$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle

Abstract

Let $\mathfrak{X}$ be a formal smooth curve over a complete discrete valuation ring $\mathcal{V}$ of mixed characteristic $(0 , p)$. Let $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, \mathbb{Q}}$ be the sheaf of crystalline differential operators of level 0 (i.e., generated by the derivations). In this situation, Garnier proved that holonomic $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, \mathbb{Q}}$-modules as defined by Berthelot have finite length. In this article, we address this question for the sheaves $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ of congruence level $k$ defined by Christine Huyghe, Tobias Schmidt and Matthias Strauch. Using the same strategy as Garnier, we prove that holonomic $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$-modules have finite length. We finally give an application to coadmissible modules by proving that coadmissible modules with integrable connection over curves have finite length.