Sur le topos infinit\'esimal p-adique d'un sch\'ema lisse I

TL;DR

This paper addresses the intrinsic cohomological lifting problem for smooth schemes from characteristic p to zero, enabling better understanding of p-adic de Rham cohomology and deriving the p-adic factorization of zeta functions.

Contribution

It provides a cohomological solution to lifting problems for smooth schemes, aligning geometric failures with cohomological perspectives, and applies this to factorize zeta functions over finite fields.

Findings

Cohomological lifting of smooth schemes is possible despite geometric failures.

Derived the p-adic factorization of zeta functions for varieties over finite fields.

Established a cohomological framework consistent with Grothendieck motives.

Abstract

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.