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Deformations of quasicoherent sheaves of algebras

✍ Scribed by Valery A. Lunts

Book ID: 104140467
Publisher: Elsevier Science
Year: 2003
Tongue: English
Weight: 232 KB
Volume: 259
Category: Article
ISSN: 0021-8693
DOI: 10.1016/s0021-8693(02)00555-0

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✦ Synopsis

Gerstenhaber and Schack [NATO Adv. Sci. Inst. Ser. C, Vol. 247, 1986] developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherent sheaves of algebras on a quasiprojective scheme X in terms of sheaf cohomology on X and X × X. These results are applied to the study of deformations of the sheaf D X of differential operators on X. In particular, in case X is a flag variety we show that any deformation of D X , which is induced by a deformation of O X , must be trivial. This result is used in [Lunts, Rosenberg, manuscript], where we study the localization construction for quantum groups.

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