Tuesday, January 18, 14:00
Haiping Yang (Imperial College, London), Derived ring of differential
operators on a singularity.
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.