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
DOI:
https://doi.org/10.1090/S0002-9939-1985-0773985-3
MathSciNet review:
773985
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Abstract | References | Similar Articles | Additional Information
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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1985-0773985-3
Article copyright:
© Copyright 1985
American Mathematical Society