Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories
Authors: T. J. Hodges and S. P. Smith
Journal: Proc. Amer. Math. Soc. 93 (1985), 379-386
MSC: Primary 17B35; Secondary 14A20, 16A63, 22E46, 57S25
MathSciNet review: 773985
Abstract: Let be an irreducible algebraic variety defined over a field , let be a sheaf of (noncommutative) noetherian -algebras on containing the sheaf of regular functions and let be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of obtained from the local sections of ) there is an equivalence between the category of -modules and the category of sheaves of -modules which are quasicoherent as -modules. This shows that the equivalence of categories established by Beilinson and Bernstein as the first step in their proof of the KazhdanLusztig conjectures (where is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and is a sheaf of twisted differential operators on a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.
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