﬁnitions of standard and straight apply to the tableaux of shape λ/µ, ﬁlled with the elements of G "F, and our analysis of ﬂippable and unﬂippable inversions goes through without change. Thus the straightening procedure we used to show that the straight tableaux generate goes through using <+ and <− on the multi-signed alphabet G " F. Similarly, the Taylor algorithm goes through without change as long as we use the inequalities above. In fact, one can see that the “modiﬁcations” made

Weyl–Schur complexes

191

in discussing the Taylor algorithm for Schur modules rather than Weyl modules were really unnecessary if we had used the <+ and <− inequalities throughout. As a result, we can state the following theorem without further proof: Theorem VI.8.7 Let ϕ : G → F be a morphism of free R-modules, and λ/µ an almost skew-shape. The following statements are true: 1. The Weyl–Schur complex, Kλ/µ (ϕ), is a free R-complex with basis consisting of the straight tableaux in a basis of F union a basis of G. ¯ λ/µ (ϕ) → Kλ/µ (ϕ) is an isomorphism. 2. The map θλ/µ : K

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VII SOME APPLICATIONS OF WEYL AND SCHUR MODULES

In the previous chapter we developed the basic deﬁnitions of Weyl and Schur modules, as well as Weyl–Schur complexes. In this chapter, we will give indications of how we may apply some of these ideas. In the ﬁrst two sections, we deal with very basic material, and go into some detail, as the results are fundamental for most of the applications of characteristic-free representation theory that we will see. The remainder of the chapter takes a more cavalier approach to the results considered; we include a few historical notes and rely even more heavily than heretofore on recent reference material. VII.1 The fundamental exact sequence Let ϕ : G → F be a morphism of free R-modules, and A a shape matrix. The ﬁrst thing we should do in this section is consider what our Weyl–Schur complexes look like in two very special cases: (i) G = 0 and (ii) F = 0. Remark VII.1.1 When G = 0, we see that our complex Dk (ϕ) reduces to Λk F while the complex Λk (ϕ) reduces to Sk (F ). Since the Weyl–Schur map is deﬁned in terms of place polarizations in all of our cases, we see that the complex KA (ϕ) simply is the Schur module LA (F ). Similarly, when F = 0, we ﬁnd that KA (ϕ) is just the Weyl module KA (F ). In this case, we notice that the module does not lie in degree zero as in the former case, but occurs in the highest degree term of our complex. This is simply because we have chosen the convention of considering our complexes as left, rather than right, complexes. From Remark VI.8.6, we see that the chains of KA (ϕ) for an arbitrary ϕ are the same as those for KA (ϕ ), where ϕ = ϕ1 ⊕ ϕ2 . with ϕ1 = G → 0 and ϕ2 = 0 → F . We will return to the implications of this later, to get a description of the chains when A is the shape matrix of a skew-shape. We now consider an almost skew-shape, λ/µ = (p1 , . . . , pn ; t1 , . . . , tn−1 ), where we use the notation introduced at the beginning of Section VI.6. (Remember that the integer, tn−1 , may be non-negative, or negative; in the ﬁrst case, we have a skew-shape, while in the second, we have an almost skew-shape of positive type.) From this shape, let us construct the associated almost skew-shape, λ /µ = (p1 , . . . , pn ; t1 , . . . , 1+tn−1 ). If we write λ/µ = (λ1 , . . . , λn )/(µ1 , . . . , µn ),

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Some applications of Weyl and Schur modules

we can write λ /µ = (λ1 + 1, . . . , λn−1 + 1, λn )/(µ1 + 1, . . . , µn−1 + 1, µn ). If our original shape is a skew-shape, our convention adopted in Chapter VI would have us assume that µn = 0, and in that case our skew-shape λ /µ also satisﬁes that convention. On the other hand, if our almost skew-shape we started with is of positive type, our convention would have us choose µn−1 = 0, in which case our representation of λ /µ no longer satisﬁes that convention. Of course, in this latter case, we could choose to present our new shape as λ /µ = (λ1 , . . . , λn−1 , λn − 1)/(µ1 , . . . , µn−1 , µn − 1), since our assumption of positive type implies that µn > 0. For notational convenience, we denote by M the complex Dp1 ⊗R · · · ⊗R Dpn , where we have omitted the ϕ; that is, we have written Dpi for Dpi (ϕ). Then we have the two exact sequences: (E ) Rel(λ /µ ) → M → Kλ /µ (ϕ) → 0, (E) Rel(λ/µ) → M → Kλ/µ (ϕ) → 0, and we observe that (†) Rel(λ/µ) = Rel(λ /µ ) ⊕ N, where (‡) N = Dp1 ⊗R · · · ⊗R Dpi +ti −s+1 ⊗R Dpi+1 ⊗R · · · ⊗R Dpn−1 ⊗R Dpn −ti +s−1 , with s and i as described below. This last observation may require a bit of explanation: we resort to the deﬁnitions of the terms Rel(λ/µ) and Rel(λ /µ ) found in Chapter VI, Section 6, which we will not reproduce here. If our original shape, λ/µ, is a skew-shape, then so is λ /µ , and in that case i = n − 1 and s = 0. There are no “extra” terms that have to be added to the basic relations of a skew-shape, and it is easy to see that (†) and (‡) are true. If λ/µ is an almost skew-shape of positive type, then there are a number of possibilities, which we now examine. Recall that, in this case, our type τ is equal to n − i − 1, where i is such that µi+1 < µn ≤ µi . (Here we will adopt our convention that µn−1 = 0 and µn > 0.) If it turns out that we have µi+1 < µn − 1 ≤ µi , then our almost skew-shape, λ /µ , is also of type τ , its “s” is now s − 1, and again it is easy to see that (†) and (‡) are true. The situation becomes a bit more complex when it turns out that µi+1 = µn − 1. In this case, the “s” of our shape λ/µ is equal to 1, and the type of λ /µ decreases. In fact, suppose that we take α to be the largest integer such that µi+1 = · · · = µi+α , with α ≤ τ . Then the type of λ /µ is precisely τ − α ≥ 0. In this case, we see that all the integers ti+1 , . . . , ti+α−1 are zero, and the “s” for λ /µ is zero also. Now a quick examination of the terms comprising Rel(λ/µ) and Rel(λ /µ ) will conﬁrm the statements (†) and (‡). From the exact sequences (E ) and (E), we see that the identity map on M, and the inclusion map of Rel(λ /µ ) into Rel(λ/µ), induce a surjection Kλ /µ (ϕ) → Kλ/µ (ϕ).

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195

What we intend to prove is that the kernel of this surjection is the W-S complex, Kλ /µ (ϕ), where λ /µ = (p1 , . . . , pi−1 , pi + ti − s + 1, pi+1 , . . . , pn−1 , pn − ti + s − 1; t1 , . . . , ti−2 , ti−1 + ti + s − 1, s − 1, ti+1 , . . . , tn−2 , −(tn−2 + · · · + ti )). While the above description of λ /µ is very graphic, it is also very unwieldy. Also, our insistence that in the case of a skew-shape we have µn = 0, but for an almost skew-shape we have µn−1 = 0, makes for a certain amount of distinct case analysis in our discussions. To avoid this, we will simply start with our almost skew-shape, λ/µ, deﬁne our shape λ /µ to be (λ1 + 1, . . . , λn−1 + 1, λn )/ (µ1 + 1, . . . , µn−1 + 1, µn ), and then our shape λ /µ is simply: (λ1 + 1, . . . , λn−1 + 1, λn )/(µ1 + 1, . . . , µi−1 + 1, µn , µi+1 + 1, . . . , µn−1 + 1, µi + 1).

• There is one caveat we must call attention to: if i = 1, then since µ2 < µn ≤ µ1 , µ = (µn , µ2 + 1, . . . , µn−1 + 1, µ1 + 1) is not an almost partition for µ1 + 1 µn . In this case, we set

λ = (λ1 + 1, . . . , λn−1 + 1, λn − (µ1 − µn + 1)) and µ = (µn , µ2 + 1, . . . , µn−1 + 1, µn ). All of this clearly does not aﬀect any of our discussion above. However, in the next part of this section we will be using the deﬁnition of the Weyl–Schur map as a composition of place polarizations (positive to negative), and in Chapter VI we set min(µn−1 , µn ) = 0 to deﬁne these maps. If, however, for an almost skewshape λ/µ we just set β = min(µn−1 , µn ), then we simply have to subtract β from all the λ’s and µ’s that occur in the indices of the polarization maps. With these modiﬁcations in mind, we can now state and prove the theorem on fundamental exact sequences. Theorem VII.1.2 Let λ/µ be an almost skew-shape, and let λ /µ and λ /µ be as above. Then we have the exact sequence of complexes: 0 → Kλ /µ (ϕ) → Kλ /µ (ϕ) → Kλ/µ (ϕ) → 0, where the map Kλ /µ (ϕ) → Kλ/µ (ϕ) is the one induced by the identity map on the generators of these complexes, and the map Kλ /µ (ϕ) → Kλ /µ (ϕ) is induced by the map of the generators of Kλ /µ (ϕ) into those of Kλ /µ (ϕ) via (t −s+1) . the (positive to positive) place polarization ∂n,ii Proof That the map Kλ /µ (ϕ) → Kλ/µ (ϕ) is surjective, is trivial to see. It is easy to see that the ranges of the Weyl–Schur maps ∆λ /µ and ∆λ /µ are the

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Some applications of Weyl and Schur modules

same, say W. It is also clear that the domain of ∆λ /µ is N, where N is as deﬁned above. Therefore, if we can prove that the diagram: (t −s+1)

∂n,ii

N

/ M

W

/ W

∆λ /µ

∆λ /µ

id

is commutative, where M too is as deﬁned above, we will be done. But the commutativity is easily demonstrated by application of the Capelli identities. 2 We point out here that the original proof of this theorem made use of a rather complicated spectral sequence argument. This was due to the fact that at that time, there was no general presentation theorem for almost skew-shapes; the presentation theorem was originally proven only for skew-shapes. However, with the advent of B. Taylor’s notion of straight tableaux, the presentation theorem for almost skew-shapes was possible, and now we have the rather elementary proof above of the theorem on fundamental exact sequences. We can use this theorem, along with an induction assumption, to prove the following result. Theorem VII.1.3 Let ϕ : F → F be the identity map, and λ/µ an almost skew-shape. Then, unless λ/µ = 0, in which case the homology is simply R, the complex Kλ/µ (ϕ) is exact, that is, all the homology groups of the complex are zero. Proof We prove the nontrivial case by a double induction on the the number of rows of the diagram, and the sum, σ, of the number of overlaps of the last row of the diagram with each of the other rows. Note ﬁrst that if λ/µ consists of only one row, our result follows from the discussion immediately preceding Deﬁnition III.1.3. Thus, we may assume our theorem true for shapes having at most n − 1 rows, and prove it for those having n rows. If σ = 0, then our complex Kλ/µ (ϕ) must be of the form Dλn /µn ⊗R Kλ /µ (ϕ), where λ /µ is our shape λ/µ with its last row removed. But by our induction hypothesis on the number of rows, we see that we have the tensor product of two free exact complexes, and this is again exact. If we now start with an almost skew-shape, λ/µ, with σ > 0, then we notice that the shapes λ /µ and λ /µ of our Theorem VII.1.2 above both have smaller values of σ, and hence the associated complexes are exact. It is now trivial to conclude that our original complex is exact. 2 Theorem VII.1.2 deals only with what happens when one slides the last row of the diagram one move to the left. There is, however, a special situation in which

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197

one can move much more to the left (still by only one step). In fact, a similar, but much easier argument than that used to prove Theorem VII.1.2 yields the following result. Theorem VII.1.4

Let α be the skew-shape represented by

(p1 , . . . , pk , pk+1 , . . . , pn ; t1 , . . . , tk−1 , pk+1 − 1, tk+1 , . . . , tn−1 ). Then we have the exact sequence 0 → Kδ (ϕ) → Kβ (ϕ) ⊗ Kγ (ϕ) → Kα (ϕ) → 0 , where β = (p1 , . . . , pk ; t1 , . . . , tk−1 ), γ = (pk+1 , . . . , pn ; tk+1 , . . . , tn−1 ) and δ = (p1 , . . . , pk−1 , pk + pk+1 , pk+2 , . . . , pn ; t1 , . . . , tk−2 , tk−1 + pk+1 , tk+1 , . . . , tn−1 ).

VII.2 Direct sums and ﬁltrations for skew-shapes In this section we are going to consider just the class of skew-shapes. When we discuss the W–S complex associated to a direct sum of maps and a skew-shape λ/µ, we will take a brief look also at the kind of additional problem that arises if the shape is assumed to be almost skew. Even for skew-shapes we will not give full proofs, since a good reference is available (Reference [3], section 5.1, where our W–S complexes are called Schur complexes and denoted by Lλ/µ (ϕ)), and here it will be enough to just give an indication of the main ideas. Theorem VII.2.1 ([3], Theorem 5.1.13) For i = 1, 2, let ϕi : Gi → Fi be two maps, and let ϕ = ϕ1 ⊕ ϕ2 . Then for any skew-shape λ/µ, the W–S complex, Kλ/µ (ϕ), has a ﬁltration by subcomplexes the terms of whose associated graded complex are Kν/µ (ϕ1 ) ⊗R Kλ/ν (ϕ2 ), where ν runs over all partitions such that µ ⊆ ν ⊆ λ. Discussion. We will only discuss the considerations that lead to the deﬁnition of the ﬁltration. First, we order the basis for G ⊕ F , with G = G1 ⊕ G2 and F = F1 ⊕ F2 , by making the basis elements of Gi precede those of Fi , and the basis elements of G1 ⊕ F1 precede those of G2 ⊕ F2 . For notational convenience, let us denote the basis of G1 ⊕ F1 by U, and the basis of G2 ⊕ F2 by V. Since for a skewshape, “straight” tableaux are the same as “standard” tableaux, we see that row-standardness of a tableau for Kλ/µ (ϕ) places the elements of U before those of V, while column-standardness requires that the “separators” between the ﬁrst

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Some applications of Weyl and Schur modules

and the second set of elements in each row together form a partition ν that contains µ and is contained in λ. We may illustrate this situation as follows: u

v

u

(S)

v

u

.

v

This leads us to deﬁne the ﬁltration on Kλ/µ (ϕ) as follows. For each partition ν with µ ⊆ ν ⊆ λ, deﬁne Dσ/µ (ϕ1 ) ⊗R Dλ/σ (ϕ2 ) → Dλ/µ (ϕ) Mν (Dλ/µ (ϕ)) = im ν⊆σ⊆λ

˙ ν (Dλ/µ (ϕ)) = im Dσ/µ (ϕ1 ) ⊗R Dλ/σ (ϕ2 ) → Dλ/µ (ϕ) M ν⊂σ⊆λ

Mν (Kλ/µ (ϕ)) = ∆λ/µ (ϕ) Mν (Dλ/µ (ϕ)) ˙ ν (Dλ/µ (ϕ)) , ˙ ν (Kλ/µ (ϕ)) = ∆λ/µ (ϕ) M M where the top two maps are essentially the inclusion maps, and ∆λ/µ (ϕ) is the usual map. The proof that the above is our desired ﬁltration and that ˙ ν (Kλ/µ (ϕ)) ∼ Mν (Kλ/µ (ϕ)) / M = Kν/µ (ϕ1 ) ⊗R Kλ/ν (ϕ2 ) is actually an isomorphism is found in Reference [3]; the reader should be warned that there the bold-faced D is instead written as Λ. Let us now look at what the situation would be if instead of a skew-shape we had started with an almost skew-shape (still denoted by λ/µ). Keeping our same ordering of the basis, and looking at a straight tableau, we see that as in the previous case, our row-standardness has all the basis elements of U preceding those of V. An analysis of straightness in the columns, however, tells us that the line of demarcation in each row gives us, until we reach the penultimate row, what looks like a subpartition of λ which contains µ. But straightness allows the possibility that in the last row, we have some basis elements of U lying under some elements of V in a higher row, provided that this higher row protrudes beyond the bottom row. As a result, these demarcations delineate an almost partition, rather than a partition. Again, it is useful to illustrate this situation with the following diagram: u (AS)

v

u

v

u

v u

.

v

One approach to dealing with this could be to consider a larger class of shapes than the almost skew-shapes. For instance, we could consider shapes λ/µ, with

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199

λ and µ both almost partitions of length n, such that µ ⊆ λ. It would then have to be checked in detail that everything found in Chapter VI carries through for these shapes, and that all shapes manufactured in the ﬁltration are of the same kind. In order to be able to apply to these shapes the same inductive argument used in the proof of Theorem VII.1.3, we could require in addition that τλ + τµ < n − 1, where τλ and τµ are the types of the almost partitions λ and µ. However, as we will have no use in the rest of the book for a more general form of Theorem VII.2.1, we leave this as an open question. We next list three immediate corollaries of Theorem VII.2.1. Corollary VII.2.2 Let ϕ : G → F be any map of free modules, and let µ ⊆ λ be partitions. Denote by (Kλ/µ (ϕ))j the component in degree j of the complex Kλ/µ (ϕ). There is a natural ﬁltration on (Kλ/µ (ϕ))j whose associated graded module is Lν/µ (F ) ⊗R Kλ/ν (G), µ⊆ν⊆λ, |λ|−|ν|=j

where |λ| stands for the sum,

λi , and similarly for ν.

Proof Cf. Reference [3], corollary 5.1.14.

2

We remind the reader that a complex is acyclic if all the positive homology modules vanish. Corollary VII.2.3 Let ϕ : G → F be a split injection with (projective) cokernel E. Then Kλ/µ (ϕ) is acyclic and, when E is free, H0 (Kλ/µ (ϕ)) = Lλ/µ (E). Proof Since it is suﬃcient to prove acyclicity locally, we may assume R is local, that F = G ⊕ E, with E free, and that our map ϕ is the direct sum of ϕ1 = id : G → G and ϕ2 : 0 → E. Hence by Theorem VII.2.1, we see that up to ﬁltration, the homology of Kλ/µ (ϕ)) is the direct sum of H Kν/µ (ϕ1 ) ⊗R Lλ/ν (E). By Theorem VII.1.3, it follows that all of these homology modules are zero except when ν = µ, and this gives the acyclicity as well as the result that H0 (Kλ/µ (ϕ)) = Lλ/µ (E). If E is free to begin with, the result that H0 (Kλ/µ (ϕ)) = Lλ/µ (E) is also true globally. 2 By essentially the same method of proof one obtains the following result. Corollary VII.2.4 If ϕ = ϕ1 ⊕ ϕ2 , where ϕ1 is an isomorphism, then the complex Kλ/µ (ϕ) is homotopically equivalent to Kλ/µ (ϕ2 ). VII.3 Resolution of determinantal ideals When, in the early 1960s, the hunt was on for the generalized Koszul complex, the hope was that it would be possible, and not too diﬃcult, to ﬁnd resolutions of the ideals of p × p minors of the generic m × n matrix (m ≥ n). In fact, the generalized Koszul complexes described in Chapter III do the job for the n × n,

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Some applications of Weyl and Schur modules

or maximal, minors of the matrix. But if, using the notation of Chapter III, we have the map f : A → B, with A = Rm , B = Rn , then we can see that the cokernel of the map mq (f ) : Λq A ⊗R Λr B → Λq+r B has support equal to that of the ideal generated by the minors of order n − r (where mq (f ) is the composition Λq f ⊗R id

Λq A ⊗R Λr B −→ Λq B ⊗R Λr B → Λq+r B). Thus there was the hope that one could ﬁnd some way of resolving this cokernel for all r in the same way as one did the case when r = 0, that is, the case of the maximal order minors. Alas, the problem turned out to be far more complex than imagined, but fortunately it did lead to a number of interesting developments, some of which we will try to cover in the subsequent subsections. VII.3.1 The Lascoux resolutions We saw in Chapter III that the hooks had come into play in connection with the generalized Koszul complexes, and it became clear to a number of those working in this area that the terms of a resolution of an ideal generated by minors should be representations of the appropriate general linear groups. Before we proceed further, let us explain why this is so by examining in some detail just what we really mean by “generic map.” (Also cf. Section V.2.) Consider two free R-modules, F and G of ranks m and n. What do we mean by the generic map from one to the other? For notational convenience, it will be easier to talk about a “generic map” from F to G∗ , rather than from F to G (obvious modiﬁcations can be made to make it work from F to G). Let S = S(F ⊗R G) be the symmetric algebra associated to the free R-module F ⊗R G, and let F = S ⊗R F, G∗ = S ⊗R G∗ . In order to deﬁne a map from F to G∗ , it is enough to deﬁne a map from F to F ⊗R G ⊗R G∗ . This is so because S0 = R and S1 = F ⊗R G. Therefore, if we have the map from F to F ⊗R G⊗R G∗ , this means we have a map F → S1 ⊗R G∗ . Hence, to deﬁne the map Sν ⊗R F → Sν+1 ⊗R G∗ , we simply take the composite Sν ⊗R F → Sν ⊗R S1 ⊗R G∗ → Sν+1 ⊗R G∗ , where the last map is just the multiplication in the symmetric algebra. But the map F to F ⊗R G ⊗R G∗ simply sends an element x ∈ F to the element x ⊗ cG ∈ F ⊗R G ⊗R G∗ , where cG ∈ G ⊗R G∗ is the element (as in Chapter V) corresponding to the identity map in HomR (G, G) under the natural identiﬁcation of G⊗R G∗ with HomR (G, G). If we now choose bases {x1 , . . . , xm } for F , {y1 , . . . , yn } for G, and set Xij = xi ⊗ yj ∈ S1 , it is easy to see that the matrix corresponding to our map, using {1 ⊗ xi } and {1 ⊗ yj } as bases for F and G∗ , is simply (Xij ). The upshot of all of this is that the “generic” map is deﬁned without recourse to a basis, so that it seemed natural to suppose that the modules that would occur in a resolution of the ideal, Ip , generated by minors of order p (which also can be deﬁned without recourse to basis), should be independent of a choice of basis, and hence should be representations of GL(F ) × GL(G).

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201

However, it was not until the mid-1970s that A. Lascoux found a way to explicitly use representation theory to describe the resolutions of these determinantal ideals [64]. Working in characteristic zero, where every representation of the general linear group splits into a direct sum of irreducible representations, he was able to write down the GL(F ) × GL(G)-irreducible components of the representation modules Xkp that occur as the chains in dimension k of the resolutions of the determinantal ideal Ip , the ideal of p × p minors of the generic m × n matrix. Since the irreducible representations of GL(F ) × GL(G) are of the form M ⊗R N , where M and N are irreducible representations of GL(F ) and GL(G), resp., and the irreducible representations would be of the form Lλ F and Lµ G for partitions λ, µ, the question to be answered is: for which pairs of partitions λ, µ do the tensor products Lλ F ⊗R Lµ G appear as direct summands in Xkp ? A. Lascoux determined what this family of pairs is by using a clever geometric argument. First he “desingularized” the variety of Ip , by going up to the appropriate Grassmannian; the ideal of this desingularized variety has a Koszul complex as its resolution. Then, by Bott’s theorem on the vanishing of the cohomology of homogeneous vector bundles ([14], theorem 4), the higher direct image functors on the Koszul complex can be computed, and the resulting complex is the minimal resolution of Ip . While the desingularization technique holds in arbitrary characteristic, Bott’s theorem is valid only in characteristic zero, which partially explains why the Lascoux approach seems to be limited to that case. (We will return to this point in a later section.) This geometric approach was originated by G. Kempf in the 1970s [57]; a full and very clear discussion of this and similar techniques can be found in J. Weyman’s book [86]. It should be pointed out here, for the sake of historical accuracy, that an explicit description of the boundary maps of the Lascoux resolutions was not given by A. Lascoux. Subsequent work by H. Nielsen [68] and P. Pragacz and J. Weyman [70] was needed to clear this up. To describe the partitions that do appear in the Lascoux resolutions, we need some additional notation and terminology. We say that a partition, λ, has Durfee square of size k if λi ≥ k for i = 1, . . . , k, and λk+1 ≤ k. We denote so that we write the transpose partition of λ as usual (see Chapter VI), by λ, = (λ 1 , . . . , λ t ) where λ j = #{i| λi ≥ j}. It is clear that λ and λ have the same λ Durfee square size. Finally, if λ and µ are partitions, we denote by λ + µ the partition (λ1 + µ1 , . . . , λq + µq ). We are now in a position to state (without proof) what the terms of the Lascoux resolution are.

Theorem VII.3.1 Ip . Then

Let Xkp denote the module of k-chains in the resolution of

Xkp =

λ∈Λk

Lλ+(p−1)k(λ) F ⊗R Lλ+(p−1) k(λ) G,

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Some applications of Weyl and Schur modules

where k(λ) is the Durfee square size of λ, (a)k(λ) denotes the rectangular partition (a, . . . , a), and Λk is the set of partitions, λ, such that |λ| = k. % &# $ k(λ)

If one chooses p = 1, that is, the ideal generated by the “indeterminates” Xij , one sees that the terms of the resolution are λ∈Λk Lλ F ⊗R Lλ G and, in characteristic zero, this is the classical Cauchy decomposition of Λ(F ⊗R G). What we see is that the terms for arbitrary p are the terms that one would get for p = 1, modiﬁed by the adjunction of rectangular pieces of dimensions p − 1 by k(λ). VII.3.2 The submaximal minors While there are clearly a few technicalities connected to the preceding resolutions whose clariﬁcation would be desired, it is nevertheless evident that this work of A. Lascoux aroused a great deal of interest and hope. The hope resided in the expectation that the Schur modules that he used were replicable in a characteristic-free setting, and that, while Bott’s theorem is not true in positive characteristic, its non-validity would not adversely aﬀect the outcome of Lascoux’ construction, even if it obviated the use of the same geometric approach that he had employed so eﬀectively. There had been some work already on reproducing Schur and Weyl modules in arbitrary characteristic (see [34], and [84]), but the approach in Reference [3], and presented in this book, proved to be most convenient for the purposes at hand. One of the most, if not the most, important reason for this is that these modules can be described by means of generators and relations, that is, an explicit presentation can be given, and this makes it easier to compute them and with them. In addition, as one saw in the preceding chapter, one can deﬁne these modules for a larger category of shapes than just partitions, and this allows for the use of homological methods, as illustrated by the theorem on fundamental exact sequences (Theorem VII.1.2). In any event, the terms of classical representation theory were constructible in a characteristic-free setting, and one could begin to carry out a “Lascouxiﬁcation” of the problem of resolving the determinantal ideals. While the case p = 1 gave us the Koszul complex (or at least its terms), and the case p = n gave us the Eagon–Northcott complex [40], both of which could be deﬁned in a characteristic-free way, it was not clear just how to produce a resolution for the intermediate values of p, nor was it clear that the boundary map of such a resolution, assuming it could be built out of nice representation modules, would be “natural,” as is true in the extreme cases. Some simple examples sufﬁced to show that a direct parroting of the Lascoux complexes over the integers did not lead to acyclicity; there was torsion in the integral homology. But it was possible to construct a characteristic-free resolution of the ideal of submaximal minors (see [4]), that is, the minors of order n − 1 (recall that we are assuming throughout that m ≥ n). In this case, though, for arithmetic considerations anticipated due to the previously mentioned torsion, one had to choose Z-forms

Resolution of determinantal ideals

203

of the terms that appear in characteristic zero (the term Z-form will be explained in the next subsection) in order to push the construction through. As in the previous discussion, we will skip the details, but two observations emerged from this project: one was that the boundary map which was explicitly constructed was not “natural,” that is it was not equivariant with respect to the action of the general linear groups involved; another was that it was nevertheless possible to build the terms of the resolution in a way suggested by the Lascoux approach (although many more terms might have to be involved— more about this in a later section). An important additional observation was that fairly delicate arithmetic considerations were involved in the construction of these complexes, and this led to an independent study of the Z-forms which we will discuss in the next subsection. VII.3.3 Z-forms Let F be a free abelian group of rank m, and let F be its extension to the rationals, that is, F = Q ⊗Z F . We know that D2 F and S2 F are both GL(F )representations , where GL(F ) means the general linear group over the integers, Z. Furthermore, D2 F = Q ⊗Z D2 F, S2 F = Q ⊗Z S2 F , and these are GL(F )representations. We know that there is a GL(F )-equivariant map from D2 F to S2 F , namely the composition: ∆

m

D2 F −→ F ⊗Z F −→ S2 F, where ∆ is the diagonal map, and m is the usual multiplication map. However, this is not an isomorphism of the two integral representations. Nevertheless, the corresponding map over the rationals is an isomorphism. We therefore say that D2 F and S2 F are Z-forms of the same rational representation (in this case, S2 F ). So, we are led to make the following deﬁnition. Deﬁnition VII.3.2 Let F be a free abelian group. Two GL(F )-representations are Z-forms of the same representation if, when tensored with the rationals, Q, they are isomorphic GL(F )-representations. In Subsection VII.3.2, we indicated that in the construction of the resolution of the ideal of submaximal minors, one had to choose certain Z-forms of the terms in the Lascoux resolution in order to get an integral complex which was acyclic. These representations that arose were Z-forms of certain hooks, and they came about in the following way. We know that for all l > 0 the complexes 0 → Λl F → Λl−1 F ⊗Z F → · · · → Λl−t F ⊗Z St F → · · · → F ⊗Z Sl−1 F → Sl F → 0 are exact, where the boundary map is given by diagonalizing the exterior powers and multiplying the symmetric powers. But what happens if we replace the

204

Some applications of Weyl and Schur modules

symmetric powers by divided powers, that is, if we consider the complex 0 → Λl F → Λl−1 F ⊗Z F → · · · → Λl−t F ⊗Z Dt F → · · · → F ⊗Z Dl−1 F → Dl F → 0 where we still diagonalize the exterior powers and multiply, this time, into the divided powers (something that we have avoided doing so far throughout this book)? As the reader may strongly suspect, this complex is no longer exact; the surprising thing, however, is that, counting from the left, it is exact up to the middle of the complex, that is, from t = 0 to t = [(l − 1)/2] where [x] indicates the integral part of x ([4], proposition 2.22). As a result, the cycles of this complex are Z-forms of the corresponding cycles of the complex above, involving the symmetric powers, and these, as indicated in Chapter III, are just the hooks. Another, simpler, way to construct non-isomorphic Z-forms is the following: Consider the short exact sequence (†)

0 → Dk+2 → Dk+1 ⊗Z D1 → K(k+1,1) → 0

where K(k+1,1) is the Weyl module associated to the hook partition (k + 1, 1). (We are leaving out the module F , as that is understood throughout.) If we take an integer, t, and multiply Dk+2 by t, we get an induced exact sequence and a commutative diagram: 0 →

Dk+2 ↓t 0 → Dk+2

→ →

Dk+1 ⊗Z D1 → K(k+1,1) ↓ ↓ E(t; k + 1, 1) → K(k+1,1)

→

0

→

0,

where E(t; k + 1, 1) stands for the coﬁber product of Dk+2 and Dk+1 ⊗Z D1 . Each of these modules is a Z-form of Dk+1 ⊗Z D1 , but for t1 and t2 , two such are isomorphic if and only if t1 ≡ t2 mod k + 2 (see [2]). This says that Ext1A (K(k+1,1) , Dk+2 ) ∼ = Z/(k + 2), where A is the Schur algebra of appropriate degree (we will discuss the calculation of Ext in more detail in Subsection VII.4.1 on intertwining numbers). Although we have not introduced the Schur algebra until now, it plays an important role in representation theory as, classically, the Schur algebra of degree k is the universal enveloping algebra for the homogeneous polynomial representations of degree d. For example, Dd and Λd are such representations of the general linear group. As a result, when we speak of “equivariant maps” of representation modules, what we really mean are linear maps over the Schur algebra; the Schur algebra is the underlying ring over which all of our representations are modules. Deﬁnition VII.3.3 Let F be a free R-module of rank n, and d a positive integer. The symmetric group, Sd , acts on the left of F ⊗d = F ⊗R · · · ⊗R F by % &# $ d

permuting factors. The Schur algebra of degree d is the algebra, AR (n, d), of

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205

endomorphisms of F ⊗d which commute with the action of Sd . That is, AR (n, d) = EndSd (F ⊗d ). We will usually just write A for AR (n, d) when the context is clear. It can be seen without too much diﬃculty that this algebra is, as a module, simply Dd (EndR (F )) [2]. (It is enough to recall that for any free module, V , Dd (V ) is the submodule of V ⊗d invariant under the operation of the symmetric group Sd .) The description of the ring structure on A in this guise is a bit less intuitive. It is based on the fact that for any free module, V , and any integer d, there is a map of Dd (V ) ⊗R Dd (V ) into Dd (V ⊗R V ) (the dual map of the projection Sd (V ⊗R V ) → Sd (V ) ⊗R Sd (V ) associated with the ﬁltration in Reference [3], Section 3.1). If V happens to be a ring, as is the case when V = EndR (F ), we have a map of Dd (V ⊗R V ) into Dd (V ) induced by the multiplication map V ⊗R V → V . The composite map Dd (EndR (F )) ⊗R Dd (EndR (F )) → Dd (EndR (F ) ⊗R EndR (F )) → Dd (EndR (F )) gives the multiplicative structure on the Schur algebra. All of this is discussed in detail in Reference [2], as well as Reference [45]. In both of the indicated references, it is actually proved that the Schur algebra, AR (n, d), is the direct sum of the modules Da1 (F ) ⊗R · · · ⊗R Dan (F ), as (a1 , . . . , an ) runs through all integral weights of rank d (that is, all n-tuples of non-negative integers adding up to d). As a result, the modules Da1 (F ) ⊗R · · · ⊗R Dan (F ) are projective modules over AR (n, d). (In Reference [2] it is shown that if we take the tensor product of more than n factors, this tensor product need not be projective over AR (n, d).) This, then, is the deﬁnition of the Schur algebra; when we talk about functors like Hom or Ext for representation modules, the base ring will usually be the Schur algebra, and we will write HomA or ExtA as one does for modules over rings. The Schur algebra, by the way, is not commutative. One more observation about the Schur algebra 3before we go on to other Da1 (F ) ⊗R · · · ⊗R Dan (F ), applications. If M is an A-module, then since A ∼ = and M = HomA (A, M ), we have a decomposition of M as the direct sum M∼ HomA (Da1 (F ) ⊗R · · · ⊗R Dan (F ), M ). = The modules HomA (Da1 (F ) ⊗R · · · ⊗R Dan (F ), M ) are no longer A-modules; they are R-modules. However, these are known as weight submodules of M . In terms of classical representation theory, these are deﬁned to be the submodules of M consisting of those elements m ∈ M with the property that if T is the diagonal matrix:   t1 0 · · · 0  0 t2 · · · 0    T = . .. . . . ,  .. . ..  . 0 then T m = ta1 1 ta2 2 · · · tann m.

0

· · · tn

206

Some applications of Weyl and Schur modules

It is not diﬃcult to see that if M is either a Weyl or Schur module associated with a skew-shape, then the weight submodule corresponding to the weight α = (a1 , . . . , an ) is generated by all standard tableaux of content α. By a standard tableau of content α, we mean a standard tableau in which the entries consist of ai copies of the ith basis element for every i. Thus, for example, the weight submodule of Λ2 corresponding to the weight (2, 0, . . . , 0) is zero (i.e., there are no maps from D2 to Λ2 ), while the weight submodule of Λ2 corresponding to the weight (1, 1, 0, . . . , 0) is the submodule generated by the element x1 ∧x2 , assuming that we have chosen the basis {x1 , . . . , xn } for the underlying free module, F . It is understood here that by “maps” we mean A-maps, where A = AR (n, 2) in this case (since Λ2 is a representation of degree 2). VII.4 Arithmetic considerations We have already seen that the Weyl and Schur modules are universal, so that if we deﬁne them over the integers and then tensor them with a commutative ring, R, we get the corresponding module over the ring R. We also see that there is some interest in looking at the Ext groups of certain Weyl and/or Schur modules. The Universal Coeﬃcient Theorem tells us how the Ext groups behave under ring extension. The proof of this theorem is found in Reference [2], theorem 5.3. Theorem VII.4.1 (Universal Coeﬃcient Theorem) Let R be a commutative hereditary ring, R → R a homomorphism of commutative rings, and A an R-algebra. Let M and N be left A-modules which are free R-modules. Furthermore, assume that M has a resolution, P, over A by ﬁnitely generated projective A-modules. Then there is a short exact sequence of R-modules i+1 0 → R ⊗R ExtiA (M, N ) → ExtiA (M , N ) → TorR 1 (R, ExtA (M, N )) → 0

for each i ≥ 0 where we have set Y = R ⊗R Y , for Y = A, M and N . In particular, then, if R = Z and R = Z/(p) for some prime p, we have the exact sequence: 0 → Z/(p) ⊗Z ExtiA (M, N ) → ExtiA (M , N ) → TorZ1 (Z/(p), Exti+1 A (M, N )) → 0, where A denotes the appropriate Schur algebra over Z, and A denotes Z/(p)⊗Z A. VII.4.1 Intertwining numbers The exact sequence immediately above is particularly applicable in modular representation theory in relation to the problem of calculating intertwining numbers. A standard question in modular representation theory is the following. If we let λ = (λ1 , . . . , λq ) be a partition, and let µ = (µ1 , . . . , µq ) be the partition obtained from λ by attaching d boxes from the bottom row of λ to its top row, that is, µ1 = λ1 + d, µ2 = λ2 , . . . , µq = λq − d, then what is the

Arithmetic considerations

207

Z/(p)-dimension of the Z/(p)-vector space ExtiA (Kλ , Kµ ) where the notation is as indicated above. These numbers are called intertwining numbers. From the exact sequence above, we see that it suﬃces to calculate the integral Ext groups, since the modular ones are simply the p-torsion part of one integral Ext plus the reduction modulo p of another. Conjectures about these numbers were ﬁrst made by D. Kazhdan and G. Lusztig [56] (in this case, the conjectures concerned coeﬃcients of the so-called Kazhdan–Lusztig polynomials), then by G. Lusztig [65] for modular representations (where the connection was made between the Kazhdan–Lusztig coeﬃcients and the intertwining numbers for modular representations). A ﬁne presentation of these conjectures and results on their partial solution can be found in Reference [79]. As is so often the case in conjectures about characteristic p, the solutions that have been obtained relate to “suﬃciently large primes.” One might hope that the study of these numbers through the use of the integral representations would throw more light on whether one must really limit the size of the primes and, if so, what that limit is due to. To indicate how the universal coeﬃcient theorem (UCT) can be used in just a small special case, let us suppose that we want to evaluate the dimension of Ext0A (Kλ , Kµ ), that is, of HomA (Kλ , Kµ ). The UCT tells us that we should ﬁrst look at HomA (Kλ , Kµ ). But HomA (Kλ , Kµ ) is a subgroup of HomZ (Kλ , Kµ ), and since these Weyl modules are free abelian groups, their Hom over Z is free. Thus HomA (Kλ , Kµ ) is a free abelian group, and has the same rank as its extension to the rationals. But over the rationals, the Weyl modules are irreducible, so that there are no nontrivial AQ -maps between them. Thus, the Hom group over the integers is also zero, and what is left for us to do is calculate Ext1A (Kλ , Kµ ). For once we have that, we simply have to take its p-torsion to get our result. To ﬁnd the p-torsion of a ﬁnite abelian group is indeed fairly simple, so the rub in the above discussion is how to ﬁnd Ext1A (Kλ , Kµ ). The straightforward answer of any halfway respectable homologist is, ﬁnd an A-projective resolution, P, of Kλ , Hom it into Kµ , and take the resulting homology in dimension one. We saw in Subsection VII.3.3 that certain tensor products of divided powers are A-projective, and resolutions by such modules have the advantage that one can calculate (in principle) the result of taking Hom of such modules into our Weyl modules. In Reference [2], it was proven that projective resolutions, P, of this desired type exist (we will return to this in Section VII.6), but to actually calculate the homology of the Hom complex, we need explicit information about the boundary maps of these resolutions as well as the explicit description of the terms. One simpliﬁcation we meet in this calculation comes from the fact that HomA (P, Kµ ) is a complex of free abelian groups, and that its cohomology is torsion (since over the rationals the cohomology is zero). If we write down the boundary maps in HomA (P, Kµ ) as integral matrices, elementary arguments show us that the cohomology groups are determined by the invariant factors of those matrices (the non-zero invariant factors). To calculate those

208

Some applications of Weyl and Schur modules

matrices, we clearly must know the maps in the projective resolution, P. We will do one example to illustrate how such calculations may be done. The reader is referred to References [2], [26], and [30] for more complete discussion of this problem. In Subsection VII.3.3, we made the assertion that the exact sequence (†) is exact. This is a resolution of the degree k + 2 representation, K(k+1,1) ; but how do we know that? In this case, it is fairly simple to see it, since we do have the exactness of the complex 0 → Dk+2 → Dk+1 ⊗Z D1 → Dk ⊗Z Λ2 → · · · → D1 ⊗Z Λk+1 → Λk+2 → 0 and, in Chapter III, we saw that the hooks were the cycles in this complex. (We will soon see other ways of arriving at this fact.) In any event, to compute Ext1A (K(k+1,1) , Dk+2 ) (this, after all, is the situation for λ = (k + 1, 1), d = 1 and µ = (k + 2, 0) of our discussion), we simply look at 0 → HomA (Dk+1 ⊗Z D1 , Dk+2 ) → HomA (Dk+2 , Dk+2 ) → 0 and compute the invariant factors of the matrix of the map HomA (Dk+1 ⊗Z D1 , Dk+2 ) → HomA (Dk+2 , Dk+2 ). From our discussion of weight submodules, we see that both of these abelian groups are free of rank one; the question is to identify bases for these groups and make the map explicit in terms of these bases. A simple calculation shows that the basis for HomA (Dk+2 , Dk+2 ) is the identity map, α, say, while the basis for HomA (Dk+1 ⊗Z D1 , Dk+2 ) is the multiplication map, say β. The map from Dk+2 to Dk+1 ⊗Z D1 , call it δ, is just the diagonal map. Thus, the map β is carried (k+2) (k+2) ) = (k + 2)x1 = (k + 2)α. The matrix to βδ, and we see that βδ(x1 in question, then, is the one-by-one matrix, (k + 2), and so we get the result announced earlier, namely that Ext1A (K(k+1,1) , Dk+2 ) ∼ = Z/(k + 2). While this calculation was quite simple, this is not the case in general. As of the current writing, the only complete results that are known are for two- and threerowed partitions, and only for Ext1A . Again we refer the reader to References [2], [26], and [30] for explicit calculations leading to the known results, but as we see, there is a strong incentive to get our hands on an explicit description of projective resolutions of Weyl modules. The reader may be interested to know that a newer approach to calculating the invariant factors of the matrices that we do encounter involves the use of letter-place methods, but this development is still in its infancy (see [22], section 4). VII.4.2 Z-forms again The example that we did in the previous subsection shows that we have at least k + 2 distinct Z-forms for Dk+1 ⊗Z D1 . Of course, over the rationals, the divided

Hashimoto counterexample

209

powers are the symmetric powers, and Dk+1 ⊗Z D1 splits up into the direct sum of K(k+1,1) and Dk+2 . Over the integers, the exact sequence (†) not only does not split, but that short exact sequence turns out to be the generator of Ext. Clearly there is a great deal of arithmetic involved in our integral representations of the general linear group over Z. The following example will also illustrate how these Z-forms are directly connected with resolutions of Weyl modules. In Section VII.6 we will see that a projective resolution of K(k,2) is given by 0 → Dk+2 →

Dk+1 ⊗Z D1 ⊕ → Dk ⊗Z D2 → 0, Dk+2

where the map from each component of the split term is the appropriate diagonalization (possibly followed by multiplication), the map Dk+2 → Dk+1 ⊗Z D1 is the diagonal, and the map Dk+2 → Dk+2 is multiplication by 2. Thus the cokernel of the left-hand map is the module we denoted by E(2; k + 1, 1) in Subsection VII.3.3. We therefore have the short exact sequence 0 → E(2; k + 1, 1) → Dk ⊗Z D2 → K(k,2) → 0, and over the rationals, the sequence would look the same, except that there we could replace E(2; k +1, 1) by Dk+1 ⊗Z D1 , which gives the anticipated resolution of the Weyl module in characteristic zero (see [64]). From the point of view of the Grothendieck ring of representations, this tells us that while over the rationals the class of K(k,2) may be represented as [Dk ⊗Z D2 ] − [Dk+1 ⊗Z D1 ], over the integers it would be represented by [Dk ⊗Z D2 ] − [E(2; k + 1, 1)]. The thrust of the above discussions is that there are many reasons for trying to ﬁnd an explicit description of projective resolutions of Weyl modules. VII.5 Resolutions revisited; the Hashimoto counterexample When we put the material of the preceding two sections together, we see that the construction of resolutions of determinantal ideals, particularly for the minors of lower order, may involve arithmetic subtleties. In fact, they arose very explicitly in the construction of the resolution of the submaximal minors and, in subsequent work on the minors of order n − 2, that is the subsubmaximal minors, the intricacies of the Z-form arithmetic became so onerous that there was no attempt to push through the full construction in that case. It was obviously the time to sit back and rethink the approaches to the problem. It had always been clear that the existence of a universal (that is, deﬁned over Z; also called generic) minimal free resolution of the ideal, Ip , of p × p minors would imply that the Betti numbers of that ideal are independent of characteristic (by Betti numbers we mean the dimensions over the ground ﬁeld—Z/(p) or Q in our case—of the vector spaces of chains of a minimal resolution). But since

210

Some applications of Weyl and Schur modules

these ideals were known to be perfect (i.e., their homological dimension is equal to their depth) in every characteristic (see [38], [39]), the implications about Betti numbers seemed perfectly (sic) natural. That is, while certain constructions were universal, they lacked universal minimal free resolutions, but those that were known “violators” were not perfect in all characteristics (such as the projective plane, which is perfect in every characteristic but two [51], [72]). Some attempts were made to prove this apparently true fact about the Betti numbers by less cumbersome techniques and, in 1987, K. Kurano [62], proved that the ﬁrst Betti numbers of Ip were independent of the characteristic. But in 1988, M. Hashimoto announced (in a preprint which later became Reference [47]) the surprising fact that for the generic 5 × 5 matrix, the second Betti number of I2 in characteristic three is one greater than it is in characteristic zero. (His paper [47] actually proved that the ideal Iq of the (q + 3) × (q + 3) matrix has its second Betti number in characteristic three larger than in characteristic zero.) Immediately following M. Hashimoto’s announcement, J. Roberts and J. Weyman [73] published another proof that had the same geometric ﬂavor as the Lascoux construction of his resolutions of determinantal ideals (see also [86]). We will not go into their proof in detail, but a few remarks are worth discussing. The ﬁrst observation made by Roberts and Weyman is that the geometric part of the Lascoux approach is characteristic-free. That is, the desingularization that is used over the Grassmannian, and the fact that the ideal of this desingularization is resolved by a Koszul complex, are valid over any ground ring. It is in the application of Bott’s theorem, that is, the second step of the Lascoux program, that the procedure breaks down (as one might suspect, since the Bott theorem is not true in positive characteristic). But what Roberts and Weyman do is identify the higher direct images that come up in the Bott theorem with the homology of certain canonical subcomplexes of the Weyl–Schur complexes. The homology of these complexes is directly related to the Betti numbers of the determinantal ideals, and can be computed by means of representation theory (at least theoretically). In the Hashimoto example, starting with such facts as that the representation Λ3 is contained in the representation K(2,1) in characteristic three, it is possible to see that there is a linear two-cycle that cannot be covered by a degree one map. Hence the second Betti number goes up in characteristic three. This observation makes sense in the light of the speciﬁc construction of the resolutions involved; we will forbear to go into those details here. It suﬃces to say that this example of M. Hashimoto took the steam out of the project of looking for universal minimal free resolutions of determinantal ideals. It did, however, open up a few new areas of investigation, and did use in an essential way the characteristic-free approach to Weyl modules and their resolutions. Finally, the relationship between Bott’s theorem and the homology of certain subcomplexes of Weyl–Schur complexes that is evident in the Roberts–Weyman approach to M. Hashimoto’s example should be studied carefully.

Resolutions of Weyl modules

211

VII.6 Resolutions of Weyl modules In the discussions of intertwining numbers and of Z-forms, as well as in the above section, we have seen references to and examples of resolutions of Weyl modules. As the reader may well imagine, the history of the search for a description of these resolutions is a fairly long one. In this section, we will deal with some of this history, as well as give a brief description of some of the latest results. A good deal of what appears here can be found in Reference [30]. In the early 1980s, work was begun [1] on the problem of resolving Schur modules in terms of direct sums of tensor products of exterior powers (or the so-called fundamental representations). In those years, Schur modules and fundamental representations were the focus since the terms of the Lascoux resolutions of determinantal ideals were expressed in terms of Schur modules, and the Lascoux description in characteristic zero of the “resolutions” of Schur modules was given in terms of the fundamental representations. Therefore, in the early references, most of the literature is written in those terms. It transpired that as work progressed and diﬃculties became more numerous, the question arose as to whether these resolutions really existed. To settle this question without having to actually construct the resolutions, a natural isomorphism Ω : HomA (Dλ (F ), Dµ (F )) → HomA (Λλ (F ), Λµ (F )) was deﬁned in Reference [2] which allowed one to dualize the resolutions of the Kλ (F ), given in terms of sums of tensor products of divided powers, to get the resolutions of the Lλ (F ) in terms of tensor products of exterior powers. (By Dλ we mean Dλ1 ⊗R · · ·⊗R Dλn , where λ = (λ1 , ..., λn ). By Λλ we mean Λλ1 ⊗R · · · ⊗R Λλn .) We will therefore focus, as we have throughout the book, on Weyl modules and resolutions in terms of tensor products of divided powers. We add here, as a purely philosophical comment, that we have grown accustomed to regarding the Weyl modules as “more fundamental” than the Schur modules because of the fact that the tensor products of divided powers are projective, and the weight submodules of representations are expressible in terms of them. For two-rowed Weyl modules, using the fundamental exact sequence,

0→

p+t+1 t+1 → q−t−1

p t → q

p → 0, q

and an induction argument on the number of overlaps between the two rows, namely, the integer q − t as read from the above diagram, it was possible to give a description of the desired type of resolution [1, 28] . This will be described in Subsection VII.6.2. The three-rowed case presented, of course, quite another aspect, for as we know, the kernel of the analogous surjective map is an almost skew-shape about which nothing was known at the time. In particular, a presentation of such shapes was not even guessed at. To be precise, the analogue of the above sequence is

212

Some applications of Weyl and Schur modules

the following: t1

0→ t2 +1

p q+t2 +1 → r−t2 −1 →

t1 t2 +1 t1 t2

p q r p q → 0. r

and while one can try to do an induction on the integer r − t1 − t2 , the type of shape has changed on us, that is, it has become an almost skew-shape rather than remain a skew-shape, as it so conveniently did in the two-row situation. Hence one must also try to resolve these “new” shapes. The approach to this in Reference [2] was to resort to a fairly complicated spectral sequence to prove a slightly weaker form of Theorem VII.1.2. Despite the encouragement provided by the proven existence of resolutions of the type desired, and the fundamental exact sequences which theoretically provide a mechanism for an induction proof as in the two-rowed case, the methods at hand still made the calculations almost impossible to deal with. As a result, it was not until the 1990s that this problem was looked at again from the point of view of letter-place methods, where the Weyl map and the presentation map were seen to be place polarizations, and the Capelli identities could play a role in simplifying and systematizing these calculations. And then the introduction by B. Taylor of straight tableaux made it possible to develop our theory as we have in this book, and to prove the theorem on fundamental exact sequences in more general form. One other tool that was essential in describing the resolutions was the bar complex (see [29]). As the reader can see, the strategy in building the resolutions lies in assuming that one knows what the terms look like for shapes that are “less complex,” with “complexity” being measured by the number of overlaps between rows. Theorem VII.1.2 allows us to reduce this complexity by presenting our given shape in terms of shapes that are less complex (as we did in the proof of Theorem VII.1.3). Assuming we have resolutions of these less complex shapes, we have a map between these resolutions (this is where the projectivity of the resolutions comes in), and the mapping cone of this map is then a resolution of the shape we started with. By describing the resolutions in terms of the bar complex, a complex which behaves well with respect to forming mapping cones, this procedure of putting together resolutions by forming mapping cones becomes manageable. In the rest of this section we will describe some of these techniques in more detail. VII.6.1 The bar complex We made heavy use of the bar complex in Section III.2. It turns out that this complex is really a special case of a whole class of complexes, which we call diﬀerential bar complexes as in Reference [29].

Resolutions of Weyl modules

213

The main components of the construction of such complexes are the exterior algebra over Z, Λ(S), on a set of free generators, S, called the separators, an algebra, A, an A-module, M , and the free product of Λ(S) with A. The algebra Λ(S) has a natural Z2 -grading: if m is a monomial in Λ(S), that is, a product of generators, we set |m| = 0 if m is the product of an even number of generators, and |m| = 1 if m is the product of an odd number of generators. Deﬁnition VII.6.1 The free product of the algebra A and the algebra Λ(S) will be called the bar algebra on the algebra A, with set of separators S, and denoted by Bar(A; S). The algebra Bar(A; S) inherits a Z2 -grading as follows. Every element of Bar(A; S) is a linear combination of elements of the form W = w1 m1 w2 m2 · · · wk mk

(∗)

where the mi are non-zero monomials in Λ(S), and where the wi are elements of A. (The monomial mk can, of course, be equal to a scalar, and w1 may be the identity element of A.) We set |W | = 0 if |m1 m2 · · · mk | = 0 and |W | = 1 if |m1 m2 · · · mk | = 1. One extends this deﬁnition by linearity to a Z2 -grading of the algebra Bar(A; S). Notice that Bar(A; S) is a two-sided A-module in a natural way. For every ﬁnite subset T of the set S of separators, the underlying module of the algebra Bar(A; S) has a grading, which will be called the T -grading of Bar(A; S), and which is deﬁned as follows. The submodule Bar(A; S; T, i) of T -degree i is spanned by all elements of the form (∗) where the integer i equals the total number of occurrences of separators in the set T in the sequence (m1 , m2 , . . . , mk ). Clearly, the submodule Bar(A; S; T, i) is a two-sided A-module. For every separator x, there exists a unique antiderivation, ∂x , of the algebra Λ(S), such that ∂x (x) = 1 (where 1 is the identity of the exterior algebra Λ(S)), and ∂x (y) = 0 for y in S not equal to x. We also have (∂x )2 = 0 and ∂x ∂y = −∂y ∂x (see [29] and [30]). The antiderivation ∂x uniquely extends to an antiderivation of the Z2 -graded algebra Bar(A; S), again denoted by ∂x , deﬁned as follows. If W is as in (∗), set ∂x (x) = 1 (where 1 is now the identity of the algebra Bar(A; S)), and ∂x (W ) = w1 ∂x (m1 )w2 m2 · · · wk mk + (−1)|m1 | w1 m1 w2 ∂x (m2 ) · · · wk mk + · · · k−1

+ (−1)

i=1

|mi |

w1 m1 w2 m2 · · · wk ∂x (mk ).

The antiderivation ∂x is well deﬁned. Again, we have (∂x )2 = 0 and ∂x ∂y = −∂y ∂x . Deﬁnition VII.6.2 If T is a non-empty ﬁnite subset of S, the operator ∂T = x∈T ∂x , is called the T -boundary operator on Bar(A; S).

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Some applications of Weyl and Schur modules

The boundary operator ∂T maps Bar(A; S; T, i + 1) into Bar(A; S; T, i), for i = 0, 1, 2, .... Now let M be a left A-module. If w is an element of A, we denote the action of w on an element v ∈ M by w(v). Deﬁnition VII.6.3 The free bar module of the A-module M , with set of separators S, denoted by Bar(M, A; S), is the Bar(A; S)-module Bar(A; S)⊗A M. Remark VII.6.4 lining:

The following two observations are easy, but worth under-

1. The module Bar(M, A; S) is spanned by all elements of the form w1 m1 w2 m2 · · · wk mk ⊗ v. If mk = 1, then w1 m1 w2 m2 · · · wk ⊗ v = w1 m1 w2 m2 · · · mk−1 ⊗ wk (v), since the tensor product is taken over A. 2. For each separator, x, there is a well-deﬁned antiderivation on the free bar module, Bar(M, A; S), again denoted by ∂x , deﬁned as follows: ∂x (w1 m1 w2 m2 · · · wk mk ⊗ v) = ∂x (w1 m1 w2 m2 · · · wk mk ) ⊗ v. It is worth noting from the equation above that, when mk = x, we get (setting W = w1 m1 w2 m2 · · · wk x) : ∂x (W ⊗ v) = w1 ∂x (m1 )w2 m2 · · · wk x ⊗ v + (−1)|m1 | w1 m1 w2 ∂x (m2 ) · · · wk x ⊗ v + · · · k−1

+ (−1)

i=1

|mi |

w1 m1 w2 m2 · · · mk−1 ⊗ wk (v).

That the antiderivation ∂x on Bar(M, A; S) is well deﬁned, is clear. Again, we have (∂x )2 = 0 and ∂x ∂y = −∂y ∂x . If T is a non-empty ﬁnite subset of S, the operator ∂T = x∈T ∂x , is called the T -boundary operator on Bar(M, A; S). As with Bar(A; S; T, i), one can also deﬁne Bar(M, A; S; T, i). The boundary operator ∂T maps Bar(M, A; S; T, i + 1) into Bar(M, A; S; T, i), for i = 0, 1, 2, . . . . Example VII.6.5 Let S be a one-element set, containing the element x. Then the module Bar(M, A; S, i) is spanned by all elements of the form w1 xw2 x · · · wi x ⊗ v

(∗∗)

and the derivation ∂x is computed as follows: ∂x (w1 xw2 x . . . wi x ⊗ v) = w1 w2 x...wi x ⊗ v − w1 xw2 w3 x . . . wi x ⊗ v + · · · + (−1)i−1 w1 xw2 x . . . x ⊗ wi (v).

Resolutions of Weyl modules

215

Thus, the free bar module on M with a single separator gives rise to the classical bar complex, as one sees by replacing the symbol x by the symbol “|”. What we have called Bar(M, A; S, i) are simply the chains in dimension i. Example VII.6.6 Let A be a connected graded algebra with identity (where connected means that the component in degree zero is the ground ring over which we are taking the algebra). The submodule of the module of the example above, spanned by elements of the form (∗∗) where w1 , w2 , ..., wi are all of positive degree, gives the classical normalized bar construction of A and M (see Section III.2) with set of separators S = {x}. Example VII.6.7 With A and M as above, assume that the degree of w1 is greater than t. One then obtains the (t+ )-submodule (and hence, also, the (t+ )-complex) of the normalized bar construction. If in addition M is a graded module, if a positive integer n is ﬁxed, and if in (∗∗) it is assumed that deg(w1 )+ deg(w2 ) + · · · + deg(wi ) + deg(v) = n, one obtains a bar module. Deﬁnition VII.6.8 The bar module which we obtain at the end of Example VII.6.7 is called the (t+ )-graded strand of degree n. VII.6.2 The two-rowed case In this subsection, we will construct the resolution for the two-rowed Weyl module much as it is described in Reference [28]. We do this for two reasons: to illustrate the use of the classical bar complex in our context, and to describe a contracting homotopy for the non-negative part of the resolution and a basis for the syzygies. Recall that the Weyl module associated to the skew-shape t

(A)

p q

is the image of Dp ⊗R Dq under the Weyl map. The “box map” referred to at the very beginning of Section VI.6, and denoted by λ/µ , was described there as the sum of place polarizations, (k) ∂2,1 : Dp+k ⊗R Dq−k → Dp ⊗R Dq . k>t

k>t

If we let Z2,1 stand for the generator of a divided power algebra in one free (k) generator, we see that Z2,1 acts on Dp+k ⊗R Dq−k and carries it to Dp ⊗R Dq . Thus, we may take the (t+ )-graded strand of degree q of the normalized bar complex of this algebra acting on Dp+k ⊗R Dq−k (where the degree of the

216

Some applications of Weyl and Schur modules

second factor determines the grading) to get a complex over the Weyl module: (t+k ) (k ) (k ) Z2,1 1 xZ2,12 x · · · xZ2,1l+1 x ⊗R (Dt+p+|k| ⊗R Dq−t−|k| ) → ··· → ki >0

(t+k1 )

Z2,1

(k )

(k )

xZ2,12 x · · · xZ2,1l x ⊗R (Dt+p+|k| ⊗R Dq−t−|k| ) → · · ·

ki >0

→

(t+k)

Z2,1

x ⊗R (Dt+p+k ⊗R Dq−t−k ) → Dp ⊗R Dq → 0,

k>0

where the symbol “x” is our separator variable as described above, and |k| stands for the sum of the indices ki . Here, the boundary operator is ∂x or, what is the same thing, is obtained by polarizing the variable x to the element 1. This, then, describes a left complex over the Weyl module in terms of bar complexes and letter-place algebra. We also know from the fact that the Weyl module is the cokernel of the box map, that the zero-dimensional homology of this complex is the Weyl module itself. Now the question is: how do we show that this complex is an exact left complex over the Weyl module? In other words, that it is in fact a resolution. One way, is to produce a splitting contracting homotopy, which is what we will do here. Another way is to use our fundamental exact sequences and a mapping cone argument; we refer the reader to Reference [2] for this approach. Deﬁnition VII.6.9 follows:

With our complex given as above, deﬁne the homotopy as

s0 : Dp ⊗R Dq →

k>0

(t+k)

Z2,1

x ⊗R Dt+p+k ⊗R Dq−t−k

4 w 44 1(p) 2(k) to zero if k ≤ t, and to w 4 2(q−k)

sends the double standard tableau 4 w 44 1(p+k) (k) if k > t. For higher dimensions (l > 0), Z2,1 x ⊗ w 4 2(q−k) (t+k ) (k ) (k ) sl : ki >0 Z2,1 1 xZ2,12 x · · · xZ2,1l x ⊗R Dt+p+|k| ⊗R Dq−t−|k| → (kl+1 ) (t+k1 ) (k2 ) xZ2,1 x · · · xZ2,1 x ⊗R Dt+p+|k| ⊗R Dq−t−|k| ki >0 Z2,1 4 w 44 1(t+p+|k|) 2(m) (t+k1 ) (k2 ) (kl ) to is deﬁned by sending Z2,1 xZ2,1 x · · · xZ2,1 x ⊗ w 4 2(q−t−|k|−m) 4 (t+p+|k|+m) w 44 1 (t+k ) (k ) (k ) (m) zero if m = 0, and to Z2,1 1 xZ2,12 x · · · xZ2,1l xZ2,1 x ⊗ w 4 2(q−t−|k|−m) if m > 0. The proofs of the following statements are in Reference [28]. Proposition VII.6.10 The collection of maps {sl }l≥0 provides a splitting contracting homotopy for the complex above.

Resolutions of Weyl modules

217

Theorem VII.6.11 The complex above is a projective resolution of the Weyl module associated to the shape A, over the Schur algebra of appropriate weight. VII.6.3 A three-rowed example In this subsection, we describe the resolution of the three-rowed skew-shape having one triple overlap. A full understanding of this case will help understand the diﬃculties still inherent in the general case, but will also anticipate the procedure by which one discovers the terms (if not the boundary maps) of the general resolution. For the sake of completeness, we will include the description of the general case at the end of the subsection. We start with the skew-shape t1 t2

p1 p2 , p3

where we assume that the number of triple overlaps is ≤ 1. This means that p3 ≤ t1 + t2 + 1. The important thing to notice about this class of shapes is that our fundamental exact sequence is very special in that it does not entail any almost skew-shapes. Namely, the exact sequence we are interested in reduces to: 0 → p1 , p2 + t2 + 1, p3 − t2 − 1; t1 + t2 + 1, −(t2 + 1)) → (p1 , p2 , p3 ; t1 , t2 + 1) → (p1 , p2 , p3 ; t1 , t2 ) → 0.

(VII.6.1)

(The notation we are using above is that used in Section VI.6.) But, since the third row of (p1 , p2 + t2 + 1, p3 − t2 − 1; t1 + t2 + 1, −(t2 + 1)) does not overlap with the top row, we see that the Weyl module associated to this shape is isomorphic to that associated to the skew-shape (p1 , p2 + t2 + 1, p3 − t2 − 1; t1 + t2 + 1, 0). Hence we have the exact sequence of skew-shapes: 0 → (p1 , p2 + t2 + 1, p3 − t2 − 1; t1 + t2 + 1, 0) → (p1 , p2 , p3 ; t1 , t2 + 1) → (p1 , p2 , p3 ; t1 , t2 ) → 0.

(VII.6.2)

We know that the Weyl module is presented by the box map: k>0 Dp1 +t1 +k ⊗R Dp2 −t1 −k ⊗R Dp3 ⊕ → Dp1 ⊗R Dp2 ⊗R Dp3 , : l>0 Dp1 ⊗R Dp2 +t2 +l ⊗R Dp3 −t2 −l where the maps Dp1 +t1 +k ⊗R Dp2 −t1 −k ⊗R Dp3 → Dp1 ⊗R Dp2 ⊗R Dp3 are the kth divided power of the place polarization from place 1 to place 2, and the maps l>0 Dp1 ⊗R Dp2 +t2 +l ⊗R Dp3 −t2 −l → Dp1 ⊗R Dp2 ⊗R Dp3 are the lth divided power of the place polarization from place 2 to place 3. Again taking our cue from the two-rowed case, but recognizing that we now need two separators instead of one, we introduce two generators, Z2,1 and Z3,2 , with their divided

218

Some applications of Weyl and Schur modules

powers, and we write, in place of the above, (t1 +k) x ⊗R Dp1 +t1 +k ⊗R Dp2 −t1 −k ⊗R Dp3 k>0 Z2,1 ⊕ (t2 +l) y ⊗R Dp1 ⊗R Dp2 +t2 +l ⊗R Dp3 −t2 −l l>0 Z3,2

→ Dp1 ⊗R Dp2 ⊗R Dp3 ,

where x and y stand for separator variables, and the boundary map is the sum of polarizing x and y to 1. In short, we see that we have the makings of the free bar module on the set, S, consisting of two separators x and y, the free associative (non-commutative) algebra, A, generated by Z2,1 and Z3,2 , and their divided powers, and the module, M, which is the appropriate direct sum of tensor products of divided power modules Dp ⊗R Dq ⊗R Dr for suitable p, q and r, with the action of Z2,1 and Z3,2 (and their powers) being that of the indicated place polarizations. The boundary map that we use is then ∂x + ∂y . However, we do not take the free bar module on these data, but the quotient after dividing out by the following identities (and it is here that we have the ﬁrst stirrings of the Capelli identities): Z32 x = x Z32 , (b−k) (a−k) (k) = Z21 xZ32 ∂31 ,

(a) (b) Z32 Z21 x

(VII.6.3) (VII.6.4)

k
where the symbol ∂31 means the divided power of the usual place polarization, and its presence on the right means that it is to be brought to operate on the element of M that is to the right of the tensor product sign. For example, if (k) (k) W = W ⊗ m, with m ∈ M, then ∂31 W means W ⊗ ∂31 (m). To illustrate: if (c) we have W = Z21 x ⊗ m, then (b−k) (a−k) (c) (a) (b) (k) Z32 Z21 xW = Z21 xZ32 Z21 x ⊗ ∂31 (m). k
Notice that this in turn becomes (b−k) (c−l) (a−k−l) (l) (k) Z21 xZ21 xZ32 ∂31 ⊗ ∂31 (m) k
=

k + l (b−k) (c−l) (a−k−l) (k+l) ∂31 (m), Z21 xZ21 x ⊗ ∂32 l

k
the last equality being due to the ﬁrst observation of Remark VII.6.4 (as well as to multiplication of divided powers). Remark VII.6.12 The reader may wonder at the fact that we are restricting the index of summation, k, to be less than b, and l < c. We do this because in the applications, we want our divided powers of the Z21 terms to be positive (so that we remain in a normalized complex).

Resolutions of Weyl modules

219

(a)

(b)

Deﬁnition VII.6.13 If we denote by V the element Z32 Z21 xW, where W is (c ) (c ) in the submodule spanned by all terms of the form Z211 x · · · xZ21q x ⊗ m, denote (b−k) (a−k) (k) by V the term k
=

(b−k)

Z21

(c−l)

xZ21

(a−k−l) (l) (k) ∂31 ∂31 (m).

x ⊗ ∂32

k
In order to show that we still get a complex if we factor out by the submodule, Cap(1, 2), generated by the relations of the type V − V we have just been discussing, we must show that these relations are carried into Cap(1, 2) by the boundary map. Since the boundary map, ∂, is just ∂x + ∂y , and since the only separator in the terms of the relations is x, we need only show that the relations are stable under ∂x . To this end we have the following proposition, whose proof is in Reference [30], proposition 5.1. Proposition VII.6.14 Let W be an element of the bar module of the form (c ) (c ) (a) (b) Z211 x · · · xZ21q x ⊗ m, and let V = Z32 Z21 xW. Denote by V the element (b−k) (a−k) (k) xZ32 ∂31 W . Then we have k
(b)

(a)

(b)

∂(V − V ) = Z32 ∂(Z21 xW ) − Z32 ∂(Z21 xW ) = ∂(V ) − ∂(V ), and we have the necessary stability of the relations. Deﬁnition VII.6.15 We deﬁne the module Mn ((p1 , p2 , p3 ; t1 , t2 )) , for n > 0, to be the submodule of the quotient module described above, freely spanned by all elements of the form (l ) (l1 ) (l2 ) (k ) (k ) y · · ·4 yZ3,2p yZ2,11 xZ2,12 x · · · l>0 Z3,2 yZ3,2   w 44 1(π) 2(σ1 ) 3(ρ1 ) (kq )  · · · xZ2,1 x ⊗  w 44 2(σ2 ) 3(ρ2 ) 4 (ρ3 ) w 3

220

Some applications of Weyl and Schur modules

where l1 > t2 ; k1 > t1 + |l|; li > 0; kj > 0; 0 < i ≤ p; 0 < j ≤ q |l| = li ; |k| = kj ; π = p1 + |k|; σ1 + σ2 = p2 + |l| − |k|; ρ1 + ρ2 + ρ3 = p3 − |l|; p + q = n. Further, we deﬁne M0 ((p1 , p2 , p3 ; t1 , t2 )) = Dp1 ⊗R Dp2 ⊗R Dp3 , and we set M−1 ((p1 , p2 , p3 ; t1 , t2 )) = (p1 , p2 , p3 ; t1 , t2 ). As special cases, it should be remarked that when p = 0, we have the terms of the resolution of (p1 , p2 ; t1 ) tensored with Dp3 . When q = 0, we have Dp1 tensored with the terms of the resolution of (p2 , p3 ; t2 ). Notice that the tableau, with the conditions on the π, σi and ρj , simply represents an element of Dp1 +t1 +t2 +|l|+|k| ⊗R Dp2 −t1 −|k| ⊗R Dp3 −t2 −|l| . Notice too that the map ∂x + ∂y maps Mn ((p1 , p2 , p3 ; t1 , t2 )) to Mn−1 ((p1 , p2 , p3 ; t1 , t2 )) for n > 0. To see that, we observe that when we apply ∂x + ∂y , we are removing either an x or a y between the Z1,j or immediately to the left of the tensor product. It is easy to see that in the latter case, our term remains one of the type we allow. Also, when we remove an x between two terms of type Z21 or a y between two of type Z32 , we simply multiply the like terms, and we still have a term of the type allowed. The only questionable term is one in which the y between a Z32 and a Z21 has been removed. It is precisely in this case that we invoke identity (VII.6.4) above. Of course, we have to show that when we apply that identity, the resulting summands are of the allowed type again. But notice that (l )

(l )

(l )

(k )

(k )

Z3,21 yZ3,22 y · · · yZ3,2p Z2,11 xZ2,12 x 4   w 44 1(π) 2(σ1 ) 3(ρ1 ) (kq )  · · · xZ2,1 x ⊗ w 44 2(σ2 ) 3(ρ2 ) 4 (ρ3 ) w 3

Resolutions of Weyl modules

221

is simply lp α=0

(l )

(k −α)

(l )

Z3,21 yZ3,22 y · · · yZ2,11 

w (k ) · · · xZ2,1q x ⊗ w w

(l −α)

xZ3,2p

4 (π−α) 4 1 4 4 2(σ2 ) 4 4 3(ρ3 )

2(σ1 ) 3(ρ2 )

(k )

Z2,12 x 3(α) 3(ρ1 )

 .

p−1 If we set |l | = i=1 li , we see that the exponent on the ﬁrst Z21 is greater than t1 + |l |, so that condition is preserved. The remaining ones are easy to check; we point out that it is here that we use the fact that the index of summation in identity (VII.6.4) is constrained. Theorem VII.6.16 ([30], theorem 5.2) Let (p1 , p2 , p3 ; t1 , t2 ) be a Weyl module with p3 ≤ t1 + t2 + 1. The sequence of modules and maps · · · → Mn ((p1 , p2 , p3 ; t1 , t2 )) → Mn−1 ((p1 , p2 , p3 ; t1 , t2 )) → · · · · · · → M1 ((p1 , p2 , p3 ; t1 , t2 )) → M0 ((p1 , p2 , p3 ; t1 , t2 )) is a projective resolution of (p1 , p2 , p3 ; t1 , t2 ), where the map from M0 ((p1 , p2 , p3 ; t1 , t2 )) to (p1 , p2 , p3 ; t1 , t2 ) is the Weyl map, and the arrows are the maps described above, that is, the maps induced by ∂ = ∂x + ∂y . We will denote this resolution by Res((p1 , p2 , p3 ; t1 , t2 )), and continue to denote the boundary maps by ∂. • The condition that p3 ≤ t1 + t2 + 1 is critical to this argument because otherwise we would not be able to invoke the exactness of the sequence (VII.6.2). To see how important this is the reader should look at the examples in Reference [30], section 5.

Since the description of the terms of our complex, Deﬁnition VII.6.15, seems a bit arbitrary and cumbersome, let us take a little time to give an alternate description. To do this we ﬁrst introduce some more notation. In general we let Zn,m stand for polarizations from place m to place n, and we are going to suppress the various separators that come into play (as a rule, there will be a new separator introduced for each operator Zn,m ; in the case at hand, it will be Z32 with separator y). For a ﬁxed choice of n, m, ξ and l, we will denote by (l) ξ Z n,m Zn,m

the homogeneous strand of the bar complex of total degree ξ + l with initial (0) ξ term of degree ≥ ξ. For example, Z32 Z 32 is just the complex concentrated

222

Some applications of Weyl and Schur modules

(ξ)

(2)

ξ in dimension zero consisting of the element Z32 , while Z32 Z 32 is the complex (ξ+1)

0→

(ξ) Z32 yZ32 yZ32

→

yZ32 (ξ+2) ⊕ → Z32 → 0. (ξ) (2) Z32 yZ32

Z32

If we simply write Z (l) n,m ;

l > 0,

this will mean the homogeneous strand of the normalized bar complex of degree l. Now a typical term of Deﬁnition VII.6.15 arises in the following way. We choose (l) t2 +1 ξ = t2 +1, and any non-negative integer l. We then form the complex Z32 Z 32 and tensor it with Res ((p1 , p2 + t2 + 1 + l; t1 + t2 + 1 + l)) ⊗R Dp3 −(t2 +1+l) . If one looks at the terms of the outcome, one sees that these are the terms described in Deﬁnition VII.6.15. So our complex, Res ((p1 , p2 , p3 ; t1 , t2 )), may be described as Res ((p1 , p2 ; t1 )) ⊗R Dp3 ⊕ (t +1) (l) Z322 Z 32 ⊗R Res((p1 , p2 +t2 +1+l; t1 +t2 +1+l))⊗R Dp3 −(t2 +1+l) . l≥0

The foregoing example illustrates how one goes about constructing resolutions of Weyl modules in an “easy” case. There are many more examples and techniques given in the often-cited paper, [30], which are beyond the scope of this monograph. However, in that paper, as well as at the end of this paragraph, the reader will ﬁnd an explicit description of all the terms of the resolutions of Weyl modules associated to any almost skew-shape. A discussion of the obstacles to describing the boundary maps explicitly is also in that article, as are alternative approaches that may prove to be more eﬀective in solving this problem. Here, then, is the explicit, inductive description of the terms of the resolutions. Notation (for resolutions) For an almost skew-shape, written (p1 , . . . , pm ; t1 , . . . , tm−1 ), we denote its resolution by Res(p1 , . . . , pm ; t1 , . . . , tm−1 ). We assume that we know Res(p1 , . . . , pm ; t1 , . . . , tm−1 ) for m < n, and we describe Res(p1 , . . . , pn ; t1 , . . . , tn−1 ) by induction on n.

Resolutions of Weyl modules

223

Description of Terms of Resolutions (1) For a skew-shape (p1 , . . . , pn ; t1 , . . . , tn−1 ), the resolution looks like this: Res(p1 , . . . , pn ; t1 , . . . , tn−1 ) = Res(p1 , . . . , pn−1 ; t1 , . . . , tn−2 ) ⊗ D(pn ) ⊕

+1

t

(l

)

n−1 n−1 Zn,n−1 Z n,n−1

ln−1 ≥0

⊗ Res(p1 , . . . pn−2 , pn−1 + tn−1 ; t1 , . . . tn−3 , tn−2 + tn−1 ) ⊗ D(pn −tn−1 ) ⊕

+1

t

(l

)

t

+1

(l

)

n−1 n−1 n−2 n−2 (Zn,n−1 Z n,n−1 ) ⊗ (Zn,n−2 Z n,n−2 )

ln−j ≥0

⊗ Res(p1 , . . . pn−2 +tn−2 , pn−1 +tn−1 ; t1 , . . . tn−3 +tn−2 , tn−2 +tn−1 − tn−2 ) ⊗ D(pn −tn−1 −tn−2 ) ⊕ .. .

t

+1

(l

)

t

+1

(l

)

(l )

t1 +1 n−1 n−1 n−2 n−2 1 (Zn,n−1 Z n,n−1 )⊗(Zn,n−2 Z n,n−2 ) ⊗ · · · ⊗(Zn,1 Z n,1 )

ln−j ≥0

⊗ Res(p1 + t1 , . . . , pn−1 + tn−1 ; t1 + t2 − t1 , . . . , tn−2 + tn−1 − tn−2 ) ⊗ D(pn −1≤j≤n−1 tj ) , where tj = tj + lj + 1 for j = 1, . . . , n − 1. (2) For an almost skew-shape (p1 , . . . , pn ; t1 , . . . , tn−1 ) of positive type k, let us denote by s the positive integer −(tn−1 + · · · + tn−k ), and by τ, the integer (tn−1 + · · · + tn−k−1 ). With this notation, the terms of the resolution are Res(p1 , . . . , pn ; t1 , . . . , tn−1 ) = Res(p1 , . . . , pn−1 ; t1 , . . . , tn−2 ) ⊗ D(pn ) (ln−1 ) ⊕ Z n,n−1 ⊗ Res(p1 , . . . pn−2 , pn−1 + ln−1 ; t1 , . . . tn−3 , tn−2 + ln−1 ) ln−1 >0

⊗ D(pn −ln−1 ) (ln−2 ) (ln−1 ) ⊕ (Z n,n−2 ) ⊗ (Z n,n−1 ) ln−2 >0,ln−1 ≥0

224

Some applications of Weyl and Schur modules

⊗ Res(p1 , . . . pn−2 +ln−2 , pn−1 +ln−1 ; t1 , . . . , tn−3 +ln−2 , tn−2 +ln−1 −ln−2 ) ⊗ D(pn −ln−1 −ln−2 ) ⊕ .. .

(l

)

(l

)

(l

)

τ +1 n−k−1 n−k n−1 (Zn,n−k−1 Z n,n−k−1 ) ⊗ Z n,n−k ⊗ · · · ⊗ Z n,n−1 )

ln−j ≥0

⊗ Res(p1 , p2 , . . . , pn−k−1 +τ +1+ln−k−1 , pn−k +ln−k , . . . , pn−1 +ln−1 ; t1 , . . . , tn−k−2 +τ +1+ln−k−1 , s−1+ln−k −ln−k−1 , . . . , tn−2 +ln−1 −ln−2 ) ⊗ D(pn −n−1

n−1

i=n−k−1 ln−i −

⊕

t

+1

(l

j=n−k−1 tj −1)

)

(l

)

(l

)

(l

)

n−k−2 n−k−2 τ+1 n−k−1 n−k n−1 (Zn,n−k−2 Z n,n−k−2 )⊗(Zn,n−k−1 Z n,n−k−1 )⊗Z n,n−k ⊗· · ·⊗Z n,n−1

ln−j ≥0

⊗ Res(p1 , p2 , . . . , pn−k−2 + tn−k−2 , pn−k−1 + τ + 1 + ln−k−1 , pn−k + ln−k , . . . , pn−1 +ln−1 ; t1, . . . tn−k−3 +tn−k−2 , tn−k−2 +τ +1+ln−k−1 −tn−k−2 , . . . , tn−2 + ln−1 − ln−2 ) ⊗ D(pn −n−1

n−1

i=n−k−2 ln−i −

j=n−k−2 tj −2)

⊕

.. .

t

(l )

+1

(l

)

(l

)

t1+1 n−k−2 n−k−2 τ+1 n−k−1 1 (Zn,1 Z n,1 )⊗· · ·⊗(Zn,n−k−2 Z n,n−k−2 )⊗(Zn,n−k−1 Z n,n−k−1 )

ln−j ≥0 (l

)

(l

)

n−k n−1 ⊗· · ·⊗Z n,n−1 ⊗ Z n,n−k

⊗ Res(p1 + t1 , . . . , pn−k−2 + tn−k−2 , pn−k−1 + τ + 1 + ln−k−1 , pn−k + ln−k , . . . , pn−1 + ln−1 ; t1 + t2 − t1 , . . . , tn−k−3 + tn−k−2 − tn−k−3 , tn−k−2 + τ + 1 + ln−k−1 − tn−k−2 , . . . , tn−2 + ln−1 − ln−2 ) ⊗ D(pn −n−1 tj −k) , j=1

where tj is deﬁned as above. It must be understood that if the indicated shape is not legitimate, then that term vanishes. This puts some constraints on the parameters over which we are summing.

Resolutions of Weyl modules

225

VII.6.4 Resolutions of skew-hooks There is a class of skew-shapes for which it is possible to completely describe resolutions, namely the skew-hooks (see Reference [31] as well as Reference [21]). These shapes are essentially “hooks strung together” like this: k1 .. .

l1

.. . kt−1 .. .

. lt−1

kt .. .

lt

The one illustrated is a t-hook, and the indices ki , li are the lengths of the horizontals (arms) and verticals (legs), respectively. The arms always include the left-most box, but except for the top-most arm, exclude the right-most box. The legs always include the bottom box, but exclude the top-most (as it is already being counted as part of the arm). We insist that the ki always be positive so that if, for instance, k1 = 1, this means that we are essentially starting with a vertical piece, etc. We will denote the t-hook by (k1 , . . . , kt ; l1 , . . . , lt ). Note that if lt = 0, the skew-hook ends with an arm; we have li > 0, however, if i < t. Thus, the skew-hook represented by (1, 3; 2, 0) looks like this: , while the one represented by (1, 3; 1, 0) is: . In order to describe the resolution of K(k1 ,...,kt ;l1 ,...,lt ) , we ﬁrst introduce some general notation. Then we will apply this to our skew-hooks and indicate why this yields the sought-for resolution. Let n be a ﬁxed positive integer, and S = {s1 , . . . , sp } be a strictly increasing set of integers, with sp = n. Denote by Xn the complex consisting of terms, in

226

Some applications of Weyl and Schur modules

dimension n − α, of the form Di1 ⊗R · · · ⊗R Diα with i1 + · · · + iα = n, ij > 0, and with the boundary map, δ, deﬁned by δ(Di1 ⊗R · · · ⊗R Diα ) =

α

(−1)j+1 Di1 ⊗R · · · ⊗R ∆ (Dij ) ⊗R · · · ⊗R Diα ,

j=1

where ∆ is the “normalized” diagonal map. By “normalized” diagonal map we mean that we never allow the degrees to be zero. That is, the normalized diagonal map will map Di to 0<β p, there are no such terms. If α = p, there is only one term: Ds1 ⊗R Ds2 −s1 ⊗R · · · ⊗R Dn−sp−1 . It is this type of complex, Xn / ( S), that is going to provide the resolution of our skew-shapes. If our complex is going to be the resolution of some Weyl module, the rows of the shape of that module determine the set S uniquely. In particular, if we want such a complex to resolve a skew-hook, then we can describe explicitly what the set S looks like. As before, we will denote the skew-hook by (k1 , . . . , kt ; l1 , . . . , lt ) and the total number of boxes in this count is simply |k| + |l|. In this notation, the resolution should start oﬀ with l1 −1

Dk1

l2 −1

# $% & # $% & ⊗R D1 ⊗R · · · ⊗R D1 ⊗R Dk2 +1 ⊗R D1 ⊗R · · · ⊗R D1 ⊗R · · · lt

# $% & ⊗R Dkt +1 ⊗R D1 ⊗R · · · ⊗R D1 . Hence, the set S that we are looking at is

S=

{k1 , k1 + 1, . . . , k1 + l1 − 1, k1 + k2 + l1 , k1 + k2 + l1 + 1, . . . , k1 + k2 + l1 + l2 − 1, k1 + k2 + k3 + l1 + l2 , . . . , k1 + · · · + kt−1 + l1 + · · · + lt−1 − 1, |k| + |l| − lt , . . . , |k| + |l|}.

Resolutions of Weyl modules

227

Next we look at the following situation. Suppose we have the complex Xk , and we have two sets of integers T ⊂ S, with both of them containing k. Then since ( S) ⊂ ( T ) , we clearly have Xk / ( S) mapping surjectively onto Xk / ( T ) . Now we do not know what the kernel of this morphism is in general, but suppose that the integer n is in S (n < k), and we let T = S − {n}. Then it is easy to see that the kernel, K, of the morphism is spanned by the terms Di1 ⊗R · · · ⊗R Diα of Xk / ( S) having the property that for some β, i1 + · · · + iβ = n. If we set S = {s1 , . . . , sp , n + t1 , . . . , n + tq } , where sp = n, and tq = k − n = m, then K is isomorphic to Xn / ( {s1 , . . . , sp }) ⊗R Xm / ( {t1 , . . . , tq }) . Moreover, there is a map of Xk / ( T ) into Xn / ( {s1 , . . . , sp }) ⊗R Xm / ( {t1 , . . . , tq }) which is fairly easy to deﬁne. Namely, suppose that Di1 ⊗R · · · ⊗R Diα is a term in Xk / ( T ) . This means that for all β, we have i1 + · · · + iβ = n. Then there is a β0 with i1 + · · · + iβ0 −1 = n − c < n, and i1 + · · · + iβ0 = n + d > n. We map the term Di1 ⊗R · · · ⊗R Diα to the term Di1 ⊗R · · · ⊗R Diβ0 −1 ⊗R Dc ⊗R Dd ⊗R Diβ0 +1 ⊗R · · · ⊗R Diα , by diagonalizing Diβ0 , which makes sense since iβ0 = c + d . In fact, the mapping cone of this map (notice the shift of dimension by 1 here), is precisely Xk / ( S) . Application to skew-hooks If we now start with a skew-hook (k1 , . . . , kt ; l1 , . . . , lt ), we get by Theorem VII.1.4 the exact sequence 0 → (k1 , . . . , kt−1 , kt + 1; l1 , . . . , lt−1 − 1, lt ) → (k1 , . . . , kt−1 ; l1 , . . . , lt−1 − 1) ⊗R (kt + 1; lt ) → (k1 , . . . , kt ; l1 , . . . , lt ) → 0. We want to show, by induction on t and the number of rows, that the resolution of (k1 , . . . , kt ; l1 , . . . , lt ) is our complex X|k|+|l| / ( S) where S is as above. By induction, we know that the resolution of (k1 , . . . , kt−1 , kt + 1; l1 , . . . , lt−1 − 1, lt ) is X|k|+|l| / ( T ) , where T = S − {|k| + |l| − lt − 1} . On the other hand, we see that the resolution of (k1 , . . . , kt−1 ; l1 , . . . , lt−1 − 1) ⊗R (kt + 1; lt ) is the tensor product of two complexes which is precisely the kernel of the map of X|k|+|l|/( S) onto X|k|+|l| / ( T ) . Our discussion in the preceding part of this subsection completes the proof. VII.6.5 Comparison with the Lascoux resolutions We end this section, and in fact the whole chapter, with a very detailed comparison of the characteristic-free resolution of the Weyl module, K(2,2,2) , with that of A. Lascoux (whose maps are not put in evidence). We hope this will serve to make clear what is behind the Lascoux resolutions, and how one can deal in a “hands on” way with the characteristic-free case.

228

Some applications of Weyl and Schur modules

The terms of the Lascoux resolution of any skew-shape are read oﬀ from the determinantal expansion of the Jacobi–Trudi matrix of the shape, with the position of the terms of the complex determined by the length of the permutations to which they correspond. In K-theory notation, the Jacobi–Trudi matrix of the skew-shape λ/µ = (p1 . . . . , pn ; t1 , . . . , tn−1 ) is the n × n matrix  (J − T ) :

     

Dp1 Dp2 − t1 Dp3 − t2 .. . Dpn − tn−1

Dp1 +t1 +1 Dp2 Dp3 −t2 −1 .. . Dpn −t2

Dp1 + t2 Dp2 +t2 +1 Dp3 .. . Dpn −t3

··· ··· ··· .. . ···

Dp1 +t1 Dp2 +t2 Dp3 +t3 .. . Dpn

    ,  

where ti = ti + · · · + tn−1 + (n − i), and ti = t1 − ti+1 .

VII.6.5.1 Lascoux and non-Lascoux resolutions for (2,2,2) The Jacobi–Trudi matrix of the partition (2,2,2) is 

D2  D1 D0

D3 D2 D1

 D4 D3  , D2

and the Lascoux resolution of the Weyl module associated to the partition (2, 2, 2) looks like this:

0 → D4 ⊗R D2 ⊗R D0 →

D3 ⊗R D3 ⊗R D0 D3 ⊗R D1 ⊗R D2 → → D2 ⊗RD2 ⊗RD2 → 0. ⊕ ⊕ D4 ⊗R D1 ⊗R D1 D2 ⊗R D3 ⊗R D1

The correspondence between the terms of the resolution above, and permutations, is as follows: D2 ⊗R D2 ⊗R D2 D3 ⊗R D1 ⊗R D2 D2 ⊗R D3 ⊗R D1 D3 ⊗R D3 ⊗R D0 D4 ⊗R D1 ⊗R D1 D4 ⊗R D2 ⊗R D0

←→ ←→ ←→ ←→ ←→ ←→

identity (12) (23) (123) (213) (13).

Resolutions of Weyl modules

229

By contrast, the terms of the characteristic-free resolution of the same Weyl module, are: X0 = D2 ⊗R D2 ⊗R D2 ; (1) (2) X1 = Z2,1 x ⊗R D3 ⊗R D1 ⊗R D2 ⊕ Z2,1 x ⊗R D4 ⊗R D0 ⊗R D2 ⊕ (1) (2) Z3,2 y ⊗R D2 ⊗R D3 ⊗R D1 ⊕ Z3,2 y ⊗R D2 ⊗R D4 ⊗R D0 ; (1) (1) X2 = Z2,1 xZ2,1 x ⊗R D4 ⊗R D0 ⊗R D2 ⊕ (1) (2) (1) (3) Z3,2 yZ2,1 x ⊗R D4 ⊗R D1 ⊗R D1 ⊕ Z3,2 yZ2,1 x ⊗R D5 ⊗R D0 ⊗R D1 ⊕ (2) (3) (2) (4) Z3,2 yZ2,1 x ⊗R D5 ⊗R D1 ⊗R D0 ⊕ Z3,2 yZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ⊕ (1) (1) Z3,2 yZ3,2 y ⊗R D2 ⊗R D4 ⊗R D0 ⊕ (1) (1) Z3,2 yZ3,1 z ⊗R D3 ⊗R D3 ⊗R D0 ; (1) (2) (1) X3 = Z3,2 yZ2,1 xZ2,1 x ⊗R D5 ⊗R D0 ⊗R D1 ⊕ (2) (3) (1) Z3,2 yZ2,1 xZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ⊕ (1) (1) (3) Z3,2 yZ3,2 yZ2,1 x ⊗R D5 ⊗R D1 ⊗R D0 ⊕ (1) (1) (4) Z3,2 yZ3,2 yZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ⊕ (1) (1) (1) Z3,2 yZ3,1 zZ2,1 x ⊗R D4 ⊗R D2 ⊗R D0 ⊕ (1) (1) (2) Z3,2 yZ3,1 zZ2,1 x ⊗R D5 ⊗R D1 ⊗R D0 ⊕ (1) (1) (3) Z3,2 yZ3,1 zZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ; (1) (1) (3) (1) X4 = Z3,2 yZ3,2 yZ2,1 xZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ⊕ (1) (1) (1) (1) Z3,2 yZ3,1 zZ2,1 xZ2,1 x ⊗R D5 ⊗R D1 ⊗R D0 ⊕ (1) (1) (2) (1) Z3,2 yZ3,1 zZ2,1 xZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ⊕ (1) (1) (1) (2) Z3,2 yZ3,1 zZ2,1 xZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 ; (1) (1) (1) (1) (1) X5 = Z3,2 yZ3,1 zZ2,1 xZ2,1 xZ2,1 x ⊗R D6 ⊗R D0 ⊗R D0 , where the subscripts on the X indicate the dimension in which these terms (t) appear. The symbols Za,b are the formal “polarization” operators deﬁned in preceding subsections, following Reference [30], and the letters x, y, z are the separator variables also explained in that paper and above. The boundary map for this complex is obtained by polarizing all the separator (t) variables to one. When the separator x disappears between a Za,b and elements (t)

in the tensor product of divided powers, this means ∂a,b , or the place polarization operator, applied to that tensor product. The only essentially new terms that (1) (1) we have here are the terms that involve Z3,2 yZ3,1 z, or more generally terms of the form: (1)

(1)

(k )

(k

)

Z3,2 y Z3,1 z Z2,11 x · · · x Z2,1n−2 x ⊗R D2+1+|k| ⊗R D2+1−|k| ⊗R D0 , (of which, in this example, there are not that many). For these we have to use identities of Capelli type, or some easy variants of them (Reference [30] and

230

Some applications of Weyl and Schur modules

Section VI.4). The boundary map on such a term sends it to (k ) (1) (1) (k ) Z3,2 y Z3,1 z ∂ Z2,11 x · · · x Z2,1n−2 x ⊗R Dp+1+|k| ⊗R Dq+1−|k| ⊗R D0 (k ) (1) (k ) (1) ± Z3,2 y Z3,1 Z2,11 x · · · x Z2,1n−2 x ⊗R Dp+1+|k| ⊗R Dq+1−|k| ⊗R D0 (k ) (1) (1) (k ) ∓ Z3,2 Z3,1 z Z2,11 x · · · x Z2,1n−2 x ⊗R Dp+1+|k| ⊗R Dq+1−|k| ⊗R D0 , and we have to deﬁne the terms (k

)

(1)

(k )

(1)

(k ) (k ) Z2,11 x · · · x Z2,1n−2 x ⊗R Dp+1+|k| ⊗R Dq+1−|k| ⊗R D0

Z3,1 Z2,11 x · · · x Z2,1n−2 x ⊗R Dp+1+|k| ⊗R Dq+1−|k| ⊗R D0 and (1)

Z3,2 Z3,1 z

with n ≥ 2. If n = 2, we have (1)

Z3,1 v = ∂3,1 (v), while (1)

(1)

(1)

(2)

(2)

Z3,2 Z3,1 z ⊗ v = −Z2,1 x ⊗ ∂3,2 (v)+Z3,2 y ⊗ ∂2,1 (v). For n > 2, we have (1)

(k

(k )

)

(k +1)

(1)

(k

(k )

)

Z3,1 Z2,11 x · · · xZ2,1n−2 x ⊗ v = −Z2,11 xZ3,2 Z2,12 x · · · xZ2,1n−2 x ⊗ v k1 + k2 (k ) (1) (k +k +1) Z3,2 yZ2,11 2 x · · · xZ2,1n−2 x ⊗ v, + k2 − 1 and (1)

(1)

(k

(k )

)

(1)

(2)

(k

(k )

)

Z3,2 Z3,1 zZ2,11 x · · · xZ2,1n−2 x ⊗ v = −Z2,1 xZ3,2 Z2,11 x · · · xZ2,1n−2 x ⊗ v (2)

(k +1)

+ (k1 − 1)Z3,2 yZ2,11

(k

)

x · · · xZ2,1n−2 x ⊗ v.

With these identities in play, we can easily write down the boundary map for this resolution. VII.6.5.2 Reduction of non-Lascoux to Lascoux It is of some interest to see how we can discard the “excess” terms of the characteristic-free resolution to recover that of A. Lascoux. For example, we want (2) (2) to throw away the terms Z2,1 x⊗R D4 ⊗R D0 ⊗R D2 and Z3,2 y⊗R D2 ⊗R D4 ⊗R D0 . (2)

If we look at the image of a term Z2,1 x ⊗ v, where v ∈ D4 ⊗ D0 ⊗ D2 , we (2)

1 2 2 ∂21 (v). So, (1) ∂21 (v). Hence

see that it is ∂21 (v) = 1 (1) 2 Z2,1 x

(2)

the image of Z2,1 x ⊗ v is the same as

⊗ in characteristic zero we can rig up the the image of boundary map taking this into account. Obviously the same kind of thing holds

Resolutions of Weyl modules

231

(2)

for the term Z3,2 y ⊗R D2 ⊗R D4 ⊗R D0 that is, its image is the same as that of 1 (1) 2 Z3,2 x

(1)

⊗ ∂32 (v). Thus, any term that is sent by the “full” boundary map into one of the “redundant” terms above, should now be sent into the non-redundant term instead. For example, the following terms should now be sent as indicated: (1)

(2)

(1)

(1)

(1)

(2)

(1)

(1)

Z3,2 yZ2,1 x ⊗ v → Z3,2 y ⊗ ∂2,1 (v) − Z2,1 x ⊗ ∂3,1 (v) − 12 Z2,1 x ⊗ ∂2,1 ∂3,2 (v) and (1)

(1)

(2)

(1)

Z3,2 yZ3,1 z ⊗ v → Z3,2 y ⊗ ∂3,1 (v) − Z2,1 x ⊗ ∂3,2 (v) − 12 Z3,2 y ⊗ ∂3,2 ∂2,1 (v). With this modiﬁcation, the terms of the type (1)

(1)

(1)

(1)

Z2,1 xZ2,1 x ⊗R D4 ⊗R D0 ⊗R D2 and Z3,2 yZ3,2 y ⊗R D2 ⊗R D4 ⊗R D0 are automatically sent to zero. As a result, when we go to the last term that counts in this complex, we see that under the boundary map in the big complex we get (1)

(1)

(1)

(1)

(1)

(1)

(2)

Z3,2 yZ3,1 z Z2,1 x ⊗ v → Z3,2 yZ3,1 z ⊗ ∂2,1 (v) + Z3,2 y Z2,1 x ⊗ ∂3,2 (v) (1)

(1)

(2)

(1)

(1)

(2)

− Z3,2 yZ3,2 y ⊗ ∂2,1 (v) − Z2,1 xZ2,1 x ⊗ ∂3,2 (v), so that under the modiﬁed boundary, we can simply deﬁne the boundary map on this term to be (1)

(1)

(1)

(1)

(1)

(1)

(2)

Z3,2 yZ3,1 z Z2,1 x ⊗ v → Z3,2 yZ3,1 z ⊗ ∂2,1 (v) + Z3,2 y Z2,1 x ⊗ ∂3,2 (v). Of course it remains to prove that in characteristic zero this is exact; we will now indicate how to do this. VII.6.5.3 Question of exactness Here, we indicate more schematically how we modify the maps to take advantage of divisibility in Q. Divide the terms of our big complex into the sum of those of the Lascoux complex, and the others. That is, we look at the terms of the complex as: X0 X1 X2 X3 Xj

= A0 = A1 ⊕ B1 = A2 ⊕ B2 = A3 ⊕ B3 = Bj for j = 4, 5,

where the Ai are the sums of the “Lascoux” terms, and the Bl are the sums of the others.

232

Some applications of Weyl and Schur modules

Let σ1 be the map B1 → A1 deﬁned by (2)

Z2,1 x ⊗ v −→

1 (1) Z x ⊗ ∂2,1 (v) 2 2,1

(2)

and Z3,2 y ⊗ w −→

1 (1) Z y ⊗ ∂3,2 (w) 2 3,2

where v ∈ D4 ⊗R D0 ⊗R D2 and w ∈ D2 ⊗R D4 ⊗R D0 . This situation we have already discussed, but we should point out that the map σ1 satisﬁes the identity: (C1 )

δA1 A0 σ1 = δB1 B0 ,

where by δA1 A0 we mean the component of the boundary of the large complex that carries A1 to A0 , and similarly for δB1 B0 . Naturally, we will use notation δAi+1 Ai , δAi+1 Bi etc. in the same way. We deﬁne ∂1 : A1 → A0 as ∂1 = δA1 A0 . But now we are in position to deﬁne ∂2 : A2 → A1 by ∂2 = δA2 A1 + σ1 δA2 B1 . We get immediately Remark VII.6.17 Proof

The composition ∂1 ∂2 = 0.

We have ∂1 ∂2 (a) = δA1 A0 {δA2 A1 (a) + σ1 δA2 B1 (a)} = δA1 A0 δA2 A1 (a) + δA1 A0 σ1 δA2 B1 (a).

But since δA1 A0 σ1 = δB1 B0 , we see that δA1 A0 σ1 δA2 B1 (a) = δB1 B0 δA2 B1 (a), so δA1 A0 δA2 A1 (a) + δA1 A0 σ1 δA2 B1 (a) = δA1 A0 δA2 A1 (a) + δB1 B0 δA2 B1 (a). But this is zero since it is the boundary of the fat complex applied to a. Before we can prove exactness at A1 , we have to deﬁne a map σ2 : B2 → A2

2

Resolutions of Weyl modules

233

such that (C2 )

δB2 A1 + σ1 δB2 B1 = {δA2 A1 + σ1 δA2 B1 }σ2 .

We deﬁne this map as follows: (1)

(1)

Z2,1 x Z2,1 x ⊗ v (1) (1) Z3,2 y Z3,2 y ⊗ v (1) (3) Z3,2 y Z2,1 x ⊗ v (2) (3) Z3,2 y Z2,1 x ⊗ v (2) (4) Z3,2 y Z2,1 x ⊗ v

−→ 0 −→ 0 (1) (2) −→ 13 {Z3,2 y Z2,1 x ⊗ ∂2,1 (v)} (1) (2) (1) (1) (2) −→ 13 {Z3,2 yZ2,1 x ⊗ ∂3,1 (v)−Z3,2 yZ3,1 z ⊗ ∂2,1 (v)} (1) (1) (3) −→ − 12 {Z3,2 y Z3,1 z ⊗ ∂2,1 (v)}

where the element v is in the appropriate tensor product of divided powers as given in the description of the large complex. Remark VII.6.18 Proof

2

Trivial.

Remark VII.6.19 Proof

The map σ2 deﬁned above satisﬁes the condition (C2 ).

We have exactness at A1 . 2

This is just a diagram chase.

Using σ2 we can now also deﬁne ∂3 : A3 → A2 by ∂3 = δA3 A2 + σ2 δA3 B2 . Not too surprisingly, we have Remark VII.6.20

∂2 ∂3 = 0.

Of course, what we need now is a map σ3 : B3 → A3 similar to the maps σ above, that is, satisfying (C3 )

δB3 A2 + σ2 δB3 B2 = {δA3 A2 + σ2 δA3 B2 }σ3 .

We deﬁne such a σ3 as follows: (1)

(2)

(1)

Z3,2 y Z2,1 x Z2,1 x ⊗ v (2) (3) (1) Z3,2 y Z2,1 x Z2,1 x ⊗ v (1) (1) (3) Z3,2 y Z3,2 y Z2,1 x ⊗ v (1) (1) (4) Z3,2 y Z3,2 y Z2,1 x ⊗ v (1) (1) (1) Z3,2 y Z3,1 z Z2,1 x ⊗ v (1) (1) (3) Z3,2 y Z3,1 z Z2,1 x ⊗ v

−→ −→ −→ −→ −→ −→

0 (1) (1) (1) (2) 1 3 {Z3,2 y Z3,1 z Z2,1 x ⊗ ∂2,1 (v)} (1) (1) (1) (1) − 13 {Z3,2 y Z3,1 z Z2,1 x ⊗ ∂2,1 (v)} (1) (1) (1) (2) − 13 {Z3,2 y Z3,1 z Z2,1 x ⊗ ∂2,1 (v)} (1) (1) (1) (1) 1 3 {Z3,2 y Z3,1 z Z2,1 x ⊗ ∂2,1 (v)}

0

234

Some applications of Weyl and Schur modules

where the element v again is taken in the appropriate tensor product of divided powers as described earlier. Remark VII.6.21

The map σ3 deﬁned above satisﬁes the condition (C3 ). 2

Proof Trivial.

We have yet to prove that we have exactness at A2 and that ∂3 is a monomorphism. But ﬁrst we explicitly deﬁne the boundary maps in the complex ∂

∂

∂

0 → A3 →3 A2 →2 A1 →1 A0 . The map ∂1 is clear: it is just the operation of the indicated polarization operators on the argument. The map ∂2 is deﬁned as: (1) (2) (1) (2) (1) (1) (1) (1) (1) ∂2 Z3,2 yZ2,1 x⊗v = Z3,2 y⊗∂2,1 (v)−Z2,1 x⊗∂3,1 (v)− 12 Z2,1 x⊗∂2,1 ∂3,2 (v); (1) (1) (1) (1) (1) (1) (1) (1) (2) ∂2 Z3,2 yZ3,1 z⊗v = Z3,2 y⊗∂3,1 (v)− 12 Z3,2 y⊗∂3,2 ∂2,1 (v)+Z2,1 x⊗∂3,2 (v). And ﬁnally, the map ∂3 is deﬁned as: (1) (1) (1) (1) (1) (1) (1) (2) (1) ∂3 Z3,2 yZ3,1 z Z2,1 x⊗v = Z3,2 yZ3,1 z⊗∂2,1 (v) + Z3,2 yZ2,1 x⊗∂3,2 (v). Since one component of the map ∂3 is a diagonalization of D2 into D1 ⊗R D1 , it is clear that ∂3 is injective. Hence, the only remaining exactness to prove is at A2 . To handle this, we suppose that we have V ∈ A1 with ∂2 (V ) = 0. We want to produce an element W ∈ A3 such that ∂3 (W ) = V. It is easy to see that we can add to V an element W ∈ B2 , such that under the boundary of the characteristic-free complex, V + W goes to zero, while σ2 (W ) = 0. Using the fact that this large complex is acyclic, we know that there are elements α ∈ A3 , β ∈ B3 such that α + β goes to V + W under the boundary map of that complex. It is then easy to show that α + σ3 (β) goes to V , and our Lascoux “reduction” is exact. VII.6.5.4 The Lascoux skeleton in general We end this subsection with a description of an inductive method for ﬁnding the “Lascoux skeleton” inside the characteristic-free resolution of any skew-shape. That is, if we are given a term in the expansion of the Jacobi–Trudi determinant (the determinant of the matrix denoted by (J−T) at the beginning of this subsection) corresponding to the permutation, σ ∈ Sn , we will show where, among the terms of the characteristic-free resolution, this term will be found. Recall that Lascoux [64] has described the terms and their placement in the resolutions of Weyl modules (of skew-partitions) in terms of the lengths of the permutations corresponding to the determinantal expansion of the Jacobi–Trudi matrix for that Weyl module. If σ ∈ Sn is such that σ(n) = i, then σ can be

Resolutions of Weyl modules

235

written uniquely as a product: σ = (n, n − 1)(n, n − 2) · · · (n, i)σ , where (n, j) stands for the transposition on n and j, and σ ∈ Sn−1 . (Notice that the length of σ is n − i + length(σ ).) One then goes to the part of the characteristic-free resolution of λ/µ that involves the terms t

+1

(l

)

t

+1

(l

)

(l )

ti +1 n−1 n−1 n−2 n−2 Z n,n−1 )⊗ (Zn,n−2 Z n,n−2 ) ⊗ · · · ⊗ (Zn,i Z n,ii ), (Zn,n−1

and then ﬁnds inside the corresponding Res the term corresponding to σ . This is the way of recovering the terms of the Lascoux resolution within the resolution described at the end of Subsection VII.6.3.

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APPENDIX A APPENDIX FOR LETTER-PLACE METHODS

In this appendix, we furnish the proofs of the theorems stated in Sections VI.3 and VI.8 on letter-place methods. More precisely, Sections A.1 and A.2 give a complete proof of Theorem VI.3.2 which involves straightening considerations (Section A.1) as well as a combinatorial procedure known as the Robinson– Schensted–Knuth correspondence (Section A.2). In Section A.3 we deal in a less detailed way with the proofs of Theorems VI.3.3 and VI.3.4, mainly showing what modiﬁcations are necessary to adapt the methods of proof of Theorem VI.3.1 to the other situations. In Section A.4, we discuss the modiﬁcations necessary for the proof of Theorem VI.8.4. A.1 Theorem VI.3.2, Part 1: the double standard tableaux generate To show the generation by double standard tableaux, we ﬁrst of all can assume that they are row standard all the time, since we can always row-standardize them without changing their values. We can also assume that the lengths of the rows are decreasing, since we can always arrange that without changing their values. So we are talking about double tableaux of the form:   w1 1(a11 ) 2(a21 ) 3(a31 ) · · ·  w2 1(a12 ) 2(a22 ) 3(a32 ) · · ·   (S)   ··· ··· ··· ··· ···  wn 1(a1n ) 2(a2n ) 3(a3n ) · · · where αi = (a1i + a2i + a3i + · · · ) ≥ αj for 1 ≤ i < j ≤ n, and the words, when written out, are increasing. Our tableaux of the form (W ):   w1 1(k1 )  w2 2(k2 )   (W )   ··· ···  wn n(kn ) are, by rearranging the order of the rows if necessary, of type (S) above. Since they form a basis for Dk1 (F )⊗· · ·⊗Dkn (F ), it certainly suﬃces to prove that any double tableau of type (S) is a linear combination of standard double tableaux. Our procedure is to put a quasi order on these double tableaux. The object is to show that we can express a double tableau that is not standard in the places (or letters) as a linear combination of others which are “lower” in the quasi order.

238

Appendix for letter-place methods

Since the set of tableaux is ﬁnite, this will show that we can express any double tableau as a linear combination of such that are standard in the places. We then do the same for the letters. As this process cannot go on forever, each double tableau must eventually be a linear combination of standard double tableaux; that is, the standard ones generate. At the cost of being ultra pedantic, we shall denote a double tableau, T , by a triple: T = (λ; L, P ), to denote the diagram, the “letter,” and the “place” parts of the tableau. We stress that the diagram is always to be that of a partition in this case, that is, the lengths of the rows are decreasing from top to bottom. In Subsection VI.1.1, we described a well-known partial order on partitions: we said that λ > λ if the ﬁrst row of λ (from the top) which diﬀers in length from the corresponding row of λ, is longer than that of λ. We also deﬁned in Subsection VI.1.2, a quasi order on single tableaux. Using these orderings, we now deﬁne a quasi ordering of our double tableaux. Deﬁnition A.1.1 Given tableaux, T = (λ; L, P ) and T = (λ ; L , P ), we deﬁne (λ ; L , P ) ≤ (λ; L, P ) if λ ≥ λ, Lij ≥ Lij for all i, j, and Pij ≥ Pij for all i, j. We then say that (λ ; L , P ) < (λ; L, P ) if (λ ; L , P ) ≤ (λ; L, P ), and either λ > λ or λ = λ and either Lij > Lij for some i, j, or Pij > Pij for some i, j. To give an example, and also to see how the general “straightening” proceeds, consider an elementary, yet prototypical situation: w1 1(a1 ) 2(b1 ) , T = w2 1(2) 2(b2 ) where w1 and w2 are words of degrees a1 + b1 and 2 + b2 respectively. (We are now dropping the cumbersome designation of a tableau as a triple.) Our problem in the “place” section of the tableau is that we have 1 in the bottom row, so that no matter what the value of a1 , that section is not standard. Recall that T = w1 (a1 )w2 (2) ⊗ w1 (b1 )w2 (b2 ) where the letters in parentheses indicate the degrees of the wi under appropriate diagonalization. (We will continue to use this notation to indicate that we have diagonalized our terms in the indicated degrees.) Consider now: (a1 +2) (b1 ) w w (2) 1 2 1 2 T = , w2 (b2 ) 2(b2 ) w1 1(a1 +1) 2(b1 −1) , T = w2 1 2(b2 +1) w1 1(a1 +2) 2(b1 −2) T = . w2 2(b2 +2)

Theorem VI.3.2, Part 1

Then it is straightforward to check that

239

b2 + 2 T = T − (b2 + 1)T − T . 2

• It is extremely important to observe at this point that the diagram for T has changed shape, but it is bigger than that of our original T . The shapes of T and T are the same as our original, the letter part of the tableau has not changed (that is, L = L = L), but we have both P and P which are diﬀerent from P . In both cases, though, we have decreased these tableaux (in the quasi order). Notice that if we were to “straighten” our letter tableau similarly, the changes would either lead to a bigger diagram, in which case we go down in the order, or the places would remain unchanged, and the letter tableaux that intervened would be lower in the quasi order. • When checking this type of calculation, it is most often convenient to assume that the words, wi , are simply divided powers. That is, we might assume that w1 = x(a1 +b1 ) and that w2 = y (2+b2 ) . This makes keeping track of the diagonalizations much easier.

The point of this exercise is to show that when we get rid of an instance of non-standardness, the tableaux that emerge in the process are all less, in the quasi-order we introduced, than the tableau we started with. In the case of T , this is due to the fact that the diagram of T is properly larger than that of T ; in the other two cases, the “letter” side of the tableau has not changed, but the “place” side has properly decreased. Of course, the tableau, T , is manifestly non-standard (the others may or may not be, depending on the relative values of a1 , b1 and b2 ). To see how a greater number of places aﬀect the straightening procedure, we will look at one more example. To this end, consider: w1 1(a1 ) 2(b1 ) 3(c1 ) . T = w2 1 2(b2 ) 3(c2 ) Then if we set:

1(a1 +1) 2(b1 ) 3(c1 ) w1 w2 (1) . w2 (b2 + c2 ) 2(b2 ) 3(c2 ) w1 1(a1 +1) 2(b1 −1) 3(c1 ) , T = w2 2(b2 +1) 3(c2 )

T =

and T

=

w1 w2

1(a1 +1) 2(b2 )

2(b1 )

3(c1 −1)

3(c2 +1)

we have T = T − (b2 + 1)T − (c2 + 1)T .

,

240

Appendix for letter-place methods

Again we see that we have eliminated the oﬀending 1 in the bottom row, and we have “replaced” T by the tableaux T , T and T which are strictly lower in our quasi-order than T . As these two examples show, it is enough to work either on the letter or place side of the tableau to push us further toward standardness. These examples are prototypical in the sense that it is enough to work on straightening two adjacent rows, for if we have a tableau that is not standard, this is because it is not strictly increasing in the columns (since we start with row-standardness). But then a violation of strict increase must also occur in two adjacent rows. This is the reason that we can focus on two-rowed double tableaux such as: w1 z11 z12 z13 · · · z1k T = w2 z21 z22 z23 · · · z2l where we write zij to represent the integers from 1 to n, with possible repeats. Suppose that the ﬁrst violation of strict increase occurs in the ith column, that is, z11 < z21 , . . . , z1i−1 < z2i−1 , but z1i ≥ z2i . (The ith column could be the ﬁrst, as it was in the above examples.) Next, suppose that z2i = · · · = z2t < z2t+1 (possibly t = i), and consider the sum of tableaux: w1 w2 (t) z11 · · · z1i−1 z21 · · · z2t z1i · · · z1k T = . w2 (l − t) z2t+1 · · · z2l Now, for each subset J of U = {z21 , . . . , z2t , z1i , . . . , z1k }, of order 0 < s < t, let J be the complement of J in U , let ZJ be the row tableau of that set, and ZJ the corresponding row tableau of J. We let I stand for an arbitrary subset of U of order precisely t, other than the subset {z21 , . . . , z2t } itself, I its complementary subset, and ZI , ZI as for J. Set w1 w2 (t − s) z11 · · · z1i−1 ZJ TJ = , w2 (l − t + s) ZJ z2t+1 · · · z2l and cJ equal plus or minus the product of binomial coeﬃcients that would be appropriate due to the multiplication in the divided power algebra, of ZJ with z2t+1 · · · z2l (see the foregoing examples to get a more concrete picture of this description). Finally, let w1 z11 · · · z1i−1 ZI , TI = w2 ZI z2t+1 · · · z2l and cI equal plus or minus the corresponding product of binomial coeﬃcients. Then it is tedious, but straightforward to prove that cJ TJ + cI TI . T = ±T + J

I

The important thing to observe in this equation is that the diagrams of T and TJ are all bigger than that of T and so all these terms are strictly lower in the quasi order. The terms TI all have the letter half of the tableau unchanged

Theorem VI.3.2 Part 2

241

from that of T . However, since our sets I must have t elements, and none can be the set {z21 , . . . , z2t } itself, at least one of these elements must remain in the top row, and one of the elements z1u > z2i must come down into the bottom row. Thus, the resulting tableaux in this case are all less in the quasi order than the original. One may ask, when dealing with the tableaux other than the TI what we do about keeping these tableaux within the class we are considering, namely, the lengths of the rows decreasing. In order to keep to this recipe, we simply push the top row of the two up as far as it has to go, and the bottom one down to where it has to go to make it into a legitimate shape. But this still makes the shape lower in our quasi order than the original.

A.2 Theorem VI.3.2 Part 2: linear independence of double standard tableaux In this section, we introduce the Robinson–Schensted–Knuth correspondence (which will be written R–S–K in the future) to set up a one-to-one count between the double standard tableaux, and the usual basis elements of Dk1 (F ) ⊗ · · · ⊗ Dkn (F ). · · x$, when there is no Let us agree, as we did in Chapter III, to write x(k) as x % ·&# k

danger of confusion. That is, we write x · · x$ to mean the row tableau consisting % ·&# k

of k copies of x in a row. Then a basis element of Dk (F ) is the same as a nondecreasing sequence of elements: xi1 ≤ xi2 ≤ · · · ≤ xik , and a basis element of Dk1 (F ) ⊗ · · · ⊗ Dkn (F ) is a long string of such sequences. For instance, the (2) (3) basis element x1 x3 ⊗ x2 x3 ∈ D3 ⊗ D4 corresponds to the long sequence β = x1 x1 x3 x2 x2 x2 x3 . Now R–S–K sets up a correspondence between such sequences and pairs of tableaux. Before we describe (loosely) this correspondence in general, let us look at the example at hand. To β we want to associate two tableaux, L(β) and P (β); ﬁrst we will describe how we get L(β). Since x1 x1 x3 is increasing, we put these elements in a row tableau: x1 x1 x3 . But now we hit up against x2 (the next term in our sequence), and if we were to put that in as the next element of the row, we would spoil row-standardness. So, we “bump” the x3 from the ﬁrst row and replace it by x2 while moving x3 to the second row of a now two-rowed tableau: x1 x1 x2 . And now we see that the remaining three terms of the sequence, x3 β, can all be placed in the ﬁrst row without necessitating any bumping, so the x1 x1 x2 x2 x2 x3 tableau we end up with is L(β) = . (Notice that x3 this tableau is standard, an end result guaranteed by the nature of the bumping process.)

242

Appendix for letter-place methods

To assign the next tableau, P (β), we will make a slight modiﬁcation of the usual R–S–K, and make use of the fact that we are looking very distinctly at D3 ⊗ D4 , namely, a two-fold tensor product of divided powers of designated degrees. This means that we have only two places to deal with, namely 1 and 2, and we want P (β) to be of the same shape as L(β), standard, and ﬁlled with the places 1 and 2. Now the ﬁrst part of the construction of our tableau involved using the ﬁrst three terms of the sequence, β, so that the ﬁrst three boxes of the ﬁrst row should be ﬁlled in with the place, 1. The second row was next produced by bumping, so we put the place 2 in that box of the second row. The next three entries in the ﬁrst row were produced by inserting entries from the second factor, so we ﬁll them in with 2, and the resulting tableau we get is P (β) = 1 1 1 2 2 2 . 2 On the other hand, given the pair of tableaux, L(β) and P (β), we can reconstruct the sequence β. We look at the tableau, P (β), and peel oﬀ the highest place from the highest row ﬁrst, with its corresponding letters in L(β). This tells us that in our second factor, we had x2 x2 x3 , and we still are left with the x x1 x2 1 1 1 pair of tableaux: 1 and . This means that we still have a x3 2 term from the second factor, and it was obtained by bumping from a single-rowed tableau with three entries. The only way that bumping could have happened was for x3 to have been in the upper row previously, and to have been bumped by x2 (for if an element of the ﬁrst row had been bumped by x1 , that element would have been x2 , and we would have had x2 in the second row). Therefore, the full second factor must be x2 x2 x2 x3 , and our ﬁrst factor is what is left, namely x1 x1 x3 . To explain the bumping procedure that leads to the construction of the ﬁrst tableau, let us consider the general situation of an ordered set, S = {s1 , s2 , . . . , st }, and a sequence β = si1 si2 . . . sik formed from these elements. To the element si1 we attach the tableau consisting of one box, and the one entry, si1 . If si2 ≥ si1 , then we attach the one-rowed tableau consisting of si1 and si2 ; if si2 < si1 , we attach the two-rowed tableau having si2 in the top row, and si1 in the second row. Suppose we have attached by this procedure a standard tableau, λ, to the ﬁrst l elements of the sequence. We then take the element sil+1 and try to add it to λ. If it exceeds (in the weak sense) every element in the top row of λ, then we stick it onto the end of that row. If it does not, then we look for the ﬁrst element of the ﬁrst row that is strictly greater than it, and replace it by sil+1 . This leaves us with the exiled element of the ﬁrst row, and we take it and look at the second row. If it ﬁts, we add it on to the second row, otherwise bump as before and continue in this way. By this procedure, we arrive at the tableau, L(β), associated to the sequence, β. What if, one may ask, the ﬁrst two rows of λ were of the same length and the bumped element exceeded all the elements of the second row? Would not sticking it onto the second row produce an inadmissible shape? But we have

Theorem VI.3.2 Part 2

243

assumed that the tableau, λ, is standard, and so we see that this situation cannot arise. To associate a “place” tableau to the sequence β, we have to be given a bit more information along with the sequence (as when we saw in our example that we were dealing with two factors—hence two places—and subsequences of ﬁxed length). So let us assume that our integer k = k1 + · · · + kn , and that our sequence β has the property that the ﬁrst k1 elements are increasing, the next k2 are increasing, and so on. We want to assign to our β a standard place tableau having n places. Notice that since the ﬁrst k1 elements are increasing, they all ﬁt into a single-row tableau with k1 boxes. We record this by attaching a singlerowed tableau with k1 boxes all ﬁlled in with the place 1. We then run through the next k2 elements, keeping track of all the new boxes created in the L(β) construction by labeling them with the place 2. Notice that if elements from the second strand bump elements from the ﬁrst row, the elements they bump grow in size, so that these elements all go into the second row (they bump no more). Also, once an element from this second strand gets placed in the ﬁrst row, all the others do. This means that the place tableau so far associated has 1’s in the ﬁrst row, and 2’s in the ﬁrst and second rows. Obviously the number of 2’s in the second row cannot exceed the number of 1’s in the ﬁrst row (namely, k1 ). We then proceed with the third strand placing 3’s, and so on. Rather than spend more words on this description, a not too trivial example may be in order. For simplicity, we will use numbers for letters as well as places; we believe that this should cause no confusion. Example A.2.1 Consider the element 223446 ⊗ 12335 ⊗ 3446 in D6 ⊗ D5 ⊗ D4 . Following the recipes above, one ends with the following pair of tableaux (the ﬁrst one the “letter” tableau, the second the “place” tableau ﬁlled with only three places): 1 2 6

2 3

2 4

3 4

3 5

3

4

4

6

1 2 3

1 2

1 2

1 2

1 2

1

3

3

3

and .

The original “sequence” (we write it in quotes, since we have taken the liberty to use the tensor product symbol to mark oﬀ where the substrands are to be seen) can be reconstructed from this pair of tableaux as we indicated in the ﬁrst example. By removing all the entries labeled 3 in the ﬁrst row, we see that the third factor has to end with 446. But then we see that the 6 must have been bumped into the third row by the ﬁrst term of the third factor. If 6 had been bumped from the second row, it would have had to be the 5 that did it, but if

244

Appendix for letter-place methods

the 5 had been the bumper, it would have already ﬁt nicely onto the ﬁrst row. So, it must have been the 3 that bumped, which means that the third factor was 3446, and that 3 bumped into the two-rowed tableau having 122335 in the top row and 23446 in the second. The corresponding place tableau now has only the places 1 and 2 with all the entries labeled 1 in the top row and all those labeled 2 in the second. Now the previous tableau had six entries in the top row, and only four in the second. The only way we could have arrived at the current stage is if we had bumped a 5 into the tableau having 122346 in the top row, and 2344 in the second. And so forth. In this way we reconstruct our original sequence of departure. This procedure establishes a one-to-one correspondence between the usual basis elements of Dk1 (F ) ⊗ · · · ⊗ Dkn (F ) and the double standard tableaux having ki places i for i = 1, . . . , n. As a result we have a complete proof of the linear independence of the double standard tableaux, and hence of Theorem VI.3.2. A.3 Modiﬁcations required for Theorems VI.3.3 and VI.3.4 In Subsections VI.3.2 and VI.3.3, we discussed the various changes we have to consider when we alter the signs of the letters or places. In this section, we will sketch the proofs that underlie the theorems stated there. We will look ﬁrst at the case of positive letters and negative places. In the same way that we proved that the double standard tableaux generated the tensor product of divided powers, we prove that the appropriate double standard tableaux generate the tensor product of exterior powers. That is, we introduce a quasi-order on double tableaux, and then show that any non-standard double tableau may be expressed as a linear combination of tableaux properly lower in that order. It is worth pointing out that if there is a violation of weak increase in the columns on the right hand side of a double tableau, the calculation that we have to do to straighten this is a bit easier than in the case of positive places, as there are no repeats in the rows (we may, as before, assume that we are starting with a row-standard double tableau). We will omit the proof of this fact as it is almost identical to the previous proof. The interested reader might consult the paper by B. Taylor, [82], for a complete proof. The proof of the linear independence of the standard double tableaux is slightly diﬀerent from that in the case of divided powers, so we will indicate the slight variant of the R–S–K that must be used here. The essential diﬀerence is that when we construct a letter tableau from a sequence of letters, we construct it column-wise rather than row-wise, and if bumping is necessary, we start a new column with the bumped element. We now insist, though, that the columns be strictly increasing, so that we must bump if our new element does not strictly exceed the largest element already in the column. The corresponding tableau of places (this time negative) is built as we did earlier. Again, an example should suﬃce to make this clear.

Theorems VI.3.3 and VI.3.4

245

Let us take the basis element x2 ∧ x3 ∧ x5 ⊗ x1 ∧ x3 ⊗ x2 ∧ x4 ⊗ x3 ∧ x5 ∧ x6 in Λ3 F ⊗ Λ2 F ⊗ Λ2 F ⊗ Λ3 F. Because the ﬁrst three terms are strictly increasing, after three steps we obtain x2 1 the tableau: x3 , with the accompanying tableau of places: 1 . x5 1 Now we go to our term in the second place (so that any boxes constructed from these two terms get labeled 2 in the tableau of places we construct), and bump in the term x1 . This has the eﬀect of knocking out the x2 from the ﬁrst column, and replacing it by x1 , while we place x2 in the second column getting: x1 x2 1 2 x3 , with accompanying tableau of places: 1 . x5 1 Proceeding step by step, we end up with the pair of tableaux: x1 x2 x2 1 2 3 x2 x3 x5 1 2 4 x3 x4 1 3 and . x5 4 x6 4 Putting these together, we get the standard double tableau:   x1 x2 x2 1 2 3 x2 x3 x5 1 2 4   x3 x4 . 1 3   x5  4 4 x6 These arguments are suﬃcient to prove Theorem VI.3.3. As for the case of negative letters and positive places, the proof that the appropriate double standard tableaux generate is just a mild modiﬁcation of the above. The proof of linear independence involves a modiﬁed R–S–K, namely, in this case, you build the letter tableau by rows, and again label the new boxes by the place corresponding to the element that you bump in. To illustrate, we will use the same element we used above, namely x2 ∧ x3 ∧ x5 ⊗ x1 ∧ x3 ⊗ x2 ∧ x4 ⊗ x3 ∧ x5 ∧ x6 in Λ3 F ⊗ Λ2 F ⊗ Λ2 F ⊗ Λ3 F. Our R–S–K applied to this yields the double tableau:   x1 x2 x3 x5 x6 1 1 1 4 4 . x2 x3 x4 2 2 3 3 4 x3 x5 The reader should be careful not to confuse this double standard tableau with the double tableau that actually equals the element x2 ∧ x3 ∧ x5 ⊗ x1 ∧ x3 ⊗ x2 ∧ x4 ⊗ x3 ∧ x5 ∧ x6 in Λ3 F ⊗ Λ2 F ⊗ Λ2 F ⊗ Λ3 F. For this element

246

Appendix for letter-place methods

is simply 

x2  x1   x2 x3

x3 x3 x4 x5

x5

x6

1 2 3 4

1 2 3 4

1

  , 

4

which is, of course, far from standard. For the case of the symmetric algebra, the proof that the appropriate double standard tableaux generate presents no new feature. As for their linear independence, the R–S–K builds the letter part of the tableau by columns, and the place part of the tableau in the usual way. To take an example, consider the element x22 x3 x4 ⊗ x31 x2 ⊗ x2 x24 ∈ S4 F ⊗ S4 F ⊗ S3 F . The corresponding standard double tableau is   x1 x2 1 2  x1 x2 1 2     x1 x3 1 2     x2 x4 1 2  .    x2  3    x4  3 3 x4 This completes the proof of our theorems.

A.4 Modiﬁcations required for Theorem VI.8.4 We just have to show, in this section, how to modify the R–S–K for D(ϕ ⊗ Rn ) and Λ(ϕ ⊗ Rm ). As usual we discuss the case of D(ϕ ⊗ Rn ) in some detail and point out that the same discussion obtains for Λ(ϕ ⊗ Rm ) where the basis of Rm is considered to be negative. If we ﬁx our degrees, k1 , . . . , kn , we know that a basis for Dk1 (ϕ)⊗· · ·⊗Dkn (ϕ) is of the form (x11 · · · x1s1 ⊗ y11 ∧ · · · ∧ y1t1 ) ⊗ · · · ⊗ (xn1 · · · xnsn ⊗ yn1 ∧ · · · ∧ yntn ) where si + ti = ki and the xij , yij are elements in G, F, respectively such that for all i, j, xij ≤ xij+1 , yij < yij+1 . We now must deﬁne the R–S–K that will assign to such an element a double standard tableau. But this modiﬁcation is simple: we use the procedure that we used before, but this time we allow repeats in the x entries, and no repeats in those of y. An example here will tell the whole story.

Theorem VI.8.4

Example A.4.1

247

Consider the element

(x2 x2 ⊗ y3 ∧ y4 ) ⊗ (x1 x2 ⊗ y2 ∧ y3 ∧ y4 ) ∈ D4 (ϕ) ⊗ D5 (ϕ). From this, we build successively the tableaux: x1 x1 x2 y3 y4 1(4) x (4) → →2 x2 x2 y3 y4 1 x2 2 y2 y3 x1 x2 x2 y4 1(4) x1 x2 x2 y2 1(4) → → x2 y3 2(2) x2 y3 y4 2(3) y4 x1 x2 x2 y2 y3 y4 1(4) 2(2) x1 x2 x2 y2 y3 1(4) 2 → . 2(3) x2 y3 y4 2(3) x2 y3 y4 Here, the overset arrows indicate we are bumping the element set over the arrow into the tableau, and producing the tableau to the right of the arrow. The reader can see that this process is essentially the same as that described in earlier sections, and is reversible. This shows that the set of generating double tableaux has the same cardinality as the module it is generating, and hence is a basis.

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INDEX acyclic complex, 70 Acyclicity Lemma, 107 almost partition, 152 almost skew-partition (-shape), 150 alternating map, 128 Artin–Rees Theorem, 48 associate, 1 associated prime ideal, 50 associator of a module, 50

bar algebra, 213 bar complex, 80 bar shape, 150 Betti numbers, 144, 209

canonical basis, 9 Capelli Identities, 166, 167 cocommutative graded R-coalgebra, 27 codimension of a module, 52 Cohen–Macaulay module, 63 Cohen–Macaulay ring, 63 Cohen–Macaulay Theorem, 55 column word of a tableau, 178 column-convex shape matrix, 150 column-standard, 175 commutative graded R-algebra, 27 complex, 13 connected graded algebra, 215 content of a tableau, 206 coprime, 2 cyclic module, 69

Dedekind domain, 4 depth of a module, 52 diagonal map, 26 diﬀerential bar complex, 212 dimension of a ring, 47 dimension of an R-module, 47 discrete valuation ring, 6 divided power algebra, 28 divisor, 1 dual basis, 28 Durfee square, 144, 201

equicharacteristic, 65 Euclidean domain, 7

exact complex, 13 Extn , 20 extension, 21 exterior algebra, 30 extreme ﬂippable inversion, 178

factorization domain, 2 ﬁnite free module, 10 ﬁnitely presented, 13 ﬁnitistic global dimension, 53 ﬁrst diﬀerence function, 46 First Structure Theorem, 110 ﬂippable inversion, 175 free bar module, 214 free dimension, 18 free module, 9 free presentation, 13 free resolution, 17

generic alternating map, 137 generic (universal) resolution, 144 global dimension, 44 Gorenstein ideal, 127 grade of an ideal, 127 graded R-algebra, 23 greatest common divisor, 1

height of a module, 50 height of a prime ideal, 48 hereditary ring, 16 Hilbert basis theorem, 7 Hilbert–Burch Theorem, 111 homogeneous element, 23 (ξ) (l) homogeneous strands Zn,m Z n,m , 221 homological conjectures (list of), 66 homological dimension, 3, 17 homology functor, 94 Hopf algebra, 26

ideal of deﬁnition, 48 integral weight, 205 intersection multiplicity, 64 intertwining numbers, 207 inversion, 175 irreducible element, 2

254

Index

Jacobi–Trudi matrix, 228

Koszul complex, 39 Krull Principal Ideal Theorem, 50

last row (column) of shape matrix, 149 least common multiple, 1 length of a partition, 151 length of a skew-shape, 151 length of almost skew-shape, 152 letter-place superalgebra, 164 letters, 158 lifting problem, 112

M-sequence, 42 mapping cone, 39 mapping cone construction, 39 maximal spectrum of R, 5 measuring identity, 88 minimal resolution, 53, 144 mixed characteristic, 66 modiﬁed column word of a tableau, 178 multi-signed alphabet, 175 multiplicity of q with respect to E, 60 multiplicity of M with respect to E, 101 multiplicity of R, 60 multipliers, 111

n-th extension module, 20 n-th torsion module, 18 Nakayama’s lemma, 15 noetherian, 3 noetherian ring, 4 normalized bar complex, 80 (k)

operators Zi,j , 215 oriented free module, 110

parameter matrix for a module, 101 partition, 150 perfect ideal, 52 pfaﬃan, 129 pfaﬃan ideal, 130 place polarization, 165–167 places, 158 polynomial, 6 polynomial function, 46 power series (formal), 6 prime element, 2 projective dimension, 3 projective module, 14

projective resolution, 17 proper divisor, 1

Quillen-Suslin theorem, 15, 114

R-algebra, 22 R-coalgebra, 24 R-maps, 9 rank of a free module, 10 rank of a map, 104 rank of a projective module, 104 regular local ring, 54 regular sequence, 42 Rel(λ/µ) (almost skew-shape), 171 Rel(λ/µ) (skew-shape), 170 reversed column word of a tableau, 178 rigidity problem, 108 Robinson–Schensted–Knuth (R–S–K), 241 row tableau, 153 row-convex shape matrix, 150 row-standard tableau, 158, 175

Samuel multiplicity, 60 saturated chain condition, 55 saturated chain of prime ideals, 55 Schur algebra, 204 Schur complexes, 197 Schur map, 155, 169 Schur modules (general), 155 Schur modules (hooks), 80 Second Structure Theorem, 122 separators, 213 shape matrix, 149 short exact sequence, 13 signed inequalities, 175, 176, 186 skew-partition (-shape), 150 spectrum of R, 5 split exact sequence, 13 standard double tableau, 159, 162, 164 standard tableau, 175 standard tableau (for hooks), 79 straight ﬁlling, 182 straight tableau, 175 subshape, 150 symmetric algebra, 23 symmetrization, 28 system of divided powers, 129 system of parameters, 48

T-boundary operator, 213, 214 t+ -complex, 215 t+ -graded strand, 215 T-grading, 213 t-hook, 225

Index

255

t+ -submodule, 215 tableau with values in S, 153 Taylor Algorithm, 182 tensor algebra, 24 tensor product, 15 Torn , 18 torsion element, 18 torsion module, 18 torsion submodule, 18 torsion-free, 18 transpose (dual) of a shape, 152 trivial extension, 21 type of almost skew-shape, 152, 170

unique factorization domain (UFD), 2, 59, 111 Universal Coeﬃcient Theorem, 206 universal freeness, 73 unmixed ideal, 56

unﬂippable inversion, 175

Z-forms, 203

weight of a shape, 153 weight submodule, 205 Weyl map, 154, 169 Weyl modules (general), 155 Weyl modules (hooks), 80 Weyl–Schur complex, 190 Weyl–Schur map, 190

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