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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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
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Abstract: Let $ X$ be an irreducible algebraic variety defined over a field $ k$, let $ \mathcal{R}$ be a sheaf of (noncommutative) noetherian $ k$-algebras on $ X$ containing the sheaf of regular functions $ \mathcal{O}$ and let $ R$ be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of $ R$ obtained from the local sections of $ \mathcal{R}$) there is an equivalence between the category of $ R$-modules and the category of sheaves of $ \mathcal{R}$-modules which are quasicoherent as $ \mathcal{O}$-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 $ R$ is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and $ \mathcal{R}$ 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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PII: S 0002-9939(1985)0773985-3
Article copyright: © Copyright 1985 American Mathematical Society

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