Methods of homological algebra同調(diào)代數(shù)方法

內(nèi)容概要

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.

書籍目錄

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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