Test modules, singularities and nearby cycles
Final Report Abstract
During my research stay in Ann Arbor I successfully completed two projects. I was able to show, using techniques of Cartier crystals, to show that the intermediate extension functor of perverse constructible p-torsion sheaves commutes with smoothbase change. There is an equivalence of three categories involved here. Namely, that of Cartier crystals, unit-R[F] modules and perverse constructible sheaves. I showed that the equivalence of Cartier crystals and unit-R[F] modules is compatible with smooth morphisms. This allowed me to solve the problem concerning intermediate extensions in the category of Cartier crystals where more tools are available. Another project that I successfully completed was concerned with Bernstein-Sato polynomials and the associated graded of the test module filtration. I showed that one may attach a Bernstein-Sato polynomial to a larger class of Cartier modules than previously known. In constructing these Bernstein-Sato polynomials one considers several Cartier structures on certain D-module quotients. We show that these Cartier structures correspond to Cartier structures naturally obtained on the associated graded of the test module filtration. I also studied a generalization of so-called non-F-pure ideals to Cartier modules and was able to show a weaker connection between these and Bernstein-Sato polynomials in complete generality. I also showed that certain pathologies of the corresponding Q-indexed filtration delta t is limited to cases where the denominator of t is divisible by p.
