In the past 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 subject. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. The general theory is applied to a number of examples and the connections with combinatorics are highlighted. Throughout each chapter, the authors have supplied many examples and exercises.
Author(s): Winfried Bruns, H. Jürgen Herzog
Series: Cambridge Studies in Advanced Mathematics 39
Edition: 2
Publisher: Cambridge University Press
Year: 1998
Language: English
Pages: 465
Cover......Page 1
Copyright......Page 2
Contents......Page 5
Preface to the revised edition......Page 9
Preface to the first edition......Page 10
Part 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 52
Notes......Page 67
2.1 CohenMacaulay rings and modules......Page 69
2.2 Regular rings and normal rings......Page 77
2.3 Complete intersections......Page 85
Notes......Page 97
3.1 Finite modules of finite injective dimension......Page 100
3.2 Injective hulls. Matlis duality......Page 109
3.3 The canonical module......Page 119
3.4 Gorenstein ideals of grade 3. Poincaré duality......Page 132
3.5 Local cohomology. The local duality theorem......Page 139
3.6 The canonical module of a graded ring......Page 148
Notes......Page 156
4.1 Hilbert functions over homogeneous rings......Page 159
4.2 Macaulay's theorem on Hilbert functions......Page 168
4.3 Gotzmann's regularity and persistence theorem......Page 180
4.4 Hilbert functions over graded rings......Page 185
4.5 Filtered rings......Page 194
4.6 The Hilbert-Samuel function and reduction ideals......Page 200
4.7 The multiplicity symbol......Page 204
Notes......Page 213
Part II. Classes of Cohen-Macaulay rings......Page 217
5.1 Simplicial complexes......Page 219
5.2 Polytopes......Page 235
5.3 Local cohomology of Stanley-Reisner rings......Page 241
5.4 The upper bound theorem......Page 249
5.5 Betti numbers of Stanley-Reisner rings......Page 252
5.6 Gorenstein complexes......Page 255
5.7 The canonical module of a Stanley-Reisner ring......Page 258
Notes......Page 266
6.1 Affine semigroup rings......Page 268
6.2 Local cohomology of affine semigroup rings......Page 275
6.3 Normal semigroup rings......Page 282
6.4 Invariants of tori and finite groups......Page 290
6.5 Invariants of linearly reductive groups......Page 304
Notes......Page 310
7.1 Graded Hodge algebras......Page 312
7.2 Straightening laws on posets of minors......Page 318
7.3 Properties of determinantal rings......Page 323
Notes......Page 330
Part III. Characteristic p methods......Page 333
8.1 The annihilators of local cohomology......Page 335
8.2 The Frobenius functor......Page 339
8.3 Modifications and non degeneracy......Page 343
8.4 Hochster's finiteness theorem......Page 348
8.5 Balanced big Cohen-Macaulay modules......Page 354
Notes......Page 358
9.1 Grade and acyclicity......Page 360
9.2 Regular rings as direct summands......Page 366
9.3 Canonical elements in local cohomology modules......Page 369
9.4 Intersection theorems......Page 373
9.5 Ranks of syzygies......Page 379
9.6 Bass numbers......Page 384
Notes......Page 388
10.1 The tight closure of an ideal......Page 390
10.2 The Briançon-Skoda theorem......Page 399
10.3 F-rational rings......Page 405
10.4 Direct summands of regular rings......Page 418
Notes......Page 422
Appendix: A summary of dimension theory......Page 424
References......Page 432
Notation......Page 451
Index......Page 456
