Le théorème du symbole total d’un opérateur différentiel $p$-adique d’échelon $h$ ≥ 0

Résumé 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.