ISBN: 3-540-65378-3
TITLE: Homological Algebra
AUTHOR: Gelfand, S.I.; Manin, Yu.I.
TOC:

Introduction 4
Chapter 1. Complexes and Cohomology 8
 1. Complexes and the Exact Sequence 8
 2. Standard Complexes in Algebra and in Geometry 9
 3. Spectral Sequence 17
Bibliographic Hints 21
Chapter 2. The Language of Categories 22
 1. Categories and Functors 22
 2. Additive and Abelian Categories 35
 3. Functors in Abelian Categories 42
 4. Classical Derived Functors 47
Bibliographic Hints 52
Chapter 3. Homology Groups in Algebra and in Geometry 52
 1. Small Dimensions 52
 2. Obstructions, Torsors, Characteristic Classes 56
 3. Cyclic (Co)Homology 60
 4. Non-Commutative Differential Geometry 67
 5. (Co)Homology of Discrete Groups 71
 6. Generalities on Lie Algebra Cohomology 76
 7. Continuous Cohomology of Lie Groups 77
 8. Cohomology of Infinite-Dimensional Lie Algebras 81
Bibliographic Hints 85
Chapter 4. Derived Categories and Derived Functors 86
 1. Definition of the Derived Category 86
 2. Derived Category as the Localization of Homotopic Category 92
 3. Structure of the Derived Category 97
 4. Derived Functors 102
 5. Sheaf Cohomology 110
Bibliographic Hints 120
Chapter 5. Triangulated Categories 121
 1. Main Notions 121
 2. Examples 128
 3. Cores 133
Bibliographic Hints 139
Chapter 6. Mixed Hodge Structures 140
 0. Introduction 140
 1. The Category of Hodge Structures 142
 2. Mixed Hodge Structures on Cohomology with Constant Coefficients 145
 3. Hodge Structures on Homotopic Invariants 148
 4. Hodge-Deligne Complexes 153
 5. Hodge-Deligne Complexes for Singular
and Simplicial Varieties 155
 6. Hodge-Beilinson Complexes and Derived Categories of Hodge Structures 157
 7. Variations of Hodge Structures 159
Bibliographic Hints 162
Chapter 7. Perverse Sheaves 163
 1. Perverse Sheaves 163
 2. Glueing 168
Bibliographic Hints 172
Chapter 8. D-Modules 173
 0. Introduction 173
 1. The Weyl Algebra 175
 2. Algebraic D-Modules 182
 3. Inverse Image 188
 4, Direct Image 190
 5. Holonomic Modules 195
 6. Regular Connections 202
 7. D-Modules with Regular Singularities 205
 8. Equivalence of Categories (Riemann-Hilbert Correspondence) 208
Bibliographic Hints 210
References 211
Author Index 217
Subject Index 219
END
