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内容简介

　　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.

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目录

Ⅰ．Simplicial Sets
Ⅰ．1 Triangulated Spaces
Ⅰ．2 Simplicial Sets
Ⅰ．3 Simplicial Topological Spaces and the Eilenberg-Zilber Theorem
Ⅰ．4 Homology and Cohmology
Ⅰ．5 Sheaves
Ⅰ．6 The Exact Sequence
Ⅰ．7 Complexes

Ⅱ．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 Categories

Ⅲ．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 Cohomology

Ⅳ．Triangulated Categories
Ⅳ．1 Triangulated Categories
Ⅳ．2 Derived Categories Are Triangulated
Ⅳ．3 An Example： The Triangulated Category of A-Modules
Ⅳ．4 Cores

Ⅴ．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 Categories

References
Index

前言/序言

　　要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。
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　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
国外数学名著系列（续一 影印版）37：同调代数方法（第二版） [Methods of Homological Algebra Second Edition] 电子书 下载 mobi epub pdf txt

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