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. Th...
| Main Authors: | , |
|---|---|
| Format: | eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1997, 1997
|
| Edition: | 2nd ed. 1997 |
| Series: | Graduate Texts in Mathematics
|
| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- 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!
- 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