Haiping Yang (Imperial College, London), Derived ring of differential operators on a singularity.

Abstract:

Using the theory of compact generators, 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 Diff(X). When the variety is cuspidal, we show that this is just the usual ring 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 Diff(X) and where it is not. ArXiv: 2110.03100.