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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2508.08348 |
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| _version_ | 1866909915221590016 |
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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 of mixed characteristic and let $\mathfrak{X}\_K$ be its generic fiber. We consider respectively over $\mathfrak{X}$ and $\X\_K$ the sheaves of differential operators $\mathcal{D}\_{\mathfrak{X}, \infty}$ and $\wideparen{\D}\_{\mathfrak{X}\_K}$ 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 $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules. We deduce a notion of ''sub-holonomicity'' for coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic $\wideparen{\D}\_{\mathfrak{X}\_K}$ which are weakly-holonomic. Moreover, we prove that a coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules is sub-holonomic if and only if the corresponding coadmissible $\Di$-module is. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves Hallopeau, Raoul Algebraic Geometry 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 $\X\_K$ the sheaves of differential operators $\mathcal{D}\_{\mathfrak{X}, \infty}$ and $\wideparen{\D}\_{\mathfrak{X}\_K}$ 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 $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules. We deduce a notion of ''sub-holonomicity'' for coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic $\wideparen{\D}\_{\mathfrak{X}\_K}$ which are weakly-holonomic. Moreover, we prove that a coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules is sub-holonomic if and only if the corresponding coadmissible $\Di$-module is. |
| title | Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2508.08348 |