Sheaves of Noncommutative Algebras and the Beilinson-bernstein Equivalence of Categories

Let Xbe an irreducible algebraic variety defined over a field k. let ¡X be a sheaf of (noncommutative) noetherian A-algebras on X containing the sheaf of regular functions 0 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 ifî) there is an equivalence between the category of ft-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 R is a primitive factor ring of the enveloping algebra of a complex semisimple Lie algebra, and ¿ft is a sheaf of twisted differential operators on • a generalised flag variety) is valid for more fundamental reasons than is apparent from their work.