We have inserted, in this edition, an extra chapter (Chapter X) entitled "Some Applications and Recent Developments." The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Naturally, the treatments in these five sections are somewhat cursory, the intention being to give the flavor of the homological methods rather than the details of the arguments and results.

本书为英文版。

Preface to the Second Edition

Introduction

I. Modules

1. Modules

2. The Group of Homomorphisms

3. Sums and Products

4. Free and Projective Modules

5. Projective Modules over a Principal Ideal Domain

6. Dualization, Injective Modules

7. Injective Modules over a Principal Ideal Domain

8. Cofree Modules

9. Essential Extensions

II. Categories and Functors

1. Categories

2. Functors

3. Duality

4. Natural Transformations

5. Products and Coproducts; Universal Constructions

6. Universal Constructions (Continued); Pull-backs and Push-outs

7. Adjoint Functors

8. Adjoint Functors and Universal Constructions

9. Abelian Categories

10. Projective, Injective, and Free Objects

III. Extensions of Modules

1. Extensions

2. The Functor Ext

3. Ext Using Injectives

4. Computation of some Ext-Groups

5. Two Exact Sequences

6. A Theorem of Stein-Serre for Abelian Groups

7. The Tensor Product

8. The Functor Tor

IV. Derived Funetors

1. Complexes

2. The Long Exact (Co)Homology Sequence

3. Homotopy

4. Resolutions

5. Derived Functors

6. The Two Long Exact Sequences of Derived Functor

7. The Functors Ext Using Projectives

8. The Functors Using Injectives

9. Extn and n-Extensions

10. Another Characterization bfDerived Functors

11. The Functor Tot

12. Change of Rings

V. The Kiinneth Formula

1. Double Complexes

2. The Kiinneth Theorem

3. The Dual KiJnneth Theorem

4. Applications of the Kiinneth Formulas

VI. Cohomology of Groups

1. The Group Ring

2. Definition of (Co) Homology

3. H, Ho

4. H, H1 with Trivial Coefficient Modules

5. The Augmentation Ideal, Derivations, and the Semi-Direct Product

6. A Short Exact Sequence

7. The (Co) Homology of Finite Cyclic Groups

8. The 5-Term Exact Sequences

9. H2, Hoprs Formula, and the Lower Central Series.

10. H2 and Extensions

11. Relative Projeetives and Relative Injectives

12. Reduction Theorems

13. Resolutions

14. The (Co)Homology of a Coproduct

15. The Universal Coefficient Theorem and the(Co) Homology of a Product

16. Groups and Subgroups

Vll. Cohomology of Lie Algebras

1. Lie Algebras and their Universal Enveloping Algebra.

2. Definition of Cohomology; H, H1

3. H2 and Extensions

4. A Resolution of the Ground Field K

5. Semi-simple Lie Algebras

6. The two Whitehead Lemmas

7. Appendix: Hilbert's Chain-of-Syzygies Theorem

VIII. Exact Couples and Spectral Sequences

1. Exact Couples and Spectral Sequences

2. Filtered Differential Objects

3. Finite Convergence Conditions for Filtered Chain Complexes

4. The Ladder of an Exact Couple

5. Limits

6. Rees Systems and Filtered Complexes

7. The Limit of a Rees System

8. Completions of Filtrations

9. The Grothendieck Spectral Sequence

IX. Satellites and Homology

I. Projective Classes of Epimorphisms

2. g-Derived Functors

3. 8-Satellites

4. The Adjoint Theorem and Examples

5. Kan Extensions and Homology

6. Applications: Homology of Small Categories,Spectral Sequences

X. Some Applications and Recent Developments

I. Homological Algebra and Algebraic Topology

2. Nilpotent Groups

3. Finiteness Conditions on Groups

4. Modular Representation Theory

5. Stable and Derived Categories

Bibliography

Index