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 Functors
I.Complexes
2.The Long Exact Co Homology Sequence
3.Homotopy
4.Resolutions
5.Derived Functors
6.The Two Long Exact Sequences of Derived Functors
7.The Functors Ext Using Projectives
8.The Functors Ext Using Injectives
9.Ext and n-Extensions
10.Another Characterization of Derived Functors
11.The Functor Tor
12.Change of Rings
Ⅴ.The Kiinneth Formula
1.Double Complexes
2.The Kunneth Theorem
3.The Dual Kunneth Theorem
4.Applications of the Kunneth Formulas
Ⅵ.Cohomology of Groups
1.The Group Ring
2.Definition of Co Homology
3.H, H
4.H, H 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, Hopfs Formula, and the Lower Central Series
10.Hz and Extensions
11.Relative Projectives and Relative Injectives
12.Reduction Theorems
13.Resolutions
14.The（Co）Homology of a Coproduct
……
Ⅶ.Cohomology of Lie Algebras
Ⅷ.Exact Couples and Spectral Sequences
Ⅸ.Satellites and Homology
Ⅹ.Some Applications and Recent Developments
Bibliography
Index
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.
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