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

Abstract : We consider the sheaf of differential operators $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$ with a congruence level $k \in \mathbb{N}$, where $\mathfrak{X}$ is a formal smooth quasi-compact scheme over a complete discrete valuation ring $V$ of mixed characteristic $(0, p)$. We define a category of holonomic $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules using characteristic varieties in the cotangent space $T^*X$ with $X$ the special fiber of $\mathfrak{X}$. In 1995, Laurent Garnier proves in its article [2] that any holonomic $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, 0, \mathbb{Q}}$-module has finite lenght when $\mathfrak{X}$ is a curve. We adapt this result for $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules. We assume that $\mathfrak{X}$ has dimension one and we prove that holonomic $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules are exactly the finite length coherent $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules.
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https://hal-cnrs.archives-ouvertes.fr/hal-03760719
Contributor : Raoul Hallopeau Connect in order to contact the contributor
Submitted on : Friday, August 26, 2022 - 3:46:11 PM
Last modification on : Tuesday, August 30, 2022 - 3:31:56 AM

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### Identifiers

• HAL Id : hal-03760719, version 1
• ARXIV : 2208.14387

### Citation

Raoul Hallopeau. $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe. 2022. ⟨hal-03760719⟩

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