Homological Questions in Local Algebra

Homological Questions in Local Algebra......Page 1
Contents......Page 6
Preface......Page 9
1. Homological preliminaries......Page 14
1.1 Acyclicity lemmas......Page 15
1.2 A few isomorphisms of complexes......Page 17
2. Adic topologies and completions......Page 21
2.1 Induced topologies and purity......Page 22
2.2 Completions......Page 28
2.3 Lifting in complete local rings......Page 32
3. Injective envelopes and minimal injective resolutions......Page 36
3.1 The injective envelope of a module......Page 37
3.2 Decomposition of injective modules......Page 40
3.3 Minimal injective resolutions......Page 47
3.4 Matlis duality......Page 53
4. Local cohomology and Koszul complexes......Page 61
4.1 Local cohomology......Page 62
4.2 Koszul complexes......Page 67
4.3 Limits of Koszul complexes and local cohomology......Page 73
5. (Pre-) regular sequences and depth......Page 79
5.1 (Pre-) regular sequences......Page 80
5.2 Pre-regularity under completion; connections with local cohomology......Page 92
5.3 Depth......Page 99
6. Exactness of complexes and linear equations over rings......Page 110
6.1 The acyclicity lemma and a few consequences......Page 111
6.2 The Buchsbaum-Eisenbud criterion......Page 117
6.3 Linear equations over rings......Page 124
7. Comparing homological invariants......Page 128
7.1 Auslander-Buchsbaum - and Bass identities generalized......Page 129
7.2 Equalities and inequalities involving grade......Page 133
7.3 Annihilators of local cohomology......Page 142
8. Dimension......Page 150
8.1 Krull\'s Hauptidealsatz......Page 151
8.2 Parameters and dimension......Page 154
8.3 Parameters and regular sequences......Page 162
8.4 Extensions of the Hauptidealsatz......Page 169
8.5 Dimension conjectures......Page 173
9.1 Cohen-Macaulay modules and rings......Page 179
9.2 Regular local rings......Page 186
9.3 Complete local rings and the direct summand conjecture......Page 189
10.1 Gorenstein rings......Page 195
10.2 Local duality......Page 202
10.3 The direct summand and monomial conjectures in equal characteristic......Page 209
11. Frobenius and big Cohen-Macaulay modules in characteristic p......Page 214
11.1 Modifications......Page 215
11.2 Relations between annihilators......Page 219
11.3 The Frobenius functor......Page 223
11.4 Pre-regular modules in characteristic p......Page 231
11.5 Balanced big Cohen-Macaulay modules in characteristic p......Page 233
12.0 Introduction......Page 234
12.1 Hochster algebras and equational constraints......Page 237
12.2 Constructible properties and transfer to finite fields......Page 243
12.3 Proof of Lemma IV......Page 253
12.4 The Weierstrass theorems......Page 262
12.5 Henselian rings and Henselization......Page 265
12.6 An approximation theorem of M. Artin......Page 281
12.7 From Noetherian to finitely generated algebras......Page 294
13. Uses of big Cohen-Macaulay modules......Page 298
13.1 New intersection theorems and a few consequences......Page 299
13.2 Nonvanishing of Bass numbers......Page 306
References......Page 310