A note on Frobenius divided modules in mixed characteristics

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If X is a smooth scheme over a perfect field of characteristic p, and if ₓ is the sheaf of differential operators on X [EGAIV], it is well known that giving an action of ₓ on an ₓ-module is equivalent to giving an infinite sequence of ₓ-modules descending via the iterates of the Frobenius endomorphism of X. We show that this result can be generalized to any infinitesimal deformation f: X → 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 ₓ-modules from characteristic p to characteristic 0 with control of the differential Galois group.