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Algebra V: Homological Algebra PDF

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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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is conccrned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication ofthis publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permis sion for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © 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.

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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 h
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