A Course in Homological Algebra

We have inserted, in this edition, an extra chapter (Chapter X) entitled "Some Applications and Recent Developments." The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Naturally, the treatments in these five sections are somewhat cursory, the intention being to give the flavor of the homo logical methods rather than the details of the arguments and results. We would like to express our appreciation of help received in writing Chapter X; in particular, to Ross Geoghegan and Peter Kropholler (Section 3), and to Jacques Thevenaz (Sections 4 and 5). The only other changes consist of the correction of small errors and, of course, the enlargement of the Index. Peter Hilton Binghamton, New York, USA Urs Stammbach Zurich, Switzerland Contents Preface to the Second Edition vii Introduction. . I. Modules.

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 Functors. - 1. Complexes. - 2. The Long Exact (Co) Homology Sequence. - 3. Homotopy. - 4. Resolutions. - 5. Derived Functors. - 6. The Two Long Exact Sequences of Derived Functors. - 7. The Functors Extn? Using Projectives. - 8. The Functors % MathType! MTEF! 2! 1! +- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa! 40A3! $$ \overline {Ext} _\Lambda ^n $$ Using Injectives. - 9. Extn and n-Extensions. - 10. Another Characterization of Derived Functors. - 11. The Functor Torn? . - 12. Change of Rings. - V. The Kiinneth Formula. - 1. Double Complexes. - 2. TheKünneth Theorem. - 3. The Dual Künneth Theorem. - 4. Applications of the Künneth Formulas. - VI. Cohomology of Groups. - 1. The Group Ring. - 2. Definition of (Co) Homology. - 3. H0, H0. - 4. H1, 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, Hopf s Formula, and the Lower Central Series. - 10. H2 and Extensions. - 11. Relative Projectives 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. - VII. Cohomology of Lie Algebras. - 1. Lie Algebras and their Universal Enveloping Algebra. - 2. Definition of Cohomology; H0, 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 : Hubert 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. - 1. Projective Classes of Epimorphisms. - 2. ? -Derived Functors. - 3. ? -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. - 1. Homological Algebra and Algebraic Topology. - 2. Nilpotent Groups. - 3. FinitenessConditions on Groups. - 4. Modular Representation Theory. - 5. Stable and Derived Categories.