Relative homological algebra / by Edgar E. Enochs, Overtoun M.G. Jenda.

Format:

Book

Author/Creator:

Enochs, Edgar E.

Contributor:

Jenda, Overtoun M. G.

Sabin W. Colton, Jr., Memorial Fund.

Series:

De Gruyter expositions in mathematics 0938-6572 ; 30.

De Gruyter expositions in mathematics, 0938-6572 ; 30

Language:

English

Subjects (All):

Algebra, Homological.

Physical Description:

xi, 339 pages ; 25 cm.

Place of Publication:

Berlin ; New York : Walter de Gruyter, 2000.

Summary:

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board -- Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany -- Honorary Editor -- Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia -- Titles in planning include -- Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Contents:

1.1 Zorn's lemma, ordinal and cardinal numbers 1

1.2 Modules 7

1.3 Categories and functors 17

1.4 Complexes of modules and homology 25

1.5 Direct and inverse limits 31

1.6 I-adic topology and completions 36

2 Flat Modules, Chain Conditions and Prime Ideals 40

2.1 Flat modules 40

2.2 Localization 44

2.3 Chain conditions 46

2.4 Prime ideals and primary decomposition 51

2.5 Artin-Rees lemma and Zariski rings 61

3 Injective and Flat Modules 68

3.1 Injective modules 68

3.2 Natural identities, flat modules, and injective modules 75

3.3 Injective modules over commutative noetherian rings 84

3.4 Matlis duality 88

4 Torsion Free Covering Modules 93

4.1 Existence of torsion free precovers 93

4.2 Existence of torsion free covers 95

4.4 Direct sums and products 101

5 Covers 105

5.1 F-precovers and covers 105

5.2 Existence of precovers and covers 107

5.3 Projective and flat covers 110

5.4 Injective covers 120

5.5 Direct sums and T-nilpotency 125

6 Envelopes 129

6.1 F-preenvelopes and envelopes 129

6.2 Existence of preenvelopes 130

6.3 Existence of envelopes 132

6.4 Direct sums of envelopes 134

6.5 Flat envelopes 136

6.6 Existence of envelopes for injective structures 139

6.7 Pure injective envelopes 144

7 Covers, Envelopes, and Cotorsion Theories 152

7.1 Definitions and basic results 152

7.2 Fibrations, cofibrations and Wakamatsu lemmas 154

7.3 Set theoretic homological algebra 160

7.4 Cotorsion theories with enough injectives and projectives 162

8 Relative Homological Algebra and Balance 167

8.1 Left and right F-resolutions 167

8.2 Derived functors and balance 169

8.3 Applications to modules 177

8.4 F-dimensions 180

8.5 Minimal pure injective resolutions of flat modules 194

8.6 [lambda] and [mu]-dimensions 203

9 Iwanaga-Gorenstein and Cohen-Macaulay Rings and Their Modules 211

9.1 Iwanaga-Gorenstein rings 211

9.2 The minimal injective resolution of R 215

9.3 More on flat and injective modules 223

9.4 Torsion products of injective modules 226

9.5 Local cohomology and the dualizing module 229

10 Gorenstein Modules 239

10.1 Gorenstein injective modules 239

10.2 Gorenstein projective modules 246

10.3 Gorenstein flat modules 253

10.4 Foxby classes 258

11 Gorenstein Covers and Envelopes 269

11.1 Gorenstein injective precovers and covers 269

11.2 Gorenstein injective preenvelopes 270

11.3 Gorenstein injective envelopes 274

11.4 Gorenstein essential extensions 277

11.5 Gorenstein projective precovers and covers 279

11.6 Auslander's last theorem (Gorenstein projective covers) 284

11.7 Gorenstein flat covers 288

11.8 Gorenstein flat and projective preenvelopes 292

12 Balance over Gorenstein and Cohen-Macaulay Rings 294

12.1 Balance of Hom (

,

) 294

12.2 Balance of

[characters not reproducible]

298

12.3 Dimensions over n-Gorenstein rings 300

12.4 Dimensions over Cohen-Macaulay rings 305

12.5 [Omega]-Gorenstein modules 307.

Notes:

Includes bibliographical references (pages [321]-330) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Sabin W. Colton, Jr., Memorial Fund.

ISBN:

311016633X

OCLC:

43167922