This book provides a self-contained systematic treatment of the subject of relative homological algebra. It is designed for graduate students as well as researchers and specialists. It contains twelve chapters with abundant supply of important results with complete proofs covering material that is essential to understanding topics in algebra, algebraic geometry, and algebraic topology. The text also contains results that are in book form for the first time and thus provides essential reading for researchers and specialists. At the end of each section of each chapter, there are exercises that provide practice problems for students as well as additional important results for specialists.
The book can be used as a text for graduate students and as a handbook for researchers and specialists. The material in the first three chapters constitute notes from lectures of the authors at their respective universities and is suitable for an introductory course in module and ring theory. The following chapters are suitable for a course in relative homological algebra and its applications to commutative and non-commutative algebra. The last three chapters give applications to ring theory. These chapters together with Chapter 7 contain recent achievements including a solution to the flat cover conjecture.
Author(s): Edgar E. Enochs, Overtoun M. G. Jenda
Series: Degruyter Expositions in Mathematics
Publisher: Walter de Gruyter
Year: 2001
Language: English
Pages: 352
Contents......Page all_21901_to_00352.cpc0009.djvu
Preface......Page all_21901_to_00352.cpc0007.djvu
1.1 Zorn's lemma, ordinal and cardinal numbers......Page all_21901_to_00352.cpc0013.djvu
1.2 Modules......Page all_21901_to_00352.cpc0019.djvu
1.3 Categories and functors......Page all_21901_to_00352.cpc0029.djvu
1.4 Complexes of modules and homology......Page all_21901_to_00352.cpc0037.djvu
1.5 Direct and inverse limits......Page all_21901_to_00352.cpc0043.djvu
1.6 I-adic topology and completions......Page all_21901_to_00352.cpc0048.djvu
2.1 Flat modules......Page all_21901_to_00352.cpc0052.djvu
2.2 Localization......Page all_21901_to_00352.cpc0056.djvu
2.3 Chain conditions......Page all_21901_to_00352.cpc0058.djvu
2.4 Prime ideals and primary decomposition......Page all_21901_to_00352.cpc0063.djvu
2.5 Artin-Rees lemma and Zariski rings......Page all_21901_to_00352.cpc0073.djvu
3.1 Injective modules......Page all_21901_to_00352.cpc0080.djvu
3.2 Natural identities, flat modules, and injective modules......Page all_21901_to_00352.cpc0087.djvu
3.3 Injective modules over commutative noetherian rings......Page all_21901_to_00352.cpc0096.djvu
3.4 Matlis duality......Page all_21901_to_00352.cpc0100.djvu
4.1 Existence of torsion free precovers......Page all_21901_to_00352.cpc0105.djvu
4.2 Existence of torsion free covers......Page all_21901_to_00352.cpc0107.djvu
4.3 Examples......Page all_21901_to_00352.cpc0109.djvu
4.4 Direct sums and products......Page all_21901_to_00352.cpc0113.djvu
5.1 F-precovers and covers......Page all_21901_to_00352.cpc0117.djvu
5.2 Existence of precovers and covers......Page all_21901_to_00352.cpc0119.djvu
5.3 Projective and flat covers......Page all_21901_to_00352.cpc0122.djvu
5.4 Injective covers......Page all_21901_to_00352.cpc0132.djvu
5.5 Direct sums and T-nilpotency......Page all_21901_to_00352.cpc0137.djvu
6.1 F-preenvelopes and envelopes......Page all_21901_to_00352.cpc0141.djvu
6.2 Existence of preenvelopes......Page all_21901_to_00352.cpc0142.djvu
6.3 Existence of envelopes......Page all_21901_to_00352.cpc0144.djvu
6.4 Direct sums of envelopes......Page all_21901_to_00352.cpc0146.djvu
6.5 Flat envelopes......Page all_21901_to_00352.cpc0148.djvu
6.6 Existence of envelopes for injective structures......Page all_21901_to_00352.cpc0151.djvu
6.7 Pure injective envelopes......Page all_21901_to_00352.cpc0156.djvu
7.1 Definitions and basic results......Page all_21901_to_00352.cpc0164.djvu
7.2 Fibrations, cofibrations and Wakamatsu lemmas......Page all_21901_to_00352.cpc0166.djvu
7.3 Set theoretic homological algebra......Page all_21901_to_00352.cpc0172.djvu
7.4 Cotorsion theories with enough injectives and projectives......Page all_21901_to_00352.cpc0174.djvu
8.1 Left and right F-resolutions......Page all_21901_to_00352.cpc0179.djvu
8.2 Derived functors and balance......Page all_21901_to_00352.cpc0181.djvu
8.3 Applications to modules......Page all_21901_to_00352.cpc0189.djvu
8.4 F-dimensions......Page all_21901_to_00352.cpc0192.djvu
8.5 Minimal pure injective resolutions of flat modules......Page all_21901_to_00352.cpc0206.djvu
8.6 \lambda and \mu-dimensions......Page all_21901_to_00352.cpc0215.djvu
9.1 Iwanaga-Gorenstein rings......Page all_21901_to_00352.cpc0223.djvu
9.2 The minimal injective resolution of R......Page all_21901_to_00352.cpc0227.djvu
9.3 More on flat and injective modules......Page all_21901_to_00352.cpc0235.djvu
9.4 Torsion products of injective modules......Page all_21901_to_00352.cpc0238.djvu
9.5 Local cohomology and the dualizing module......Page all_21901_to_00352.cpc0241.djvu
10.1 Gorenstein injective modules......Page all_21901_to_00352.cpc0251.djvu
10.2 Gorenstein projective modules......Page all_21901_to_00352.cpc0258.djvu
10.3 Gorenstein flat modules......Page all_21901_to_00352.cpc0265.djvu
10.4 Foxby classes......Page all_21901_to_00352.cpc0270.djvu
11.1 Gorenstein injective precovers and covers......Page all_21901_to_00352.cpc0281.djvu
11.2 Gorenstein injective preenvelopes......Page all_21901_to_00352.cpc0282.djvu
11.3 Gorenstein injective envelopes......Page all_21901_to_00352.cpc0286.djvu
11.4 Gorenstein essential extensions......Page all_21901_to_00352.cpc0289.djvu
11.5 Gorenstein projective precovers and covers......Page all_21901_to_00352.cpc0291.djvu
11.6 Auslander's last theorem (Gorenstein projective covers)......Page all_21901_to_00352.cpc0296.djvu
11.7 Gorenstein flat covers......Page all_21901_to_00352.cpc0300.djvu
11.8 Gorenstein flat and projective preenvelopes......Page all_21901_to_00352.cpc0304.djvu
12.1 Balance of Hom(-,-)......Page all_21901_to_00352.cpc0306.djvu
12.2 Balance of - \otimes -......Page all_21901_to_00352.cpc0310.djvu
12.3 Dimensions over n-Gorenstein rings......Page all_21901_to_00352.cpc0312.djvu
12.4 Dimensions over Cohen-Macaulay rings......Page all_21901_to_00352.cpc0317.djvu
12.5 \Omega-Gorenstein modules......Page all_21901_to_00352.cpc0319.djvu
Bibliographical Notes......Page all_21901_to_00352.cpc0331.djvu
Bibliography......Page all_21901_to_00352.cpc0333.djvu
Index......Page all_21901_to_00352.cpc0343.djvu
