内容简介

同调代数领域在20世纪后半叶己演进成为数学研究人员的一种基本工具。本书论述了关于当今同调代数的基本概念，并阐述了同调代数与拓扑学、正则局部环以及半单李代数联系的历史渊源。

本书前半部分论述了导出函子、Tor与Ext函子、透视维数及谱序列等同调代数的典范论题，群的同调和李代数解释了这些论题。其间混杂某些不甚典范的论题，如导出逆极限函子lim、周部上同调、伽罗瓦上同调以及仿射李代数。

本书后半部分论述了一些并非传统的论题，它们是现代同调数学工具箱中的重要部分，如单纯形法、霍赫希尔德和循环同调、导出范畴以及全导出函子。本书通过展示这些工具的使用方法，帮助初学者突破同调代数的技术壁垒。

作品目录

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 Chevalley-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