Étale splittings of certain Azumaya algebras on toric and hypertoric varieties in positive characteristic

For a smooth toric variety X over a field of positive characteristic, a T-equivariant étale cover Y→T∗X(1) trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to show that D is a trivial Azumaya algebra along the fibers of the moment map μ:T∗X(1)→t∗(1). This result is then extended to certain Azumaya algebras on hypertoric varieties, whose global sections are analogous to central reductions of the hypertoric enveloping algebra. A criterion for a derived Beilinson-Bernstein localization theorem is then formulated.