In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, and bounds for Bass numbers. Throughout each chapter the authors have supplied many examples and exercises, which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading.
Author(s): Winfried Bruns, Jürgen Herzog
Series: Cambridge Studies in Advanced Mathematics 39
Publisher: Cambridge University Press
Year: 1993
Language: English
Pages: 416
Front cover......Page 1
Series......Page 2
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Contents......Page 7
Preface......Page 9
I Basic concepts......Page 13
1.1 Regular sequences......Page 15
1.2 Grade and depth......Page 20
1.3 Depth and projective dimension......Page 28
1.4 Some linear algebra......Page 30
1.5 Graded rings and modules......Page 39
1.6 The Koszul complex......Page 51
Notes......Page 66
2.1 Cohen-Macaulay rings and modules......Page 68
2.2 Regular rings and normal rings......Page 76
2.3 Complete intersections......Page 84
Notes......Page 96
3.1 Finite modules of finite injective dimension......Page 99
3.2 Injective hulls. Matlis duality......Page 108
3.3 The canonical module......Page 118
3.4 Gorenstein ideals of grade 3. Poincare duality......Page 131
3.5 Local cohomology. The local duality theorem......Page 138
3.6 The canonical module of a graded ring......Page 147
Notes......Page 155
4.1 Hilbert functions of graded modules......Page 158
4.2 Macaulay's theorem on Hilbert functions......Page 166
4.3 Further constraints on Hilbert functions......Page 177
4.4 Filtered rings......Page 186
4.5 The Hilbert-Samuel function and reduction ideals......Page 192
4.6 The multiplicity symbol......Page 197
Notes......Page 206
II Classes of Cohen-Macaulay rings......Page 209
5.1 Simplicial complexes......Page 211
5.2 Polytopes......Page 227
5.3 Local cohomology of Stanley-Reisner rings......Page 233
5.4 The upper bound theorem......Page 241
5.5 Gorenstein complexes......Page 244
5.6 The canonical module of a Stanley-Reisner ring......Page 248
Notes......Page 255
6.1 Affine semigroup rings......Page 257
6.2 Local cohomology of affine semigroup rings......Page 264
6.3 Normal semigroup rings......Page 270
6.4 Invariants of tori and finite groups......Page 279
6.5 Invariants of linearly reductive groups......Page 292
Notes......Page 299
7.1 Graded Hodge algebras......Page 301
7.2 Straightening laws on posets of minors......Page 306
7.3 Properties of determinantal rings......Page 312
Notes......Page 319
III Homological theory......Page 321
8.1 The annihilators of local cohomology......Page 323
8.2 The Frobenius functor......Page 327
8.3 Modifications and non-degeneracy......Page 330
8.4 Hochster's finiteness theorem......Page 335
8.5 Balanced big Cohen-Macaulay modules......Page 342
Notes......Page 346
9.1 Grade and acyclicity......Page 348
9.2 Regular rings as direct summands......Page 354
9.3 Canonical elements in local cohomology modules......Page 357
9.4 Intersection theorems......Page 361
9.5 Ranks of syzygies......Page 367
9.6 Bass numbers......Page 371
Notes......Page 375
Appendix: A summary of dimension theory......Page 378
References......Page 386
Notation......Page 402
Index......Page 407
Back cover......Page 416
