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Preface

Chapter 1. Rings and Modules

1. Preliminaries

2. Projective modules

3. Injective modules

4. Semi-simple rings

5. Hereditary rings

6. Semi-hereditary rings

7. Noetherian rings

Exercises

Chapter 2. Additive Functors

I. Definitions

2. Examples

3. Operators

4. Preservation of exactness

5. Composite functors

6. Change of rings

Exercises

Chapter 3. Satellites

1. Definition of satellites

2. Connecting homomorphisms

3. Half exact functors

4. Connected sequence of functors

5. Axiomatic description of satellites

6. Composite functors

7. Several variables

Exercises

Chapter 4. Homology

1. Modules with differentiation

2. The ring of dual numbers

3. Graded modules, complexes

4. Double gradings and complexes

5. Functors of complexes

6. The homomorphism

7. The homomorphism （continuation）

8. Kiinneth relations

Exercises

Chapter 5. Derived Functors

1. Complexes over modules; resolutions

2. Resolutions of sequences

3. Definition of derived functors

4. Connecting homomorphisms

5. The functors ROT and LoT

6. Comparison with satellites

7. Computational devices

8. Partial derived functors

9. Sums, products, limits

10. The sequence of a map

Exercises

Chapter 6. Derived Functors of and Hom

1. The functors Tor and Ext

2. Dimension of modules and rings

3. Kiinneth relations

4. Change of rings

5. Duality homomorphisms

Exercises

Chapter 7. Integral Domains

1. Generalities

2. The field of quotients

3. Inversible ideals

4. Priifer rings

5. Dedekind rings

6. Abelian groups

7. A description of Tort （A,C）

Exercises

Chapter 8. Augmented Rings

1. Homology and cohomology o'f an augmented ring

2. Examples

3. Change of rings

……

Chapter 9. Associative Algebras

Chapter 10. Supplemented Algebras

Chapter 11. Products

Chapter 12. Finite Groups

Chapter 13. Lie Algebras

Chapter 14. Extensions

Chapter 15. Spectral Sequences

Chapter 16. Applications of Spectral Sequences

Chapter 17. Hyperhomology

Appendix: Exact categories, by David A. Buchsbaum

List of Symbols

Index of Terminology

During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to present a unified account of these developments and to lay the foundations of a full-fledged theory.
The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments. We present herein a single cohomology （and also a homology） theory which embodies all three； each is obtained from it by a suitable specialization.