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

TL;DR

This paper extends Garnier's result on the finite length of holonomic modules from level 0 to higher congruence levels on formal curves, with applications to coadmissible modules.

Contribution

It proves that holonomic modules over sheaves of crystalline differential operators at higher levels also have finite length, generalizing previous results.

Findings

Holonomic modules at higher levels have finite length.

Finite length property applies to coadmissible modules with integrable connection.

Extension of Garnier's theorem to congruence level $k$.

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.