A universal sheaf of algebras governing representations of vector fields on quasi-projective varieties

Abstract

We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in étale charts these associative algebras decompose into a tensor product of the algebra of differential operators and the universal enveloping algebra of the Lie algebra of power series vector fields vanishing at the origin.

In this revised version, we fixed a technical error related to completed tensor products