Homological Algebra

By: S. I. Gelfand, Yu I. Manin, A. I. Kostrikin (Editor)

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Paperback | 20 May 1999

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236 Pages

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This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.

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Introduction	p. 4
Complexes and Cohomology	p. 8
Complexes and the Exact Sequence	p. 8
Standard Complexes in Algebra and in Geometry	p. 9
Spectral Sequence	p. 17
Bibliographic Hints	p. 21
The Language of Categories	p. 22
Categories and Functors	p. 22
Additive and Abelian Categories	p. 35
Functors in Abelian Categories	p. 42
Classical Derived Functors	p. 47
Bibliographic Hints	p. 52
Homology Groups in Algebra and in Geometry	p. 52
Small Dimensions	p. 52
Obstructions, Torsors, Characteristic Classes	p. 56
Cyclic (Co)Homology	p. 60
Non-Commutative Differential Geometry	p. 67
(Co)Homology of Discrete Groups	p. 71
Generalities on Lie Algebra Cohomology	p. 76
Continuous Cohomology of Lie Groups	p. 77
Cohomology of Infinite-Dimensional Lie Algebras	p. 81
Bibliographic Hints	p. 85
Derived Categories and Derived Functors	p. 86
Definition of the Derived Category	p. 86
Derived Category as the Localization of Homotopic Category	p. 92
Structure of the Derived Category	p. 97
Derived Functors	p. 102
Sheaf Cohomology	p. 110
Bibliographic Hints	p. 120
Triangulated Categories	p. 121
Main Notions	p. 121
Examples	p. 128
Cores	p. 133
Bibliographic Hints	p. 139
Mixed Hodge Structures	p. 140
Introduction	p. 140
The Category of Hodge Structures	p. 142
Mixed Hodge Structures on Cohomology with Constant Coefficients	p. 145
Hodge Structures on Homotopic Invariants	p. 148
Hodge-Deligne Complexes	p. 153
Hodge-Deligne Complexes for Singular and Simplicial Varieties	p. 155
Hodge-Beilinson Complexes and Derived Categories of Hodge Structures	p. 157
Variations of Hodge Structures	p. 159
Bibliographic Hints	p. 162
Perverse Sheaves	p. 163
Perverse Sheaves	p. 163
Glueing	p. 168
Bibliographic Hints	p. 172
D-Modules	p. 173
Introduction	p. 173
The Weyl Algebra	p. 175
Algebraic D-Modules	p. 182
Inverse Image	p. 188
Direct Image	p. 190
Holonomic Modules	p. 195
Regular Connections	p. 202
D-Modules with Regular Singularities	p. 205
Equivalence of Categories (Riemann-Hilbert Correspondence)	p. 208
Bibliographic Hints	p. 210
References	p. 211
Author Index	p. 217
Subject Index	p. 219
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Homological Algebra

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