Paper detail

Sur le topos infinitésimal p-adique d'un schéma lisse I

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.