International audienceWe define the pm-curvature map on the sheaf of differential operators of level m on a scheme of positive characteristic p as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of pm-curvature on differential modules of level m. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level m. Then, using a lifting modulo p2 of Frobenius, we are able to define a Frobenius map on differential operators of level m as dual to some divided Frobenius on infinitesimal neighborhhods. We use this map to build a true Azumaya splitting of the completed sheaf of differential operators of level m (up to an automorphism of the center). From this, we derive the fact that Frobenius pull back gives, when restricted to quasi-nilpotent objects, an equivalence between Higgs-modules and differential modules of level m. We end by explaining the relation with related work of Ogus-Vologodski and van der Put in level zero as well as Berthelot's Frobenius descent