A Course in Homological AlgebraSpringer Science & Business Media, 5 sept. 2012 - 366 pages We have inserted, in this edition, an extra chapter (Chapter X) entitled "Some Applications and Recent Developments." The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Naturally, the treatments in these five sections are somewhat cursory, the intention being to give the flavor of the homo logical methods rather than the details of the arguments and results. We would like to express our appreciation of help received in writing Chapter X; in particular, to Ross Geoghegan and Peter Kropholler (Section 3), and to Jacques Thevenaz (Sections 4 and 5). The only other changes consist of the correction of small errors and, of course, the enlargement of the Index. Peter Hilton Binghamton, New York, USA Urs Stammbach Zurich, Switzerland Contents Preface to the Second Edition vii Introduction. . I. Modules. |
Table des matières
1 | |
10 | |
Categories and Functors | 40 |
Extensions of Modules | 84 |
Two Exact Sequences | 99 |
A Theorem of SteinSerre for Abelian Groups | 106 |
The Tensor Product | 109 |
The Functor Tor | 112 |
The Dual Künneth Theorem | 177 |
Applications of the Künneth Formulas | 180 |
Cohomology of Groups | 184 |
The Group Ring | 186 |
Definition of Co Homology | 188 |
H Ho | 191 |
H H with Trivial Coefficient Modules | 192 |
The Augmentation Ideal Derivations and the SemiDirect Product | 194 |
Derived Functors | 116 |
Complexes | 117 |
The Long Exact Co Homology Sequence | 121 |
Homotopy | 124 |
Resolutions | 126 |
Derived Functors | 130 |
The Two Long Exact Sequences of Derived Functors | 136 |
The Functors Ext Using Projectives | 139 |
The Functors Ext Using Injectives | 143 |
Ext and nExtensions | 148 |
Another Characterization of Derived Functors | 156 |
The Functor Tor | 161 |
Change of Rings | 162 |
The Künneth Formula | 166 |
Double Complexes | 167 |
The Künneth Theorem | 172 |
A Short Exact Sequence | 197 |
The Co Homology of Finite Cyclic Groups | 200 |
The 5Term Exact Sequences | 202 |
H2 Hopfs Formula and the Lower Central Series | 204 |
H2 and Extensions | 206 |
Relative Projectives and Relative Injectives | 210 |
Reduction Theorems | 213 |
Resolutions | 214 |
The Co Homology of a Coproduct | 219 |
Co Homology of a Product | 221 |
Cohomology of Lie Algebras | 229 |
Exact Couples and Spectral Sequences | 255 |
Satellites and Homology | 306 |
Some Applications and Recent Developments | 331 |
Bibliography | 357 |
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Expressions et termes fréquents
A₁ abelian category abelian group additive functor B₁ bifunctor C₁ chain complex chain map Chapter cochain coefficients cohomology cokernel commutative diagram construction coproduct Corollary D₁ define definition denote derived functors differential dimension direct sum dual Dualize element embedding epimorphisms exact couple example Exercise exists Ext(A extension filtered filtration finite given graded group G H¹(G Hence Hom(A homological algebra homology groups homology theory homotopy induced injective resolution isomorphism kernel Künneth left adjoint Lemma Lie algebra long exact monomorphic morphism natural equivalence natural transformation nilpotent groups object obtain P-local P₁ plainly projective presentation projective resolution proof of Theorem Proposition prove pull-back quotient reader Rees system ring Section short exact sequence Show spectral sequence splits structure subgroup submodule surjective Theorem 2.1 trivial G-module unique universal property vector space Yext zero