: Edgar E. Enochs
: Relative Homological Algebra Volume 1
: Walter de Gruyter GmbH& Co.KG
: 9783110215212
: De Gruyter Expositions in MathematicsISSN
: 1
: CHF 160.10
:
: Allgemeines, Lexika
: English
: 372
: Wasserzeichen/DRM
: PC/MAC/eReader/Tablet
: PDF
< >This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists. The book is also suitable for an introductory course in commutative and ordinary homological algebra.



< >Edgar E. Enochs, University of Kentucky, Lexington, USA;Overtoun M. G. Jenda, Auburn University, Alabama, USA.
Preface7
Preface to the Second Edition9
1 Basic Concepts15
1.1 Zorn’s Lemma, Ordinal and Cardinal Numbers15
1.2 Modules22
1.3 Tensor Products of Modules and Nakayama Lemma28
1.4 Categories and Functors33
1.5 Complexes of Modules and Homology41
1.6 Direct and Inverse Limits47
1.7 I-adic Topology and Completions52
2 Flat Modules, Chain Conditions and Prime Ideals56
2.1 Flat Modules56
2.2 Localization60
2.3 Chain Conditions63
2.4 Prime Ideals and Primary Decomposition68
2.5 Artin-Rees Lemma and Zariski Rings77
3 Injective and Flat Modules85
3.1 Injective Modules85
3.2 Natural Identities, Flat Modules, and Injective Modules92
3.3 Injective Modules over Commutative Noetherian Rings101
3.4 Matlis Duality107
4 Torsion Free Covering Modules112
4.1 Existence of Torsion Free Precovers112
4.2 Existence of Torsion Free Covers114
4.3 Examples116
4.4 Direct Sums and Products120
5 Covers124
5.1 F-precovers and covers124
5.2 Existence of Precovers and Covers126
5.3 Projective and Flat Covers129
5.4 Injective Covers139
5.5 Direct Sums and T-nilpotency145
6 Envelopes149
6.1 F-preenvelopes and Envelopes149
6.2 Existence of Preenvelopes150
6.3 Existence of Envelopes152
6.4 Direct Sums of Envelopes154
6.5 Flat Envelopes156
6.6 Existence of Envelopes for Injective Structures159
6.7 Pure Injective Envelopes164
7 Covers, Envelopes, and Cotorsion Theories172
7.1 Definitions and Basic Results172
7.2 Fibrations, Cofibrations and Wakamatsu Lemmas174
7.3 Set Theoretic Homological Algebra180
7.4 Cotorsion Theories with Enough Injectives and Projectives182
8 Relative Homological Algebra and Balance187
8.1 Left and Right F-resolutions187
8.2 Derived Functors and Balance189
8.3 Applications to Modules198
8.4 F-dimensions201
8.5 Minimal Pure Injective Resolutions of Flat Modules215
8.6 . and µ-dimensions225
9 Iwanaga-Gorenstein and Cohen-Macaulay Rings and Their Modules233
9.1 Iwanaga-Gorenstein Rings233
9.2 The Minimal Injective Resolution of R237
9.3 More on Flat and Injective Modules246
9.4 Torsion Products of Injective Modules249
9.5 Local Cohomology and the Dualizing Module252
10 Gorenstein Modules263
10.1 Gorenstein Injective Modules263
10.2 Gorenstein Projective Modules270
10.3 Gorenstein Flat Modules277
10.4 Foxby Classes282
11 Gorenstein Covers and Envelopes294
11.1 Gorenstein Injective Precovers and Covers294
11.2 Gorenstein Injective Preenvelopes295
11.3 Gorenstein Injective Envelopes299
11.4 Gorenstein Essential Extensions302
11.5 Gorenstein Projective Precovers and Covers304
11.6 Auslander’s Last Theorem (Gorenstein Projective Covers)309
11.7 Gorenstein Flat Covers314
11.8 Gorenstein Flat and Projective Preenvelopes318
11.9 Kaplansky Classes319
12 Balance over Gorenstein and Cohen-Macaulay Rings325
12.1 Balance of Hom(–, –)325
12.2 Balance of – . –329
12.3 Dimensions over n-Gorenstein Rings332
12.4 Dimensions over Cohen-Macaulay Rings337
12.5 O-Gorenstein Modules339
Bibliographical Notes351
Bibliography355
Index365