Let W be the ring of the Witt vectors of a perfect field of characteristic p, ?? a smooth formal scheme over W, ??' the base change of ?? by the Frobenius morphism of W, ??'2 the reduction modulo~p2 of ??' and X the special fiber of ??.
We lift the Cartier transform of Ogus-Vologodsky defined by ??'2 modulo pn. More precisely, we construct a functor from the category of pn-torsion ????'-modules with integrable p-connection to the category of pn-torsion ????-modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of Ogus-Vologodsky in characteristic p.
If there exists a lifting F:?????' of the relative Frobenius morphism of X, our functor is compatible with a functor constructed by Shiho from F. As an application, we give a new interpretation of Faltings' relative Fontaine modules and of the computation of their cohomology.