同調(diào)代數(shù)領(lǐng)域在20世紀(jì)后半葉己演進(jìn)成為數(shù)學(xué)研究人員的一種基本工具。本書論述了關(guān)于當(dāng)今同調(diào)代數(shù)的基本概念，并闡述了同調(diào)代數(shù)與拓?fù)鋵W(xué)、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(dǎo)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(dǎo)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(xué)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(dǎo)出范疇以及全導(dǎo)出函子。本書通過展示這些工具的使用方法，幫助初學(xué)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(xué)教授，數(shù)學(xué)系研究生項(xiàng)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。
目錄：
Introduction
1 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 Categories
2 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 Ext
3 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 Theorems
4 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 Cohomology
5 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 Couples
6 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 Groups
7 Lie Algebra Homology and Cohomology
7.1 Lie Algebras
7.2 ft-Modules
7.3 Universal Enveloping Algebras
7.4 Hl and Hi
7.5 The Hochschild-Serre Spectral Sequence
7.6 H2 and Extensions
7.7 The Cheva lley-Eilenberg Complex
7.8 Semisimple Lie Algebras
7.9 Universal Central Extensions
8 Simplicial Methods in Homological Algebra
8.1 Simplicial Objects
8.2 Operations on Simplicial Objects
8.3 Simplicial Homotopy Groups
8.4 The Dold-Kan Correspondence
8.5 The Eilenberg-Zilber Theorem
8.6 Canonical Resolutions
8.7 Cotriple Homology
8.8 Andre-Quillen Homology and Cohomology
9 Hochschild and Cyclic Homology
9.1 Hochschild Homology and Cohomology of Algebras
9.2 Derivations, Differentials, and Separable Algebras
9.3 H2, Extensions, and Smooth Algebras
9.4 Hochschild Products
9.5 Morita Invariance
9.6 Cyclic Homology
9.7 Group Rings
9.8 Mixed Complexes
9.9 Graded Algebras
9.10 Lie Algebras of Matrices
10 The Derived Category
10.1 The Category K(A)
10.2 Triangulated Categories
10.3 Localization and the Calculus of Fractions
10.4 The Derived Category
10.5 Derived Functors
10.6 The Total Tensor Product
10.7 Ext and RHom
10.8 Replacing Spectral Sequences
10.9 The Topological Derived Category
A Category Theory Language
A.1 Categories
A.2 Functors
A.3 Natural Transformations
A.4 Abelian Categories
A.5 Limits and Colimits
A.6 Adjoint Functors
References
Index
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