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

Rev. Mat. Iberoamericana, Tome 26 (2010) no. 1, p. 825-859 / Harvested from Project Euclid

Résumé

Let ${\mathcal X}^\dagger$ be a smooth $\dagger$-scheme (in the sense of Meredith) over a complete discrete valuation ring $(V, {\mathfrak m})$ of unequal characteristics $(0,p)$ and let ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ be the sheaf of $V$-linear endomorphisms of ${\mathcal O}_{{\mathcal X}^\dagger}$ 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 ${\mathcal O}_{{\mathcal X}^\dagger}$-isomorphism between the sections of ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ and the overconvergent total symbols, and we deduce a cohomological triviality property.

Détail

Publié le : 2010-09-15
Classification:  affinoid algebra,  Dwork-Monsky-Washnitzer algebra,  $\dagger$-scheme,  $\dagger$-adic differential operator,  14F30,  14F10