同調(diào)代數(shù)方法

前言

要使我國(guó)的數(shù)學(xué)事業(yè)更好地發(fā)展起來(lái)，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強(qiáng)對(duì)數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強(qiáng)數(shù)學(xué)的出版工作。從出版方面來(lái)講，除了較好較快地出版我們自己的成果外，引進(jìn)國(guó)外的先進(jìn)出版物無(wú)疑也是十分重要與必不可少的。從數(shù)學(xué)來(lái)說(shuō)，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社。科學(xué)出版社影印一批他們出版的好的新書(shū)，使我國(guó)廣大數(shù)學(xué)家能以較低的價(jià)格購(gòu)買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見(jiàn)到這些書(shū)，無(wú)疑是對(duì)推動(dòng)我國(guó)數(shù)學(xué)的科研與教學(xué)十分有益的事。這次科學(xué)出版社購(gòu)買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書(shū)，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書(shū)中，包括基礎(chǔ)數(shù)學(xué)書(shū)5本，應(yīng)用數(shù)學(xué)書(shū)6本與計(jì)算數(shù)學(xué)書(shū)12本，其中有些書(shū)也具有交叉性質(zhì)。這些書(shū)都是很新的，2000年以后出版的占絕大部分，共計(jì)16本，其余的也是1990年以后出版的。這些書(shū)可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙?，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書(shū)”的分冊(cè)。對(duì)從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點(diǎn)，基礎(chǔ)數(shù)學(xué)類的書(shū)以“經(jīng)典”為主，應(yīng)用和計(jì)算數(shù)學(xué)類的書(shū)以“前沿”為主。這些書(shū)的作者多數(shù)是國(guó)際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書(shū)的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎(jiǎng)”和“沃爾夫數(shù)學(xué)獎(jiǎng)”。這些大數(shù)學(xué)家的著作無(wú)疑將會(huì)對(duì)我國(guó)的科研人員起到非常好的指導(dǎo)作用。當(dāng)然，23本書(shū)只能涵蓋數(shù)學(xué)的一部分，所以，這項(xiàng)工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書(shū)還應(yīng)該翻譯成中文出版，使之有更大的讀者群?？傊?，我對(duì)科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績(jī)。

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.

作者簡(jiǎn)介

作者：(德國(guó))蓋爾范德 (Sergei I.Gelfand)

書(shū)籍目錄

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  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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