Lifting the Cartier Transform of Ogus-Vologodsky Modulo pn

A publication of the Societe Mathematique de France

Let W be the ring of the Witt vectors of a perfect field of characteristic p, X a smooth formal scheme over W, X’ the base change of X by the Frobenius morphism of W, X'2 the reduction modulo p2 of X’ and X the special fiber of X.

The author lifts the Cartier transform of Ogus-Vologodsky defined by X'2 modulo pn. More precisely, the author constructs a functor from the category of pn-torsion OX’-modules with integrable p-connection to the category of pn-torsion OX-modules with integrable connection, each subject to suitable nilpotence conditions. The author’s construction is based on Oyama’s reformulation of the Cartier transform of Ogus-Vologodsky in characteristic p.

If there exists a lifting F:X X’ of the relative Frobenius morphism of X, the author’s functor is compatible with a functor constructed by Shiho from F. As an application, the author gives a new interpretation of Faltings’ relative Fontaine modules and of the computation of their cohomology.