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

1．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 Extn ∧ Using Projectives

8．The Functors？Using Injectives

9．Extn and n-Extensions

10．Another Characterization of Derived Functors

11．The Functor Tor∧ n

12．Change of Rings

Ⅴ．The Künneth Formula

1．Double Complexes

2．The Künneth Theorem

3．The Dual Künneth Theorem

4．Applications of the Künneth Formulas

Ⅵ．Cohomology of Groups

1．The Group Ring

2．Definition of（Co）Homology

3．H0,H0

4．H1,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,Hopf's Formula,and the Lower Central Series

10．H2 and Extensions

11．Relative Projectives 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

Ⅶ．Cohomology of Lie Algebras

1．Lie Algebras and their Universal Enveloping Algebra

2．Definition of Cohomology；H0,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

1．Projective Classes of Epimorphisms

2．E-Derived Functors

3．E-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

1．Homological Algebra and Algebraic Topology

2．Nilpotent Groups

3．Finiteness Conditions on Groups

4．Modular Representation Theory

5．Stable and Derived Categories

Bibliography

Index