Author | Hilton, Peter J. author |
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Title | A Course in Homological Algebra [electronic resource] / by Peter J. Hilton, Urs Stammbach |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1997 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4419-8566-8 |
Descript | XII, 366 p. online resource |
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. The Kรผnneth Theorem -- 3. The Dual Kรผnneth Theorem -- 4. Applications of the Kรผnneth 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, Hopfโs 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 : Hubertโs 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. Finiteness Conditions on Groups -- 4. Modular Representation Theory -- 5. Stable and Derived Categories