同調(diào)代數(shù)導論

內(nèi)容概要

同調(diào)代數(shù)領(lǐng)域在20世紀后半葉己演進成為數(shù)學研究人員的一種基本工具。本書論述了關(guān)于當今同調(diào)代數(shù)的基本概念，并闡述了同調(diào)代數(shù)與拓撲學、正則局部環(huán)以及半單李代數(shù)聯(lián)系的歷史淵源。    本書前半部分論述了導出函子、Tor與Ext函子、透視維數(shù)及譜序列等同調(diào)代數(shù)的典范論題，群的同調(diào)和李代數(shù)解釋了這些論題。其間混雜某些不甚典范的論題，如導出逆極限函子lim、周部上同調(diào)、伽羅瓦上同調(diào)以及仿射李代數(shù)。    本書后半部分論述了一些并非傳統(tǒng)的論題，它們是現(xiàn)代同調(diào)數(shù)學工具箱中的重要部分，如單純形法、霍赫希爾德和循環(huán)同調(diào)、導出范疇以及全導出函子。本書通過展示這些工具的使用方法，幫助初學者突破同調(diào)代數(shù)的技術(shù)壁壘。

作者簡介

Charles A.Weibel 羅格斯大學教授，數(shù)學系研究生項目副主任，《Journal of Pure and Applied Algebra》雜志主編。他的研究領(lǐng)域包括代數(shù)K理論、代數(shù)幾何和同調(diào)代數(shù)等。

書籍目錄

Introduction1  Chain Complexes  1.1  Complexes of R-Modules  1.2  Operations on Chain Complexes  1.3  Long Exact Sequences  1.4  Chain Homotopies  1.5  Mapping Cones and Cylinders  1.6  More on Abelian Categories2  Derived Functors  2.1  -Functors  2.2  Projective Resolutions  2.3  Injective Resolutions  2.4  Left Derived Functors  2.5  Right Derived Functors  2.6  Adjoint Functors and Left/Right Exactness  2.7  Balancing Tor and Ext3  Tot and Ext  3.1  Tot for Abelian Groups  3.2  Tor and Flatness  3.3  Ext for Nice Rings  3.4  Ext and Extensions  3.5  Derived Functors of the Inverse Limit  3.6  Universal Coefficient Theorems4  Homological Dimension  4.1  Dimensions   4.2  Rings of Small Dimension  4.3  Change of Rings Theorems  4.4  Local Rings  4.5  Koszui Complexes  4.6  Local Cohomology5  Spectral Sequences  5.1  Introduction  5.2  Terminology  5.3  The Leray-Serre Spectral Sequence  5.4  Spectral Sequence of a Filtration  5.5  Convergence  5.6  Spectral Sequences of a Double Complex  5.7  Hyperhomology  5.8  Grothendieck Spectral Sequences  5.9  Exact Couples6  Group Homology and Cohomology  6.1  Definitions and First Properties  6.2  Cyclic and Free Groups  6.3  Shapiro's Lemma  6.4  Crossed Homomorphisms and Hi  6.5  The Bar Resolution  6.6  Factor Sets and H2  6.7  Restriction, Corestriction, Inflation, and Transfer  6.8  The Spectral Sequence  6.9  Universal Central Extensions  6.10  Covering Spaces in Topology  6.11  Galois Cohomology and Profinite Groups7  Lie Algebra Homology and Cohomology……8  Simplicial Methods in Homological Algebra9  Hochschild and Cyclic Homology10  The Derived CategoryA  Category Theory LanguageReferencesIndex

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