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

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.