Author | Mac Lane, Saunders. author |
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Title | Homology [electronic resource] / by Saunders Mac Lane |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1995 |
Connect to | http://dx.doi.org/10.1007/978-3-642-62029-4 |
Descript | X, 422 p. online resource |
I. Modules, Diagrams, and Functors -- 1. The Arrow Notation -- 2. Modules -- 3. Diagrams -- 4. Direct Sums -- 5. Free and Projective Modules -- 6. The Functor Horn -- 7. Categories -- 8. Functors -- II. Homology of Complexes -- 1. Differential Groups -- 2. Complexes -- 3. Cohomology -- 4. The Exact Homology Sequence -- 5. Some Diagram Lemmas -- 6. Additive Relations -- 7. Singular Homology -- 8. Homotopy -- 9. Axioms for Homology -- III. Extensions and Resolutions -- 1. Extensions of Modules -- 2. Addition of Extensions -- 3. Obstructions to the Extension of Homomorphisms -- 4. The Universal Coefficient Theorem for Cohomology -- 5. Composition of Extensions -- 6. Resolutions -- 7. Injective Modules -- 8. Injective Resolutions -- 9. Two Exact Sequences for Extn -- 10. Axiomatic Description of Ext -- 11. The Injective Envelope -- IV. Cohomology of Groups -- 1. The Group Ring -- 2. Crossed Homomorphisms -- 3. Group Extensions -- 4. Factor Sets -- 5. The Bar Resolution -- 6. The Characteristic Class of a Group Extension -- 7. Cohomology of Cyclic and Free Groups -- 8. Obstructions to Extensions -- 9. Realization of Obstructions -- 10. SCHURโS Theorem -- 11. Spaces with Operators -- V. Tensor and Torsion Products -- 1. Tensor Products -- 2. Modules over Commutative Rings -- 3. Bimodules -- 4. Dual Modules -- 5. Right Exactness of Tensor Products -- 6. Torsion Products of Groups -- 7. Torsion Products of Modules -- 8. Torsion Products by Resolutions -- 9. The Tensor Product of Complexes -- 10. The KรNNETH Formula -- 11. Universal Coefficient Theorems -- VI. Types of Algebras -- 1. Algebras by Diagrams -- 2. Graded Modules -- 3. Graded Algebras -- 4. Tensor Products of Algebras -- 5. Modules over Algebras -- 6. Cohomology of free Abelian Groups -- 7. Differential Graded Algebras -- 8. Identities on Horn and ? -- 9. Coalgebras and HOPF Algebras -- VII. Dimension -- 1. Homological Dimension -- 2. Dimensions in Polynomial Rings -- 3. Ext and Tor for Algebras -- 4. Global Dimensions of Polynomial Rings -- 5. Separable Algebras -- 6. Graded Syzygies -- 7. Local Rings -- VIII. Products -- 1. Homology Products -- 2. The Torsion Product of Algebras -- 3. A Diagram Lemma -- 4. External Products for Ext -- 5. Simplicial Objects -- 6. Normalization -- 7. Acyclic Models -- 8. The EILENBERG-ZILBER Theorem -- 9. Cup Products -- IX. Relative Homological Algebra -- 1. Additive Categories -- 2. Abelian Categories -- 3. Categories of Diagrams -- 4. Comparison of Allowable Resolutions -- 5. Relative Abelian Categories -- 6. Relative Resolutions -- 7. The Categorical Bar Resolution -- 8. Relative Torsion Products -- 9. Direct Products of Rings -- X. Cohomology of Algebraic Systems -- 1. Introduction -- 2. The Bar Resolution for Algebras -- 3. The Cohomology of an Algebra -- 4. The Homology of an Algebra -- 5. Homology of Groups and Monoids -- 6. Ground Ring Extensions and Direct Products -- 7. Homology of Tensor Products -- 8. The Case of Graded Algebras -- 9. Complexes of Complexes -- 10. Resolutions and Constructions -- 11. Two-stage Cohomology of DGA-Algebras -- 12. Cohomology of Commutative DGA-Algebras -- 13. Homology of Algebraic Systems -- XI. Spectral Sequences -- 1. Spectral Sequences -- 2. Fiber Spaces -- 3. Filtered Modules -- 4. Transgression -- 5. Exact Couples -- 6. Bicomplexes -- 7. The Spectral Sequence of a Covering -- 8. Cohomology Spectral Sequences -- 9. Restriction, Inflation, and Connection -- 10. The Lyndon Spectral Sequence -- 11. The Comparison Theorem -- XII. Derived Functors -- 1. Squares -- 2. Subobjects and Quotient Objects -- 3. Diagram Chasing -- 4. Proper Exact Sequences -- 5. Ext without Projectives -- 6. The Category of Short Exact Sequences -- 7. Connected Pairs of Additive Functors -- 8. Connected Sequences of Functors -- 9. Derived Functors -- 10. Products by Universality -- 11. Proper Projective Complexes -- 12. The Spectral KรNNETH Formula -- List of Standard Symbols