$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle - CNRS - Centre national de la recherche scientifique Access content directly
Preprints, Working Papers, ... Year : 2023

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


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.
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Dates and versions

hal-03760719 , version 1 (26-08-2022)
hal-03760719 , version 2 (15-05-2023)



Raoul Hallopeau. $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle. 2023. ⟨hal-03760719v2⟩
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