On the derived ring of differential operators on a singularity

TL;DR

This paper establishes an explicit DG algebra model for the derived category of D-modules on affine varieties, linking it to the ring of differential operators and exploring cases with singularities.

Contribution

It constructs an explicit DG algebra whose modules correspond to D-modules on singular affine varieties, extending known equivalences to singular cases.

Findings

Derived category of D-modules is equivalent to DG modules over an explicit DG algebra.

Computed cohomology algebra for hypersurface, curve, and quotient singularities.

Identified when D-modules are ordinary modules over the ring of differential operators.

Abstract

We show for an affine variety $X$, the derived category of quasi-coherent $D$-modules is equivalent to the category of DG modules over an explicit DG algebra, whose zeroth cohomology is the ring of Grothendieck differential operators $\operatorname{Diff}(X)$. When the variety is cuspidal, we show that this is just the usual ring $\operatorname{Diff}(X)$, and the equivalence is the abelian equivalence constructed by Ben-Zvi and Nevins. We compute the cohomology algebra and its natural modules in the hypersurface, curve and isolated quotient singularity cases. We identify cases where a $D$-module is realised as an ordinary module (in degree 0) over $\operatorname{Diff}(X)$ and where it is not.