On the derived ring of differential operators on a singularity

6 Oct 2021  ·  Haiping Yang ·

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. read more

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Algebraic Geometry Representation Theory