$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle
Abstract
Let $\mathfrak{X}$ be a formal smooth curve locally of finite type 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 connection over curves have finite length.
Domains
Algebraic Geometry [math.AG]
Origin : Files produced by the author(s)