Microlocalisation d'op{é}rateurs diff{é}rentiels arithm{é}tiques sur un sch{é}ma formel lisse

Abstract: Let $\mathfrak{X}$ be a formal smooth quasi-compact scheme 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 construct microlocalization sheaves of $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ for the different congruence levels $k$ admitting transition morphisms.Then we pass to the projective limit over $k$ in order to obtain some microlocalization sheaves of $\mathcal{D}{\mathfrak{X}, \infty}$.These sheaves will be useful later to define a characteristic variety for coadmissible $\mathcal{D}{\mathfrak{X}, \infty}$-modules when $\mathfrak{X}$ is a formal curve.