Projective generation for equivariant $D$-modules

TL;DR

This paper proves that the category of equivariant D-modules on a smooth affine variety has a finite set of compact projective generators, simplifying the understanding of its structure, with special elementary proof when the group is a torus.

Contribution

It establishes the finiteness of generators for equivariant D-modules and provides a general proof and an elementary proof for tori, advancing the structural understanding of these categories.

Findings

Finite set of generators suffices for the category

General proof via equivariant derived categories

Elementary proof available for torus actions

Abstract

We investigate compact projective generators in the category of equivariant $D$-modules on a smooth affine variety. For a reductive group $G$ acting on a smooth affine variety $X$, there is a natural countable set of compact projective generators indexed by finite dimensional representations of $G$. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof in the general case goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest. We also provide an alternative (more elementary) proof in the case that $G$ is a torus.