On the derived ring of differential operators on a singularity

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.