  We define the $p^m$-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
$p^m$-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 $p^2$ 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.
