Relative Homological Algebra (eBook)

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Edgar E. Enochs, University of Kentucky, Lexington, USA; Overtoun M. G. Jenda, Auburn University, Alabama, USA.

Preface 7
Preface to the Second Edition 9
1 Basic Concepts 15
1.1 Zorn’s Lemma, Ordinal and Cardinal Numbers 15
1.2 Modules 22
1.3 Tensor Products of Modules and Nakayama Lemma 28
1.4 Categories and Functors 33
1.5 Complexes of Modules and Homology 41
1.6 Direct and Inverse Limits 47
1.7 I-adic Topology and Completions 52
2 Flat Modules, Chain Conditions and Prime Ideals 56
2.1 Flat Modules 56
2.2 Localization 60
2.3 Chain Conditions 63
2.4 Prime Ideals and Primary Decomposition 68
2.5 Artin-Rees Lemma and Zariski Rings 77
3 Injective and Flat Modules 85
3.1 Injective Modules 85
3.2 Natural Identities, Flat Modules, and Injective Modules 92
3.3 Injective Modules over Commutative Noetherian Rings 101
3.4 Matlis Duality 107
4 Torsion Free Covering Modules 112
4.1 Existence of Torsion Free Precovers 112
4.2 Existence of Torsion Free Covers 114
4.3 Examples 116
4.4 Direct Sums and Products 120
5 Covers 124
5.1 F-precovers and covers 124
5.2 Existence of Precovers and Covers 126
5.3 Projective and Flat Covers 129
5.4 Injective Covers 139
5.5 Direct Sums and T-nilpotency 145
6 Envelopes 149
6.1 F-preenvelopes and Envelopes 149
6.2 Existence of Preenvelopes 150
6.3 Existence of Envelopes 152
6.4 Direct Sums of Envelopes 154
6.5 Flat Envelopes 156
6.6 Existence of Envelopes for Injective Structures 159
6.7 Pure Injective Envelopes 164
7 Covers, Envelopes, and Cotorsion Theories 172
7.1 Definitions and Basic Results 172
7.2 Fibrations, Cofibrations and Wakamatsu Lemmas 174
7.3 Set Theoretic Homological Algebra 180
7.4 Cotorsion Theories with Enough Injectives and Projectives 182
8 Relative Homological Algebra and Balance 187
8.1 Left and Right F-resolutions 187
8.2 Derived Functors and Balance 189
8.3 Applications to Modules 198
8.4 F-dimensions 201
8.5 Minimal Pure Injective Resolutions of Flat Modules 215
8.6 . and µ-dimensions 225
9 Iwanaga-Gorenstein and Cohen-Macaulay Rings and Their Modules 233
9.1 Iwanaga-Gorenstein Rings 233
9.2 The Minimal Injective Resolution of R 237
9.3 More on Flat and Injective Modules 246
9.4 Torsion Products of Injective Modules 249
9.5 Local Cohomology and the Dualizing Module 252
10 Gorenstein Modules 263
10.1 Gorenstein Injective Modules 263
10.2 Gorenstein Projective Modules 270
10.3 Gorenstein Flat Modules 277
10.4 Foxby Classes 282
11 Gorenstein Covers and Envelopes 294
11.1 Gorenstein Injective Precovers and Covers 294
11.2 Gorenstein Injective Preenvelopes 295
11.3 Gorenstein Injective Envelopes 299
11.4 Gorenstein Essential Extensions 302
11.5 Gorenstein Projective Precovers and Covers 304
11.6 Auslander’s Last Theorem (Gorenstein Projective Covers) 309
11.7 Gorenstein Flat Covers 314
11.8 Gorenstein Flat and Projective Preenvelopes 318
11.9 Kaplansky Classes 319
12 Balance over Gorenstein and Cohen-Macaulay Rings 325
12.1 Balance of Hom(–, –) 325
12.2 Balance of – . – 329
12.3 Dimensions over n-Gorenstein Rings 332
12.4 Dimensions over Cohen-Macaulay Rings 337
12.5 O-Gorenstein Modules 339
Bibliographical Notes 351
Bibliography 355
Index 365

lt;P>"As their book is primarily aimed at graduate students in homological algebra, the authors have made any effort to keep the text reasonably self-contained and detailed. The outcome is a comprehensive textbook on relative homological algebra at its present state of art." Zentralblatt für Mathematik (review of the first edition)

"[...] in the reviewer’s opinion it is an elegant introduction to homological methods towards applications in ring theory." Zentralblatt für Mathematik