Algebra V: Homological Algebra

229 Pages·1994·7.38 MB·English

by  A. I. Kostrikin

Encyclopaedia of
Mathematical Sciences
Volume 38
Editor-in-Chief: R. V. Gamkrelidze
A.I. Kostrikin 1. R. Shafarevich (Eds.)
Algebra V
Homological Algebra
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
AA Agrachev, AA. Gonchar, E.F. Mishchenko,
N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 38, Algebra 5
Publisher VINITI, Moscow 1989
Mathematics Subject Classification (1991):
14F05, 18-XX, 18Fxx, 18Gxx, 32C35, 32Sxx, 32S35, 32S40, 55Nxx
Library of Congress Cataloging-in-Publication Data
Algebra 5. English
Algebra V: homological algebra / A. 1. Kostrikin. 1. R. Shafarevich (eds.).
p. cm. - (Encyclopaedia of mathematical sciences; v. 38)
Includes bibliographical references and indexes.
ISBN 978-3-540-65378-3 ISBN 978-3-642-57911-0 (eBook)
DOI 10.1007/978-3-642-57911-0
1. Algebra. Homological. 1. Kostrikin, A. 1. (Aleksel Ivanovich)
II. Shafarevich, 1. R. (Igor' Rostislavovich), 1923 - .
1I1. Title. IV. Title: Algebra 5. V. Series.
QA169.A3713 1994 512'.55-dc20 93-33442
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© Springer-Verlag Berlin Heidelberg 1994
Softcover reprint ofthe hardcover 1s t edition 1994
Typesetting: Springer TEX in-house system
41/3140 -5 43210 -Printed on acid-free paper
List of Editors, Authors and Translators
Editor-in-Chief
R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITI),
ul. Usievicha 20a, 125219 Moscow, Russia
Consulting Editors
A. I. Kostrikin, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
I. R. Shafarevich, Steklov Mathematical Institute, ul. Vavilova 42,
117966 Moscow, Russia
Authors and Translators
S.I. Gelfand, American Mathematical Society, P. O. Box 6248, Providence,
RI 02940-6248, USA
Yu. I. Manin, Max-Planck-Institut fUr Mathematik, Gottfried-Claren-Str. 26,
D-53225 Bonn, FRG
Homological Algebra
S. 1. Gelfand, Yu. 1. Manin
Contents
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
2 Contents
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
§O. 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. V-Modules 173
§O. Introduction . . . 173
§1. The Weyl Algebra 175
§2. Algebraic V-Modules 182
§3. Inverse Image ... 188
§4. Direct Image 190
§5. Holonomic Modules 195
§6. Regular Connections 202
Contents 3
§7. V-Modules with Regular Singularities . . . . . . . . . . . . 205
§8. Equivalence of Categories (Riemann-Hilbert Correspondence) 208
Bibliographic Hints ....................... 210
References . . 211
Author Index 217
Subject Index 219
4 Introduction
Introduction
1. Homological algebra is rather young. Its subject descends from two areas
of mathematics studied at the end of the previous century; these areas later
became combinatorial topology and "modern algebra" (in the sense of van
der Waerden) respectively. As the examples of main notions inherited from
this early period, we can mention Betti numbers of a topological space and
D. Hilbert's "syzygy theorem" (1890).
At present we easily recognize a general construction which underlies these
notions. A topological space X is glued from cells (or simplices) of various
dimensions i; the boundary of a cell is a linear combination of other cells.
The i-th Betti number is the number of linearly independent chains with zero
boundary modulo chains that are boundaries themselves; in other words, the
i-th Betti number is the rank of the group Ker 8d 1m 8i- ll where 8i : Ci -
Ci- 1 is the boundary operator and Ci is the group of i-dimensional chains.
"Syzygies" occur in a different problem. Let M be a graded module with a
finite number of generators over the ring A = k[Xl, .. " xnl of polynomials
with coefficients in a fixed field k. Hilbert considered the case when M is
an ideal in A generated by several forms (homogeneous polynomials). In
general, generators of M can not be chosen to be independent. Fixing a set
of TO generators we obtain a submodule in ATo consisting of coefficients of
all relations among these generators. This submodule has a natural grading
and is called ''the first syzygy module" Zo(M) of the module M. For i > 1
let Zi(M) = ZO(Zi-l(M)) (on each step we have a freedom in choosing the
generators of Zi-l(M)). The Hilbert theorem asserts that Zn-l(M) is a free
module so that we can always assume Zn(M) = O.
The algebraic framework of both constructions is the notion of a com-
plex; a complex is a sequence of modules and homomorphisms ... - Ki ~
Ki- 1 ----+ ••• with the condition 8i- 18i = O. The complex of chains of a topo
logical space determines its homology Hi(X) = Ker 8d Im8i- 1· The Hilbert
complex consists of free modules. It is acyclic everywhere but at the end:
Zi(M) is both the group of cycles and the group of boundaries in a free
resolution of the module M:
Both the complex of chains of a space X and the resolution of a module M,
are defined non-uniquely: they depend on the decomposition of X into cells
or on the choice of generators of subsequent syzygy modules. The essence
of the first theorems in homological algebra is that there is something that
does not depend on this ambiguity in the choice of a complex, namely the
Betti numbers (or the homology groups themselves) in the first case, and the
maximal length of a complex (the last non-zero place) in the second case.
Introduction 5
The first stage of homological algebra was marked by the acquisition of
data. Combinatorial and, later, homotopic topology supplied plentiful exam
ples of
- types of complexes;
- operations over complexes that reflect some geometrical constructions:
the product of spaces led to the tensor product of complexes, the multiplica
tion in cohomology led to the notion of a differential graded algebra, homo
topy resulted in the algebraic notion of a homotopy between morphisms of
complexes, the algebraic framework of the geometrical study of fiber spaces
is the notion of a spectral sequence associated to a filtered complex, and so
on and so forth;
- algebraic constructions imitating topological ones; examples are coho
mology of groups, of Lie algebras, of associative algebras, etc.
2. The famous "Homological algebra" by H. Cartan and S. Eilenberg, pub
lished in 1956 (and written some time between 1950 and 1953) summarized
the achievements of this first period, and introduced some very important
new ideas which determined the development of this branch of algebra for
many years ahead. It seems that the very name "homological algebra" became
generally accepted only after the publication of this book.
First of all, this book contains a detailed study of the main algebraic
formalism of (co)homology groups and of working instructions that do not
depend on the origin of the complex. Second, this book gave a concepti ally
important answer to the question about the nature of homological invariants
(as opposed to complexes themselves, which cannot be considered as invari
ants). This answer can be formulated as follows. The application of some
basic operations over modules, such as tensor products, the formation of the
module of homomorphisms, etc., to short exact sequences violates the exact
ness; for example, if the sequence 0 ---; M' ---; M ---; Mil ---; 0 is exact, the
sequence 0 ---; N ® M' ---; N ® M ---; N ® Mil ---; 0 can have non-trivial coho
mology at the left term. One can define the "torsion product" Torl(N, Mil)
in such a way that the complex
TorI (N, M') ---; TorI (N, Mil) ---; TorI (N, Mil) ---;
---; N ® M' ---; N ® M ---; N ® Mil ---; 0
is acyclic. However, to extend this complex further to the left one must in
troduce Tor2(N, Mil), etc.
These modules Tori(N, M) are the derived functors (in one of the ar
guments) of the functor ®. They are uniquely determined by the require
ment that the exact triples are mapped to acyclic complexes. To compute
these functors one can use, say, free resolutions of the module M and define
Tori(M, N) as homology groups of the tensor product of such a resolution
with the module N.