## ＜電子ブック＞A Course in Homological Algebra

責任表示 by Peter J. Hilton, Urs Stammbach Hilton, Peter J Stammbach, Urs SpringerLink (Online service) Second Edition. English (英語) Springer New York Imprint: Springer 1997- New York, NY, United States シリーズ Graduate Texts in Mathematics ; 4 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 abstr...action 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.続きを見る I. Modules1. Modules2. The Group of Homomorphisms3. Sums and Products4. Free and Projective Modules5. Projective Modules over a Principal Ideal Domain6. Dualization, Injective Modules7 Injective Modules over a Principal Ideal Domain8. Cofree Modules9. Essential ExtensionsII. Categories and Functors1. Categories2. Functors3. Duality4. Natural Transformations5. Products and Coproducts; Universal Constructions6. Universal Constructions (Continued); Pull-backs and Push-outs7. Adjoint Functors8. Adjoint Functors and Universal Constructions9. Abelian Categories10. Projective, Injective, and Free ObjectsIII. Extensions of Modules1. Extensions2. The Functor Ext3. Ext Using Injectives4. Computation of some Ext-Groups5. Two Exact Sequences6. A Theorem of Stein-Serre for Abelian Groups7. The Tensor Product8. The Functor TorIV. Derived Functors1. Complexes2. The Long Exact (Co) Homology Sequence3. Homotopy4. Resolutions5. Derived Functors6. The Two Long Exact Sequences of Derived Functors7. The Functors Extn? Using Projectives8. The Functors % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa!40A3! $$\overline {Ext} _\Lambda ^n$$ Using Injectives9. Extn and n-Extensions10. Another Characterization of Derived Functors11. The Functor Torn?12. Change of RingsV. The Kiinneth Formula1. Double Complexes2. The Künneth Theorem3. The Dual Künneth Theorem4. Applications of the Künneth FormulasVI. Cohomology of Groups1. The Group Ring2. Definition of (Co) Homology3. H0, H04. H1, H1 with Trivial Coefficient Modules5. The Augmentation Ideal, Derivations, and the Semi-Direct Product6. A Short Exact Sequence7. The (Co) Homology of Finite Cyclic Groups8. The 5-Term Exact Sequences9. H2, Hopf's Formula, and the Lower Central Series10. H2 and Extensions11. Relative Projectives and Relative Injectives12. Reduction Theorems13. Resolutions14. The (Co) Homology of a Coproduct15. The Universal Coefficient Theorem and the (Co)Homology of a Product16. Groups and SubgroupsVII. Cohomology of Lie Algebras1. Lie Algebras and their Universal Enveloping Algebra2. Definition of Cohomology; H0, H13. H2 and Extensions4. A Resolution of the Ground Field K5. Semi-simple Lie Algebras6. The two Whitehead Lemmas7. Appendix : Hubert's Chain-of-Syzygies TheoremVIII. Exact Couples and Spectral Sequences1. Exact Couples and Spectral Sequences2. Filtered Differential Objects3. Finite Convergence Conditions for Filtered Chain Complexes4. The Ladder of an Exact Couple5. Limits6. Rees Systems and Filtered Complexes7. The Limit of a Rees System8. Completions of Filtrations9. The Grothendieck Spectral SequenceIX. Satellites and Homology1. Projective Classes of Epimorphisms2. ?-Derived Functors3. ?-Satellites4. The Adjoint Theorem and Examples5. Kan Extensions and Homology6. Applications: Homology of Small Categories, Spectral SequencesX. Some Applications and Recent Developments1. Homological Algebra and Algebraic Topology2. Nilpotent Groups3. Finiteness Conditions on Groups4. Modular Representation Theory5. Stable and Derived Categories.続きを見る Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

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レコードID 3208995 QA612.33 512.66 Mathematics. K-theory. Mathematics. K-Theory. ssj0001296330 9781461264385[1461264383](print) 9781441985668[1441985662] 2020.06.27 2020.06.28