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

TL;DR

This paper constructs a sheaf of associative algebras that governs vector field representations on quasi-projective varieties, revealing a local structure akin to differential operators and Lie algebra enveloping algebras.

Contribution

It introduces a universal sheaf of algebras that unify the algebraic structures controlling vector field modules on quasi-projective varieties.

Findings

Local decomposition of the algebra into differential operators and Lie algebra enveloping algebra

Construction of a sheaf controlling AV-modules on quasi-projective varieties

Establishment of a local structure theorem in étale charts

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 \'etale 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.