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On semi-infinite cohomology of finite-dimensional graded algebras

Part of: Lie algebras and Lie superalgebras Homological methods Groups and algebras in quantum theory

Published online by Cambridge University Press:  23 February 2010

Roman Bezrukavnikov  and
Leonid Positselski
Roman Bezrukavnikov
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (email: bezrukav@math.mit.edu)
Leonid Positselski
Affiliation:
Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow 127994, Russia (email: posic@mccme.ru)
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Abstract

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We describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

Keywords

semi-infinite cohomologysmall quantum groups

MSC classification

Secondary: 16E30: Homological functors on modules (Tor, Ext, etc.) 17B37: Quantum groups (quantized enveloping algebras) and related deformations 81R50: Quantum groups and related algebraic methods
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 480 - 496
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Andersen, H. H., Jantzen, J. C. and Soergel, W., Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque 220 (1994).Google Scholar
[2]Arinkin, D. and Bezrukavnikov, R., Perverse coherent sheaves, Moscow Math. J. (2010), to appear.Google Scholar
[3]Arkhipov, S., Semiinfinite cohomology of quantum groups, Commun. Math. Phys. 188 (1997), 379–405.Google Scholar
[4]Arkhipov, S., Semi-infinite cohomology of associative algebras and bar duality, Int. Math. Res. Not. 17 (1997), 833–863.Google Scholar
[5]Arkhipov, S., Semi-infinite cohomology of quantum groups. II, in Topics in quantum groups and finite-type invariants, American Mathematical Society Translations, Series 2, vol. 185 (American Mathematical Society, Providence, RI, 1998), 3–42.Google Scholar
[6]Arkhipov, S., A proof of Feigin’s conjecture, Math. Res. Lett. 5 (1998), 403–422.Google Scholar
[7]Arkhipov, S., Bezrukavnikov, R. and Ginzburg, V., Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), 595–678.Google Scholar
[8]Bezrukavnikov, R., Perverse coherent sheaves (after Deligne), Preprint (2000), arXiv:math.AG/0005152.Google Scholar
[9]Bezrukavnikov, R., On semi-infinite cohomology of finite dimensional algebras, Preprint (2000), arXiv:math.RT/0005148.Google Scholar
[10]Bezrukavnikov, R., Cohomology of tilting modules over quantum groups, and t-structures on derived categories of coherent sheaves, Invent. Math. 166 (2006), 327–357.Google Scholar
[11]Bezrukavnikov, R., Finkelberg, M. and Schechtman, V., Factorizable sheaves and quantum groups, Lecture Notes in Mathematics, vol. 1691 (Springer, Berlin, 1998).Google Scholar
[12]Bezrukavnikov, R. and Lachowska, A., The small quantum group and the Springer resolution, in Quantum groups (Proceedings of the Haifa conference, 2004, in memory of J. Donin), Contemporary Mathematics, vol. 433 (American Mathematical Society, Providence, RI, 2007), 89–101.Google Scholar
[13]Brzezinski, T., The structure of corings: induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebr. Represent. Theory 5 (2002), 389–410.Google Scholar
[14]Brzezinski, T. and Wisbauer, R., Corings and comodules, London Mathematical Society Lecture Note Series, vol. 309 (Cambridge University Press, Cambridge, 2003).Google Scholar
[15]Ginzburg, V. and Kumar, S., Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), 179–198.Google Scholar
[16]Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966).Google Scholar
[17]Lusztig, G., Finite-dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257–296.Google Scholar
[18]Positselski, L., Homological algebra of semimodules and semicontramodules, in Semi-infinite homological algebra of associative algebraic structures (with appendices coauthored by S. Arkhipov and D. Rumynin), Preprint (2007), arXiv:0708.3398, to appear in Birkhäuser’s series Monografie Matematyczne, vol. 70 (2010).Google Scholar
[19]Sevostyanov, A., Semi-infinite cohomology and Hecke algebras, Adv. Math. 159 (2001), 83–141.Google Scholar
[20]Voronov, A., Semi-infinite homological algebra, Invent. Math. 113 (1993), 103–146.Google Scholar