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Pré-Publication, Document De Travail Année : 2023

Microlocalisation des modules coadmissibles sur une courbe formelle

Résumé

Let $\mathfrak{X}$ be a formal smooth quasi-compact curve over a complete discrete valuation ring $\mathcal{V}$ of mixed characteristic $(0 , p)$. We consider 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 the first part, we introduce a microlocalisation of the sheaf $\mathcal{D}_{\mathfrak{X}, \infty}$. In fact, we construct microlocalisations for the sheaves $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ admitting transition morphisms. Then we pass to the projective limit. In the second part, we define a characteristic variety for coadmissible modules as a closed subset of the cotangent space $T^*\mathfrak{X}$. With such a characteristic variety, one can introduce a notion of holonomic modules : a coadmissible module is holonomic if its characteristic variety has dimension less than or equal to one. We prove that a coadmissible module is holonomic if and only if it a connection module on some open subset of $\mathfrak{X}$. At last, we bridge holonomic modules with weakly holonomic modules.
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Dates et versions

hal-03975658 , version 1 (07-02-2023)
hal-03975658 , version 2 (12-05-2023)
hal-03975658 , version 3 (02-08-2023)
hal-03975658 , version 4 (12-01-2024)
hal-03975658 , version 5 (25-01-2024)

Identifiants

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Raoul Hallopeau. Microlocalisation des modules coadmissibles sur une courbe formelle. 2023. ⟨hal-03975658v3⟩
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