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Introduction

Chapter I. Modules, Diagrams, and Functors

1. The Arrow Notation

2. Modules

3. Diagrams

4. Direct Sums

5. Free and Projective Modules

6. The Functor Horn

7. Categories

8. Functors

Chapter If. Homology of Complexes

1. Differential Groups

2. Complexes

3. Cohomology

4. The Exact Homology Sequence

5. Some Diagram Lemmas

6. Additive Relations

7. Singular Homology

8. Homotopy

9. Axioms for Homology

Chapter III. Extensions and Resolutions

1. Extensions of Modules

2. Addition of Extensions

3. Obstructions to the Extension of Homomorphisms

4. The Universal Coefficient Theorem for Cohomology

5. Composition of Extensions

6. Resolutions

7. Injective Modules

8. Injective Resolutions

9.Two Exact Sequences for Extn

10. Axiomatic Description of Ext

11. The Injective Envelope

Chapter IV. Cohomol0gy of Groups

1. The Group Ring

2. Crossed Homomorphisms

3. Group Extensions

4. Factor Sets

5. The Bar Resolution

6. The Characteristic Class of a Group Extension

7. Cohomology of Cyclic and Free Groups

8. Obstructions to Extensions

9. Realization of Obstructions

10. SCHURS Theorem

11. Spaces with Operators

Chapter V. Tensor and Torsion Products

1. Tensor Products

2. Modules over Commutative Rings

3. Bimodules

4. Dual Modules

5. Right Exactness of Tensor Products .

6. Torsion Products of Groups

7. Torsion Products of Modules

8. Torsion Products by Resolutions

9. The Tensor Product of Complexes

10. The KONNETH Formula

11. Universal Coefficient Theorems

Chapter VI. Types of Algebras

1. Algebras by Diagrams

2. Graded Modules

3. Graded Algebras

4. Tensor Products of Algebras

5. Modules over Algebras

6. Cohomology of free Abelian Groups

7. Differential Graded Algebras

8. Identities on Horn and

9.Coalgebras and HoPF Algebras

Chapter VII. Dimension

1. Homological Dimension

2. Dimensions in Polynomial Rings

3. Ext and Tor for Algebras

4. Global Dimensions of Polynomial Rings

5. Separable Algebras

6. Graded Syzygies

7. Local Rings

Chapter VIII. Products

1. Homology Products

2. The Torsion Product of Algebras

3. A Diagram Lemma

4. External Products for Ext

5. Simplicial Objects

6. Normalization

7. Acyclic Models

8. The EILENBERG-ZILBER Theorem

9. Cup Products

Chapter IX. Relative Homological Algebra

1. Additive Categories

2. Abelian Categories

3. Categories of Diagrams

4. Comparison of Allowable Resolutions

5. Relative Abelian Categories

6, Relative Resolutions

7. The Categorical Bar Resolution

8. Relative Torsion Products

9. Direct Products of Rings

Chapter X. Cohomology of Algebraic Systems

Chapter XI. Spectral Sequences

Chapter XII. Derived Functors

Bibliography

List of Standard Symbols

Index

In presenting this treatment of homological algebra， it is a pleasureto acknowledge the help and encouragement which I have had fromall sides. Homological algebra arose from many sources in algebra andtopology. Decisive examples came from the study of group extensionsand their factor sets， a subject I learned in joint work with OTTO SCHIL-LING. A further development of homological ideas， with a view to theirtopological applications， came in my long collaboration with SAHUELEZLENBERG; to both collaborators， especial thanks. For many yearsthe Air Force Office of Scientific Research supported my researchprojects on various subjects now summarized here; it is a pleasure toacknowledge their lively understanding Of basic science.