The Beilinson Equivalence for Differential Operators and Lie Algebroids

TL;DR

This paper generalizes the Beilinson equivalence to the setting of differential operators and Lie algebroids, establishing a derived category equivalence with a sheaf of algebras on the base variety.

Contribution

It extends the Beilinson equivalence to differential operators and Lie algebroids, providing a new categorical framework for these algebraic structures.

Findings

Derived category of qgr(D) is equivalent to modules over a sheaf of algebras E on X.

Generalizes Beilinson equivalence from projective space to broader settings.

Connects differential operators and Lie algebroids through categorical equivalences.

Abstract

Let D be the ring of differential operators on a smooth irreducible affine variety X over the complex numbers; or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely-generated graded modules of the Rees algebra D has a natural quotient category qgr(D) which imitates the category of modules on Proj of a graded commutative ring. We show that the derived category D^b(qgr(D)) is equivalent to the derived category of finitely-generated modules of a sheaf of algebras E on X which is coherent over X. This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins to describe the moduli space of left ideals in D.