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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2208.14387 |
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| _version_ | 1866910298575732736 |
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| author | Hallopeau, Raoul |
| author_facet | Hallopeau, Raoul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14387 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle Hallopeau, Raoul Algebraic Geometry 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. |
| title | $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2208.14387 |