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. Language: English