Homological Algebra: Homological Algebra.

Front Cover
A.I. Kostrikin, Igorʹ Rostislavovich Shafarevich, I.R. Shafarevich
Springer Science & Business Media, Mar 29, 1994 - Mathematics - 222 pages
0 Reviews
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.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Complexes and Cohomology
8
2 Standard Complexes in Algebra and in Geometry
9
3 Spectral Sequence
17
Bibliographic Hints
21
The Language of Categories
22
2 Additive and Abelian Categories
35
3 Functors in Abelian Categories
42
4 Classical Derived Functors
47
Bibliographic Hints
139
Mixed Hodge Structures
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 HodgeDeligne Complexes
153
5 HodgeDeligne Complexes for Singular and Simplicial Varieties
155
6 HodgeBeilinson Complexes and Derived Categories of Hodge Structures
157

Homology Groups in Algebra and in Geometry
52
2 Obstructions Torsors Characteristic Classes
56
3 Cyclic CoHomology
60
4 NonCommutative Differential Geometry
67
5 CoHomology of Discrete Groups
71
6 Generalities on Lie Algebra Cohomology
76
7 Continuous Cohomology of Lie Groups
77
8 Cohomology of InfiniteDimensional Lie Algebras
81
Bibliographic Hints
85
Derived Categories and Derived Functors
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
Triangulated Categories
121
2 Examples
128
3 Cores
133
7 Variations of Hodge Structures
159
Perverse Sheaves
163
2 Glueing
168
Bibliographic Hints
172
PModules
173
1 The Weyl Algebra
175
2 Algebraic PModules
182
3 Inverse Image
188
4 Direct Image
190
5 Holonomic Modules
195
6 Regular Connections
202
7 PModules with Regular Singularities
205
8 Equivalence of Categories RiemannHilbert Correspondence
208
Bibliographic Hints
210
References
211
Author Index
217
Subject Index
219
Copyright

Common terms and phrases

Bibliographic information