Relative Homological Algebra

Description

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. In this new edition the authors have added well-known additional material in the first three chapters, and added new material that was not available at the time the original edition was published. In particular, the major changes are the following: Chapter 1: Section 1.2 has been rewritten to clarify basic notions for the beginner, and this has necessitated a new Section 1.3. Chapter 3: The classic work of D. G. Northcott on injective envelopes and inverse polynomials is finally included. This provides additional examples for the reader. Chapter 11: Section 11.9 on Kaplansky classes makes volume one more up to date. The material in this section was not available at the time the first edition was published. The authors also have clarified some text throughout the book and updated the bibliography by adding new references. The book is also suitable for an introductory course in commutative and ordinary homological algebra. Preface Preface to the Second Edition 1 Basic Concepts 1.1 Zorn’s Lemma, Ordinal and Cardinal Numbers 1.2 Modules 1.3 Tensor Products of Modules and Nakayama Lemma 1.4 Categories and Functors 1.5 Complexes of Modules and Homology 1.6 Direct and Inverse Limits 1.7 I-adic Topology and Completions 2 Flat Modules, Chain Conditions and Prime Ideals 2.1 Flat Modules 2.2 Localization 2.3 Chain Conditions 2.4 Prime Ideals and Primary Decomposition 2.5 Artin-Rees Lemma and Zariski Rings 3 Injective and Flat Modules 3.1 Injective Modules 3.2 Natural Identities, Flat Modules, and Injective Modules 3.3 Injective Modules over Commutative Noetherian Rings 3.4 Matlis Duality 4 Torsion Free Covering Modules 4.1 Existence of Torsion Free Precovers 4.2 Existence of Torsion Free Covers 4.3 Examples 4.4 Direct Sums and Products 5 Covers 5.1 ℱ-precovers and covers 5.2 Existence of Precovers and Covers 5.3 Projective and Flat Covers 5.4 Injective Covers 5.5 Direct Sums and T-nilpotency 6 Envelopes 6.1 ℱ-preenvelopes and Envelopes 6.2 Existence of Preenvelopes 6.3 Existence of Envelopes 6.4 Direct Sums of Envelopes 6.5 Flat Envelopes 6.6 Existence of Envelopes for Injective Structures 6.7 Pure Injective Envelopes 7 Covers, Envelopes, and Cotorsion Theories 7.1 Definitions and Basic Results 7.2 Fibrations, Cofibrations and Wakamatsu Lemmas 7.3 Set Theoretic Homological Algebra 7.4 Cotorsion Theories with Enough Injectives and Projectives 8 Relative Homological Algebra and Balance 8.1 Left and Right ℱ-resolutions 8.2 Derived Functors and Balance 8.3 Applications to Modules 8.4 ℱ-dimensions 8.5 Minimal Pure Injective Resolutions of Flat Modules 8.6 λ and μ-dimensions 9 Iwanaga-Gorenstein and Cohen-Macaulay Rings and Their Modules 9.1 Iwanaga-Gorenstein Rings 9.2 The Minimal Injective Resolution of R 9.3 More on Flat and Injective Modules 9.4 Torsion Products of Injective Modules 9.5 Local Cohomology and the Dualizing Module 10 Gorenstein Modules 10.1 Gorenstein Injective Modules 10.2 Gorenstein Projective Modules 10.3 Gorenstein Flat Modules 10.4 Foxby Classes 11 Gorenstein Covers and Envelopes 11.1 Gorenstein Injective Precovers and Covers 11.2 Gorenstein Injective Preenvelopes 11.3 Gorenstein Injective Envelopes 11.4 Gorenstein Essential Extensions 11.5 Gorenstein Projective Precovers and Covers 11.6 Auslander’s Last Theorem (Gorenstein Projective Covers) 11.7 Gorenstein Flat Covers 11.8 Gorenstein Flat and Projective Preenvelopes 11.9 Kaplansky Classes 12 Balance over Gorenstein and Cohen-Macaulay Rings 12.1 Balance of Hom(–, –) 12.2 Balance of – ⊗ – 12.3 Dimensions over n-Gorenstein Rings 12.4 Dimensions over Cohen-Macaulay Rings 12.5 Ω-Gorenstein Modules Bibliographical Notes Bibliography Index