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Homological Algebra : Encyclopaedia of Mathematical Sciences - S. I. Gelfand

Homological Algebra

Encyclopaedia of Mathematical Sciences


Published: 20th May 1999
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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.

Introductionp. 4
Complexes and Cohomologyp. 8
Complexes and the Exact Sequencep. 8
Standard Complexes in Algebra and in Geometryp. 9
Spectral Sequencep. 17
Bibliographic Hintsp. 21
The Language of Categoriesp. 22
Categories and Functorsp. 22
Additive and Abelian Categoriesp. 35
Functors in Abelian Categoriesp. 42
Classical Derived Functorsp. 47
Bibliographic Hintsp. 52
Homology Groups in Algebra and in Geometryp. 52
Small Dimensionsp. 52
Obstructions, Torsors, Characteristic Classesp. 56
Cyclic (Co)Homologyp. 60
Non-Commutative Differential Geometryp. 67
(Co)Homology of Discrete Groupsp. 71
Generalities on Lie Algebra Cohomologyp. 76
Continuous Cohomology of Lie Groupsp. 77
Cohomology of Infinite-Dimensional Lie Algebrasp. 81
Bibliographic Hintsp. 85
Derived Categories and Derived Functorsp. 86
Definition of the Derived Categoryp. 86
Derived Category as the Localization of Homotopic Categoryp. 92
Structure of the Derived Categoryp. 97
Derived Functorsp. 102
Sheaf Cohomologyp. 110
Bibliographic Hintsp. 120
Triangulated Categoriesp. 121
Main Notionsp. 121
Examplesp. 128
Coresp. 133
Bibliographic Hintsp. 139
Mixed Hodge Structuresp. 140
Introductionp. 140
The Category of Hodge Structuresp. 142
Mixed Hodge Structures on Cohomology with Constant Coefficientsp. 145
Hodge Structures on Homotopic Invariantsp. 148
Hodge-Deligne Complexesp. 153
Hodge-Deligne Complexes for Singular and Simplicial Varietiesp. 155
Hodge-Beilinson Complexes and Derived Categories of Hodge Structuresp. 157
Variations of Hodge Structuresp. 159
Bibliographic Hintsp. 162
Perverse Sheavesp. 163
Perverse Sheavesp. 163
Glueingp. 168
Bibliographic Hintsp. 172
D-Modulesp. 173
Introductionp. 173
The Weyl Algebrap. 175
Algebraic D-Modulesp. 182
Inverse Imagep. 188
Direct Imagep. 190
Holonomic Modulesp. 195
Regular Connectionsp. 202
D-Modules with Regular Singularitiesp. 205
Equivalence of Categories (Riemann-Hilbert Correspondence)p. 208
Bibliographic Hintsp. 210
Referencesp. 211
Author Indexp. 217
Subject Indexp. 219
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540653783
ISBN-10: 3540653783
Series: Encyclopaedia of Mathematical Sciences
Audience: General
Format: Paperback
Language: English
Number Of Pages: 222
Published: 20th May 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.55 x 15.62  x 1.37
Weight (kg): 0.35