Methods of homological algebra同调代数方法
2003-1
Springer
Sergei I. Gelfand，Yuri I. Manin 著
372
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Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections.
I. Simplicial Sets 1.1 Triangulated Spaces 1.2 Simplicial Sets 1.3 Simplicial Topological Spaces and the Eilenberg-Zilber Theorem 1.4 Homology and Cohomology 1.5 Sheaves 1.6 The Exact Sequence 1.7 ComplexesII. Main Notions of the Category Theory II.1 The Language of Categories and Functors II.2 Categories and Structures, Equivalence of Categories II.3 Structures and Categories. Representable Functors II.4 Category Approach to the Construction of Geometrical Objects II.5 Additive and Abelian Categories II.6 Functors in Abelian CategoriesIII. Derived Categories and Derived Functors III.1 Complexes as Generalized Objects III.2 Derived Categories and Localization III.3 Triangles as Generalized Exact Triples III.4 Derived Category as the Localization of Homotopic Category III.5 The Structure of the Derived Category III.6 Derived Functors III.7 Derived Functor of the Composition. Spectral Sequence III.8 Sheaf CohomologyIV. Triangulated Categories IV.1 Triangulated Categories IV.2 Derived Categories Are Triangulated IV.3 An Example: The Triangulated Category of A-Modules IV.4 CoresV. Introduction to Homotopic Algebra V.1 Closed Model Categories V.2 Homotopic Characterization of Weak Equivalences V.3 DG-Algebras as a Closed Model Category V.4 Minimal Algebras V.5 Equivalence of Homotopy CategoriesReferencesIndex
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