A NOTE ON FROBENIUS DIVIDED MODULES IN MIXED CHARACTERISTICS

摘要

If X is a smooth scheme over a perfect field of characteristic p, and if D_X~(∞) is the sheaf of differential operators onX[7], it is well known that giving an action of D_X~(∞)on an Ox-module ξis equivalent to giving an infinite sequence of Ox-modules descending ξ via the iterates of the Frobenius endomorphism of X [5]. 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 [12], dos Santos used this result to lift D_x~(∞) -modules from characteristic p to characteristic 0 with control of the differential Galois group.