Revista Matemática Iberoamericana

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Volume 26, Issue 3, 2010, pp. 825–859
DOI: 10.4171/RMI/618

Published online: 2010-12-31

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

Zoghman Mebkhout[1] and Luis Narváez Macarro[2]

(1) Université Paris 7 Denis Diderot, France
(2) Universidad de Sevilla, Spain

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.

Keywords: Affinoid algebra, Dwork-Monsky-Washnitzer algebra, †-scheme, †-adic differential operator

Mebkhout Zoghman, Narváez Macarro Luis: Le Théorème du symbole total d'un opérateur différentiel $p$-adique. Rev. Mat. Iberoam. 26 (2010), 825-859. doi: 10.4171/RMI/618