A note on Frobenius divided modules in mixed characteristics

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\dix{X}$ is the sheaf of differential operators on $X$, it is well known that giving an action of $\dix{X}$ 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 $\dix{X}$-modules from characteristic $p$ to characteristic $0$ with control of the differential Galois group.