$\mathcal{D}_{\mathfrak{X}, \infty$-modules coadmissibles holonomes 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 this article, 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 $\mathcal{D}_{\mathfrak{X}, \infty}$-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 is a connection module on some open subset of $\mathfrak{X}$.
At last, we bridge holonomic modules with weakly holonomic modules.
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