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Linear Koszul duality

Part of: Homological methods Rings and algebras arising under various constructions Abelian categories

Published online by Cambridge University Press:  16 December 2009

Ivan Mirković  and
Simon Riche
Ivan Mirković
Affiliation:
University of Massachusetts, Amherst, MA, USA (email: [email protected])
Simon Riche
Affiliation:
Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu (UMR 7586 du CNRS), Équipe d’Analyse Algébrique, 175, rue du Chevaleret, 75013 Paris, France (email: [email protected])
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Abstract

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In this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.

Keywords

dg-schemessheaves of dg-algebrasderived intersectionKoszul duality

MSC classification

Secondary: 18E30: Derived categories, triangulated categories 16E45: Differential graded algebras and applications 16S37: Quadratic and Koszul algebras
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 233 - 258
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Aubert, A.-M., Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique, Trans. Amer. Math. Soc. 347 (1995), 2179–2189.Google Scholar
[2]Beilinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527.CrossRefGoogle Scholar
[3]Bernstein, J., Gelfand, I. and Gelfand, S., Algebraic vector bundles on ℙn and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 212–214.CrossRefGoogle Scholar
[4]Bernstein, J. and Lunts, V., Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578 (Springer, Berlin, 1994).CrossRefGoogle Scholar
[5]Bezrukavnikov, R., Finkelberg, M. and Mirković, I., Equivariant homology and K-theory of affine Grassmannians and Toda lattices, Compositio Math. 141 (2005), 746–768.CrossRefGoogle Scholar
[6]Borel, A.et al., Algebraic D-modules (Academic Press, New York, 1987).Google Scholar
[7]Ciocan-Fontanine, I. and Kapranov, M., Derived quot schemes, Ann. Sci. École Norm. Sup. (4) 34 (2001), 403–440.CrossRefGoogle Scholar
[8]Deligne, P., Exposé XVII, in Cohomologie à support propre, SGA 4.3, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, vol. 305 (Springer, Berlin, 1973).CrossRefGoogle Scholar
[9]Evens, S. and Mirković, I., Fourier transform and the Iwahori–Matsumoto involution, Duke Math. J. 86 (1997), 435–464.CrossRefGoogle Scholar
[10]Garland, H. and Grojnowski, I., Affine Hecke algebras associated to Kac-Moody groups, Preprint (1995), arXiv: q-alg/9508019.Google Scholar
[11]Godement, R., Topologie algébrique et théorie des faisceaux (Hermann, Paris, 1964).Google Scholar
[12]Goresky, M., Kottwitz, R. and MacPherson, R., Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1993), 25–83.CrossRefGoogle Scholar
[13]Grothendieck, A., EGA II, Étude globale élémentaire de quelques classes de morphismes, avec la collaboration de J. Dieudonné, Publ. Math. Inst. Hautes Études Sci. 8 (1961).Google Scholar
[14]Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).CrossRefGoogle Scholar
[15]Hartshorne, R., Algebraic geometry (Springer, Berlin, 1977).CrossRefGoogle Scholar
[16]Keller, B., Derived categories and their uses, in Handbook of algebra, Vol. 1 ed. Hazewinkel, M. (Elsevier, Amsterdam, 1996).Google Scholar
[17]Lurie, J., Derived algebraic geometry V: strutured spaces, Preprint (2009), arXiv: 0905.0459.Google Scholar
[18]Mirković, I. and Riche, S., Linear Koszul duality and affine Hecke algebras, Preprint (2009), arXiv: 0903.0678.Google Scholar
[19]Riche, S., Koszul duality and modular representations of semi-simple Lie algebras, Preprint (2008), arXiv: 0803.2076, Duke Math. J., to appear.Google Scholar
[20]Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121–154.Google Scholar
[21]Töen, B., Higher and derived stacks: a global overview, in Algebraic geometry, Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 435–487.Google Scholar
[22]Vasserot, E., Induced and simple modules of double affine Hecke algebras, Duke Math. J. 126 (2005), 251–323.CrossRefGoogle Scholar