Le théorème du symbole total d’un opérateur différentiel p-adique

Abstract

Let X † be a smooth †-scheme (in the sense of Meredith) over a complete discrete valuation ring (V,m) of unequal characteristics (0, p) and let D†X†/V be the sheaf of V -linear endomorphisms of OX† whose reduction modulo ms is a linear differential operator of order bounded by an affine function in s. In this paper we prove that locally there is an OX† -isomorphism between the sections of D†X†/V and the overconvergent total symbols, and we deduce a cohomological triviality property.