A note on Frobenius divided modules in mixed characteristics

TL;DR

This paper generalizes the equivalence between differential operator modules and Frobenius descent from smooth schemes over perfect fields to more general infinitesimal deformations and formal schemes in characteristic p.

Contribution

It extends the classical Frobenius descent equivalence to infinitesimal deformations with Frobenius liftings and to adic formal schemes, broadening the scope of Frobenius module theory.

Findings

Generalization of Frobenius descent to infinitesimal deformations

Extension to adic formal schemes with p in an ideal of definition

Application to lifting D-modules from characteristic p to zero

Abstract

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\sD_X$ is the sheaf of differential operators on $X$ [EGAIV], it is well known that giving an action of $\sD_X$ on an $\sO_X$-module $\sE$ is equivalent to giving an infinite sequence of $\sO_X$-modules descending $\sE$ via the iterates of the Frobenius endomorphism of $X$. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that $p$ belongs to an ideal of definition. In a recent preprint, dos Santos used this result to lift $\sD_X$-modules from characteristic $p$ to characteristic 0 with control of the differential Galois group.