Edited by A. Dold and B. Eckmann

Winfried Bruns

Udo Vetter

Determinantal Rings

Springer-Verlag

Berlin Heidelberg New York London Paris Tokyo

Authors

Winfried Bruns

FB Mathematik/Informatik

Universit¨t Osnabr¨ck

a u

49069 Osnabr¨ck

u

Germany

Udo Vetter

FB Mathematik

Universit¨t Oldenburg

a

26111 Oldenburg

Germany

This book is now out of print. The authors are grateful to Springer-Verlag for the

permission to make this postscript ¬le accessible.

ISBN 3-540-19468-1 Springer-Verlag Berlin Heidelberg New York

ISBN 0-387-19468-1 Springer-Verlag New York Berlin Heidelberg

c Springer-Verlag Berlin Heidelberg 1988

Preface

Determinantal rings and varieties have been a central topic of commutative algebra

and algebraic geometry. Their study has attracted many prominent researchers and

has motivated the creation of theories which may now be considered part of general

commutative ring theory. A coherent treatment of determinantal rings is lacking however.

We are algebraists, and therefore the subject will be treated from an algebraic point

of view. Our main approach is via the theory of algebras with straightening law. Its

axioms constitute a convenient systematic framework, and the standard monomial theory

on which it is based yields computationally e¬ective results. This approach suggests

(and is simpli¬ed by) the simultaneous treatment of the coordinate rings of the Schubert

subvarieties of Grassmannians, a program carried out very strictly.

Other methods have not been neglected. Principal radical systems are discussed in

detail, and one section each is devoted to invariant and representation theory. However,

free resolutions are (almost) only covered for the “classical” case of maximal minors.

Our personal view of the subject is most visibly expressed by the inclusion of Sections

13“15 in which we discuss linear algebra over determinantal rings. In particular the

technical details of Section 15 (and perhaps only these) are somewhat demanding.

The bibliography contains several titles which have not been cited in the text. They

mainly cover topics not discussed: geometric methods and ideals generated by minors of

symmetric matrices and Pfa¬ans of alternating ones.

We have tried hard to keep the text as self-contained as possible. The basics of

commutative algebra supplied by Part I of Matsumura™s book [Mt] (and some additions

given in Section 16) su¬ce as a foundation for Sections 3“7, 9, 10, and 12. Whenever

necessary to draw upon notions and results not covered by [Mt], for example divisor

class groups and canonical modules in Section 8, precise references have been provided.

It is no surprise that multilinear algebra plays a role in a book on determinantal rings,

and in Sections 2 and 13“15 we expect the reader not to be frightened by exterior and

symmetric powers. Even Section 11 which connects our subject and the representation

theory of the general linear groups, does not need an extensive preparation; the linear

reductivity of these groups is the only essential fact to be imported. The rudiments on

Ext and Tor contained in every introduction to homological algebra will be used freely,

though rarely, and some familiarity with a¬ne and projective varieties, as developped in

Chapter I of Hartshorne™s book [Ha.2], is helpful.

We hope this text will serve as a reference. It may be useful for seminars following

a course in commutative ring theory. A vast number of notions, results, and techniques

can be illustrated signi¬cantly by applying them to determinantal rings, and it may even

be possible to reverse the usual sequence of “theory” and “application”: to learn abstract

commutative algebra through the exploration of the special class which is the subject of

this book.

Each section contains a subsection “Comments and References” where we have col-

lected the information on our sources. The references given should not be considered

iv Preface

assignments of priority too seriously; they rather re¬‚ect the authors™ history in learning

the subject and give credit to the colleagues in whose works we have participated. While

it is impossible to mention all of them here, it may be fair to say that we could not

have written this text without the fundamental contributions of Buchsbaum, de Concini,

Eagon, Eisenbud, Hochster, Northcott, and Procesi.

The ¬rst author gave a series of lectures on determinantal rings at the Universidade

federal de Pernambuco, Recife, Brazil, in March and April 1985. We are indebted to

Aron Simis who suggested to write an extended version for the IMPA subseries of the

Lecture Notes in Mathematics. (By now it has become a very extended version).

Finally we thank Petra D¨vel, Werner Lohmann and Matthias Varelmann for their

u

help in the production of this book. We are grateful to the sta¬ of the Computing

Center of our university, in particular Thomas Haarmann, for generous cooperation and

providing excellent printing facilities.

Vechta, January 1988 Winfried Bruns Udo Vetter

Contents

1. Preliminaries . . . . . . . . . . . . . . . . . . . . 1

A. Notations and Conventions . . . . . . . . . . . . . . . 1

B. Minors and Determinantal Ideals . . . . . . . . . . . . . 3

C. Determinantal Rings and Varieties . . . . . . . . . . . . 4

D. Schubert Varieties and Schubert Cycles . . . . . . . . . . . 6

E. Comments and References . . . . . . . . . . . . . . . 9

2. Ideals of Maximal Minors . . . . . . . . . . . . . . . 10

A. Classical Results on Height and Grade . . . . . . . . . . . 10

B. The Perfection of Im (X) and Some Consequences . . . . . . . . 13

C. The Eagon-Northcott Complex ..... . . . . . . . . 16

D. The Complex of Gulliksen and Neg˚ ard . . . . . . . . . . . 22

E. Comments and References . . . . . . . . . . . . . . . 25

3. Generically Perfect Ideals . . . . . . . . . . . . . . . 27

A. The Transfer of Perfection . . . . . . . . . . . . . . . 27

B. The Substitution of Indeterminates by a Regular Sequence . . . . . 30

C. The Transfer of Integrity and Related Properties . . . . . . . . 34

D. The Bound for the Height of Specializations ..... . . . . 36

E. Comments and References . . . . . . . . . . . . . . . 36

4. Algebras with Straightening Law on Posets of Minors . . . . . 38

A. Algebras with Straightening Law . . . . . . . .. . . . . 38

B. G(X) as an ASL . . . . . . . . . . . . .. . . . . 40

C. The Linear Independence of the Standard Monomials in G(X) . . . . 43

D. B[X] as an ASL . . . . . . . . . . . . .. . . . . 45

E. Comments and References . . . . . . . . . .. . . . . 48

5. The Structure of an ASL . . . . . . . . . . . . . . . 50

A. ASL Structures on Residue Class Rings . . . . . . . . . . . 50

B. Syzygies and the Straightening Law . . . . . . . . . . . . 53

C. Nilpotents, Regular Elements and Dimension . . . . . . . . . 54

D. Wonderful Posets and the Cohen-Macaulay Property . . . . . . . 58

E. The Arithmetical Rank of Certain Ideals ... . . . . . . . 61

F. Comments and References . . . . . . . . . . . . . . . 63

6. Integrity and Normality. The Singular Locus . . . . . . . . 64

A. Integrity and Normality . . . . . . . . . . . . . . . . 64

B. The Singular Locus . . . . . . . . . . . . . . . . . 67

C. Comments and References . . . . . . . . . . . . . . . 72

vi Contents

7. Generic Points and Invariant Theory . . . . . . . . . . . 73

A. A Generic Point for Rr+1 (X) . . . . . ... . . . . . . 73

B. Invariants and Absolute Invariants .. . ... . . . . . . 74

C. The Main Theorem of Invariant Theory for GL and SL . . . . . . 76

D. Remarks on Invariant Theory . . . . . ... . . . . . . 80

E. The Classical Generic Point for G(X; γ) . ... . . . . . . 82

F. G(X; γ) and R(X; δ) as Rings of Invariants . ... . . . . . . 84

G. The Depth of Modules of Invariants . . . ... . . . . . . 88

H. Comments and References . . . . . . ... . . . . . . 91

8. The Divisor Class Group and the Canonical Class . . . . . . . 93

A. The Divisor Class Group .. . . . . . . . . . . . . . 93

B. The Canonical Class of Rr+1 (X) . . . . . . . . . . . . . 97

C. The General Case . . . . . . . . . . . . . . . . . . 100

D. Comments and References . . . . . . . . . . . . . . . 104

9. Powers of Ideals of Maximal Minors . . . . . . . . . . . 105

A. Ideals and Subalgebras of Maximal Minors . . . . . . . . . . 105

B. ASL Structures on Graded Algebras Derived from an Ideal . . . . . 108

C. Graded Algebras with Respect to Ideals of Maximal Minors . . . . 112

D. The Depth of Powers of Ideals of Maximal Minors .. . . . . . 117

E. Comments and References . . . . . . . . . . . . . . . 120

10. Primary Decomposition . . . . . . . . . . . . . . . . 122

A. Symbolic Powers of Determinantal Ideals ...... . . . . 122

B. The Symbolic Graded Ring . . . . . . . . . . . . . . . 124

C. Primary Decomposition of Products of Determinantal Ideals . . . . 126

D. Comments and References . . . . . . . . . . . . . . . 133

11. Representation Theory . . . . . . . . . . . . . . . . 135

A. The Filtration of K[X] by the Intersections of Symbolic Powers . . . 135

B. Bitableaux and the Straightening Law Revisited . . . . . . . . 137

C. The Decomposition of K[X] into Irreducible G-Submodules . . . . 141

D. G-Invariant Ideals .............. . . . 145

E. U -Invariants and Algebras Generated by Minors . . . . . . . . 149

F. Comments and References . . . . . . . . . . . . . . . 152

12. Principal Radical Systems . . . . . . . . . . . . . . . 153

A. A Propedeutic Example. Principal Radical Systems . . . . . . . 153

B. A Principal Radical System for the Determinantal Ideals . . . . . 155

C. The Perfection of Determinantal Ideals . . . . . . . . . . . 158

D. Comments and References . . . . . . . . . . . . . . . 160

13. Generic Modules . . . . . . . . . . . . . . . . . . 162

A. The Perfection of the Image of a Generic Map . . . . . . . . . 162

B. The Perfection of a Generic Module . . . . . . . . . . . . 165

C. Homological Properties of Generic Modules . . . . . . . . . . 171

D. Comments and References . . . . . . . . . . . . . . . 173

vii

Contents

14. The Module of K¨hler Di¬erentials

a ...... . . . . . 174

A. Perfection and Syzygies of Some Determinantal Ideals . . . . . . 175

B. The Lower Bound for the Depth of the Di¬erential Module . . . . . 177

C. The Syzygetic Behaviour of the Di¬erential Module . . . . . . . 181

D. Comments and References . . . . . . . . . . . . . . . 183

15. Derivations and Rigidity ...... . .... . . . . 184

A. The Lower Bound for the Depth of the Module of Derivations . . . . 184

B. The Perfection of the Module of Derivations . .... . . . . 189

C. Syzygetic Behaviour and Rigidity . . . . . .... . . . . 197

D. Comments and References . . . . . . . .... . . . . 201

16. Appendix . . . . . . . . . . . . . . . . . . . . . 202

A. Determinants and Modules. Rank ... . . . . . . . . . 202

B. Grade and Acyclicity ....... . . . . . . . . . 206

C. Perfection and the Cohen-Macaulay Property . . . . . . . . . 209

D. Dehomogenization ........ . . . . . . . . . 211

E. How to Compare “Torsionfree” .... . . . . . . . . . 213

F. The Theorem of Hilbert-Burch . . . . . . . . . . . . . . 217

G. Comments and References . . . . . . . . . . . . . . . 218

Bibliography . . . . . . . . . . . . . . . . . . . . . 219

Index of Notations . . . . . . . . . . . . . . . . . . . 229

Subject Index . . . . . . . . . . . . . . . . . . . . 232

1. Preliminaries

This section serves two purposes. Its Subsections A and B list the ubiquitous basic

notations. In C and D we introduce the principal objects of our investigation and relate

them to their geometric counterparts.

A. Notations and Conventions

Generally we will use the terminology of [Mt] which seems to be rather standard

now. In some inessential details our notations di¬er from those of [Mt]; for example we

try to save parentheses whenever they seem dispensable. A main di¬erence is the use of

the attributes “local” and “normal”: for us they always include the property of being

noetherian. In the following we explain some notations and list the few conventions the

reader is asked to keep in mind throughout.

All rings and algebras are commutative and have an element 1. Nevertheless we

will sometimes list “commutative” among the hypotheses of a proposition or theorem in

order to signalize that the ring under consideration is only supposed to be an arbitrary

commutative ring. A reduced ring has no nilpotent elements. The spectrum of a ring A,

Spec A for short, is the set of its prime ideals endowed with the Zariski topology. The

radical of an ideal I is denoted Rad I. The dimension of A is denoted dim A, and the

height of I is abbreviated ht I.

All the modules M considered will be unitary, i.e. 1x = x for all x ∈ M . Ann M is

the annihilator of M , and the support of M is given by

Supp M = {P ∈ Spec A : MP = 0}.

We use the notion of associated prime ideals only for ¬nitely generated modules over

noetherian rings:

Ass M = {P ∈ Spec A : depth MP = 0}.

The depth of a module M over a local ring is the length of a maximal M -sequence in the

maximal ideal. The projective dimension of a module is denoted pd M . We remind the

reader of the equation of Auslander and Buchsbaum for ¬nitely generated modules over

local rings A:

if pd M < ∞

pd M + depth M = depth A

(cf. [Mt], p. 114, Exercise 4). If a module can be considered a module over di¬erent rings

(in a natural way), an index will indicate the ring with respect to which an invariant

is formed: For example, AnnA M is the annihilator of M as an A-module. Instead of

Matsumura™s depthI (M ) we use grade(I, M ) and call it, needless to say, the grade of I

with respect to M ; cf. 16.B for a discussion of grade. The rank rk F of a free module F is

the number of elements of one of its bases. We discuss a more general concept of rank in

16.A: M has rank r if M — Q is a free Q-module of rank r, Q denoting the total ring of