  In order to have cohomological operations for de Rham p-adic cohomology with
coefficients as manageable as possible, the main purpose of this paper is to
solve intrinsically and from a cohomological point of view the lifting problem
of smooth schemes and their morphisms from characteristic p > 0 to
characteristic zero which has been one of the fundamental difficulties in the
theory of de Rham cohomology of algebraic schemes in positive characteristic
since the beginning. We show that although smooth schemes and morphisms fail to
lift geometrically, it is as if this was the case within the cohomological
point of view, which is consistent with the theory of Grothendieck Motives. We
deduce the p-adic factorization of the Zeta function of a smooth algebraic
variety, possibly open, over a finite field, which is a key testing result of
our methods.
