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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2302.03959 |
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| _version_ | 1866915600453861376 |
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| author | Hallopeau, Raoul |
| author_facet | Hallopeau, Raoul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03959 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cycle caracéristique pour les D-modules coadmissibles sur une courbe formelle Hallopeau, Raoul Algebraic Geometry 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. |
| title | Cycle caracéristique pour les D-modules coadmissibles sur une courbe formelle |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2302.03959 |