$\mathcal{A}\mathcal{V}$ modules of finite type on affine space

TL;DR

This paper investigates modules with compatible actions of vector fields and polynomial functions on affine space, proving finitely generated modules are free and establishing an equivalence with modules of differential operators, leading to explicit gauge module constructions.

Contribution

It characterizes finitely generated modules with compatible actions as free and links these modules to differential operator modules, providing explicit realizations.

Findings

Finitely generated modules are free over polynomial functions.

Equivalence between compatible actions and commuting actions of differential operators and vector fields.

Explicit construction of modules as gauge modules.

Abstract

We study the category of modules admitting compatible actions of the Lie algebra $\mathcal{V}$ of vector fields on an affine space and the algebra $\mathcal{A}$ of polynomial functions. We show that modules in this category which are finitely generated over $\mathcal{A}$, are free. We also show that this pair of compatible actions is equivalent to commuting actions of the algebra of differential operators and the Lie algebra of vector fields vanishing at the origin. This allows us to construct explicit realizations of such modules as gauge modules.