Abstract
We give a new proof of the Bernstein-Lunts equivalence of ordinary and equivariant derived categories of Harish-Chandra modules. This proof requires no boundedness assumptions. It uses K-projective resolutions of equivariant complexes, which are shown to exist and constructed explicitly when the group under consideration is reductive. The general case can be obtained from the reductive case by techniques of [MP].
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References
Beilinson, A., Bernstein, J.: A proof of the Jantzen conjecture, (preprint). M.I.T. and Harvard University, 1989
Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque, 100, (1982)
Bernstein, J., Lunts, V.: Equivariant sheaves and functors. Lecture Notes in Math., vol. 1578, Springer-Verlag, 1994
Bernstein, J., Lunts, V.: Localization for derived categories of
-modules. J. Amer. Math. Soc. 8 (4), 819–856 (1995)Cartan, H., Eilenberg S.: Homological algebra. Reprint of the 1956 original, Princeton Landmarks in Mathematics, Princeton University Press, 1999
Deligne, P.: Cohomologie à supports propres. SGA 4, Lecture Notes in Math., vol. 305, Springer-Verlag, Berlin Heidelberg, 1973
Ginzburg, V.A.: Equivariant cohomology and Kähler geometry (Russian). Funktsional. Anal. i Prilozhen. 21 (4), 19–34 (1987)
Gelfand, S.I., Manin, Yu.I.: Methods of homological algebra. Springer-Verlag, Berlin, Heidelberg, New York, 1996
Illusie, L.: Complexe cotangent et déformations I. Lecture Notes in Math., vol. 239, 1971; II, Lecture Notes in Math., vol. 283, 1972, Springer-Verlag, Berlin Heidelberg
Kashiwara, M., Schapira, P.: Sheaves on manifolds. Springer-Verlag, Berlin Heidelberg, 1990
Mac Lane, S.: Categories for the working mathematician. Springer-Verlag, New York, 1971
Miličić, D.: Lectures on derived categories. http://www.math.utah.edu/~milicic/dercat.pdf
Miličić, D., Pandžić, P.: Equivariant derived categories, Zuckerman functors and localization. In: Tirao J, Vogan D, and Wolf JA (eds) Geometry and representation theory of real and p-adic Lie groups, Progress in Mathematics 158, Birkhäuser, Boston, 1996, pp 209–242
Pandžić, P.: Equivariant analogues of Zuckerman functors. Ph.D. thesis, University of Utah, 1995
Pandžić, P.: Zuckerman functors between equivariant derived categories. To appear in Trans. Amer. Math. Soc
Spaltenstein, N.: Resolutions of unbounded complexes. Compositio Math. 65, 121–154 (1988)
Verdier, J.L.: Catégories dérivées, état 0. SGA
, Lecture Notes in Math., vol. 569, Springer-Verlag, 1977Verdier, J.L.: Des catégories dérivées des catégories abéliennes. with a preface by Luc Illusie; edited and with a note by Georges Maltsiniotis. Astérisque 239, (1996)
-modules. J. Amer. Math. Soc. 8 (4), 819–856 (1995)
, Lecture Notes in Math., vol. 569, Springer-Verlag, 1977Author information
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Pandžić, P. A simple proof of Bernstein-Lunts equivalence. manuscripta math. 118, 71–84 (2005). https://doi.org/10.1007/s00229-005-0580-3
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DOI: https://doi.org/10.1007/s00229-005-0580-3