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On the derived ring of differential operators on a singularity (2110.03100v2)
Published 6 Oct 2021 in math.AG and math.RT
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.