内容

目次

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. SCHUR’S 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 KÜNNETH 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 KÜNNETH Formula.- List of Standard Symbols.