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Let Xbe an irreducible algebraic variety defined over a field k. let iX 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 ifi) 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.

Sheaves of noncommutative algebras and the Beilinson-Bernstein equivalence of categories