Le Théorème du symbole total d'un opérateur différentiel $p$-adique

Abstract

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