Equivariant Sheaves and Functors
The equivariant derived category of sheaves is introduced. All usual functors on sheaves are extended to the equivariant situation. Some applications to the equivariant intersection cohomology are given. The theory may be useful to specialists in represen
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1578
Joseph Bernstein Valery Lunts
Equivariant Sheaves and Functors
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Joseph Bernstein Department of Mathematics Harvard University Cambridge, MA 02138, USA Valery Lunts Department of Mathematics Indiana University Bloomington, IN 47405, USA
Mathematics Subject Classification (1991): 57E99, 18E30
ISBN 3-540-58071-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58071-9 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany
SPIN: 10130027
46/3140-543210 - Printed on acid-free paper
Contents Introduction
1
Part I. Derived category Da(X) and functors. . . . . .
2
1. Review of sheaves and functors
5
O. Some preliminaries Appendix A
2. Equivariant derived categories Appendix B. A simplicial description of the category Da(X) 3. Functors
13 16
32 34
4. Variants
40
5. Equivariant perverse sheaves
41
6. General inverse and direct image functors Q*, Q* 7. Some relations between functors
49
Appendix C 8. Discrete groups and functors 9. Almost free algebraic actions
43
56
57 66
Part II. DG-modules and equivariant cohomology. 10. DG-modules .
68
11. Categories D::t 12. DG-modules and sheaves on topological spaces
83
13. Equivariant cohomology 14. Fundamental example .
93 115 121
Part III. Equivariant cohomology of toric varieties. 15. Toric varieties
126
Bibliography
133
Index . . .
135
Introduction. Let f : X ---+ Y be a continuous map of locally compact spaces. Let Sh(X), Sh(Y) denote the abelian categories of sheaves on X and Y, and D(X), D(Y) denote the corresponding derived categories (maybe bounded D = Db or bounded below D = D+ if necessary). It is well known that there exist functors
1*, f., t', f!,
D, Hom, 0
between the categories D(X) and D(Y), which satisfy certain identities. Now assume that X, Yare in addition G-spaces for a topological group G, and that f is a G-map. Instead of sheaves let us consider the equivariant sheaves Sha(X), Sha(Y). One wants to have triangulated categories Da(X), Da(Y) "derived categories of equivariant sheaves" - together with all the above functors. More precisely, there should exist the forgetful functor
For: D a
---+
D,
so that the functors in categories D a are compatible with the usual ones in categories D under this forgetful functor. Simple examples show that the derived categ
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