arXiv subjects

Raoul Hallopeau

Publications and source records attributed to Raoul Hallopeau.

3 recordsLinked to original sources

arXiv2025

Cycle caracéristique pour les D-modules coadmissibles sur une courbe formelle

Let $\mathfrak{X}$ be a formal smooth quasi-compact curve over a complete discrete valuation ring of mixed characteristic. We consider over $\mathfrak{X}$ the sheaves of differential operators $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ with a congruence level $k \in \mathbb{N}$ and their projective limit $\mathcal{D}_{\mathfrak{X}, \infty} = \varprojlim_k \widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$. In this article, we define a characteristic variety for coadmissible $\mathcal{D}_{\mathfrak{X}, \infty}$-modules as a closed subset of the cotangent space $T^*\mathfrak{X}$. For this purpose, we introduce a microlocalization sheaf of $\mathcal{D}_{\mathfrak{X}, \infty}$ in which the derivation is locally invertible. We deduce a notion of "sub-holonomicity" for coadmissible $\mathcal{D}_{\mathfrak{X}, \infty}$-modules which is equivalent to being generically an integrable connection. Finally, we associate characteristic cycles to sub-holonomic modules proving that the latter are of finite length.

math.AG↗

arXiv2025

Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves

Let $\mathfrak{X}$ be a formal smooth curve over a complete discrete valuation ring of mixed characteristic and let $\mathfrak{X}_K$ be its generic fiber. We consider respectively over $\mathfrak{X}$ and $\mathfrak{X}_K$ the sheaves of differential operators $\mathcal{D}_{\mathfrak{X}, \infty}$ and Dcap with a rapid convergence condition. In this article, we define a characteristic variety as a subset of the cotangent space $T^*\mathfrak{X}_K$ together with a characteristic cycle for coadmissible Dcap-modules. We deduce a notion of ''sub-holonomicity'' for coadmissible Dcap-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic Dcap-modules which are weakly-holonomic. Moreover, we prove that a coadmissible Dcap-modules is sub-holonomic if and only if the corresponding coadmissible $\mathcal{D}_{\mathfrak{X}, \infty}$-module is.

math.AG↗

arXiv2024

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

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.

math.AG↗