arXiv subjects
Publications and source records attributed to Theodore J. Stadnik Jr.
Let $X$ be an $F$-finite smooth scheme of essentially finite type over a perfect field. This article proves the existence of $b$-functions for locally finitely generated unit $F$-modules when equipped with their induced $\mathbb{D}_X$-module structure. It is shown that the $b$-function has rational roots and is determined locally in the étale topology.
For quantum Hamiltonian reductions in arbitrary characteristics, it is known that derived localization holds if and only if the algebra of global sections has finite global dimension. In this paper we provide an alternative characterization of when derived localization holds: Derived localization holds if and only if it holds for an explicit finite set of (quantized) line bundles. As an application, we prove a new result that there are integral weights for which localization holds on in the positive characteristic hypertoric case for $p$ larger than an explicit bound. We also discuss how derived localization is a consequence of a finite number of Morita equivalences.
For a smooth toric variety X over a field of positive characteristic, a T-equivariant étale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to show that the sheaf of differential operators is a trivial Azumaya algebra along the fibers of the moment map. This result is then extended to certain Azumaya algebras on hypertoric varieties, whose global sections are central reductions of the hypertoric enveloping algebra in positive characteristic. A criteria for a derived Beilinson-Bernstein localization theorem is then formulated.