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
