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.

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 Ⅱ. 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. SCHUR''''''''''''''''S 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

 1. Introduction

 2. The Bar Resolution for Algebras

 3. The Cohomology of an Algebra

 4. The Homology of an Algebra

 5. Homology of Groups and Monoids

 6. Ground Ring Extensions and Direct Products

 7. Homology of Tensor Products

 8. The Case of Graded Algebras

 9. Complexes of Complexes

 10. Resolutions and Constructions

 11. Two-stage Cohomology of DGA-Algebras

 12. Cohomology of Commutative DGA-Algebras

 13. Homology of Algebraic Systems

Chapter Ⅺ. Spectral Sequences

 |. Spectral Sequences

 2. Fiber Spaces

 3- Filtered Modules

 4. Transgression

 5. Exact Couples

 6. Bicomplexes

 7. The Spectral Sequence of a Covering

 8. Cohomology Spectral Sequences

 9. Restriction. Inflation, and Connection

 10. The Lyndon Spectral Sequence

 11. The Comparison Theorem

Chapter Ⅻ. Derived Functors

 1. Squares

 2. Subobjects and Quotient Objects

 3. Diagram Chasing

 4. Proper Exact Sequences

 5. Ext without Projectives

 6. The Category of Short Exact Sequences

 7. Connected Pairs of Additive Functors

 8. Connected Sequences of Functors

 9. Derived Functors

 10. Products by Universality

 11. Proper Projective Complexes

 12. The Spectral KUNNETH Formula

Bibliography

List of Standard Symbols

Index