Ogus-Vologodsky equivalence via stacks

TL;DR

The paper reestablishes the Ogus-Vologodsky equivalence using stacks and the relative de Rham stack, avoiding the need for a lift of the base scheme, and extends it to algebraic stacks with a logarithmic version.

Contribution

It provides a new proof of the Ogus-Vologodsky equivalence via stacks, utilizing lifts of the family to Witt vectors and extending to algebraic stacks.

Findings

Reproves Ogus-Vologodsky equivalence using stacks and de Rham stacks.

Shows a lift of the base scheme is unnecessary, only a lift of the family to Witt vectors.

Extends the equivalence to algebraic stacks and introduces a logarithmic version.

Abstract

Using the relative de Rham stack for a family $X \to S$ in characteristic $p,$ we reprove the (local and global) Ogus-Vologodsky equivalence. Moreover, we observe that a lift of $S$ is not necessary. Instead, we use a lift of $X$ to the second Witt vectors of $S.$ The main ingredient is that, for a quasi-syntomic family $X/S,$ the relative de Rham stack admits a structure of a torsor over $X'$ which is the analogue of the Azumaya property of the algebra of differential operators. This can be applied to families of (reasonable) algebraic stacks, which gives rise to a logarithmic version of the Cartier equivalence. Along the way, we also obtain a decompleted version of the global Cartier equivalence.