Homology

This classic and much-cited book is a systematic introduction to homological algebra, starting with basic notions in abstract algebra and category theory and continuing with an up-to-date treatment of various advanced topics. Although the subject depends on the use of very general ideas, the book proceeds from the special to the general. The main ideas are introduced gradually with many examples illustrating why they are needed and what they can do. In conclusion the book treats additive functors in an abelian category relative to a proper class of exact sequences subsuming earlier results. The author has added many historical notes and also exercises which are designed both to give elementary practice in the concepts presented and to formulate additional results not included in the text.

In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL LING. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL ElLENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. Both REINHOLD BAER and JOSEF SCHMID read and commented on my entire manuscript; their advice has led to many improvements. ANDERS KOCK and JACQUES RIGUET have read the entire galley proof and caught many slips and obscurities. Among the others whose sug gestions have served me well, I note FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD, GEOFFREY HORROCKS, FRIED RICH KASCH, JOHANN LEICHT, ARUNAS LIULEVICIUS, JOHN MOORE, DIE TER PUPPE, JOSEPH YAO, and a number of my current students at the University of Chicago - not to m~ntion the auditors of my lectures at Chicago, Heidelberg, Bonn, Frankfurt, and Aarhus. My wife, DOROTHY, has cheerfully typed more versions of more chapters than she would like to count. Messrs.

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. - 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. - 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. - IV. Cohomology 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. - 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 KÜNNETH Formula. - 11. Universal Coefficient Theorems. - 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 HOPFAlgebras. - 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. - 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. - 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. - 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. - XI. Spectral Sequences. - 1. 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. - 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 KÜNNETH Formula. - List of Standard Symbols.