Systèmes inductifs surcohérents de D-modules arithmétiques logarithmiques

Abstract

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $\mathcal{P}$ be a smooth, quasi-compact, separated formal scheme over $\mathcal{V}$, $\mathcal{Z}$ be a strict normal crossing divisor of $\mathcal{P}$ and ${\mathcal{P}}^{♯}:=(\mathcal{P},\mathcal{Z})$ the induced smooth formal log-scheme over $\mathcal{V}$. In Berthelot's theory of arithmetic $\mathcal{D}$-modules, we work with the inductive system of sheaves of rings ${\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(\bullet )}:=({\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(m)}{)}_{m\in \mathbb{N}}$, where ${\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(m)}$ is the $p$-adic completion of the ring of differential operators of level $m$ over ${\mathcal{P}}^{♯}$. Moreover, he introduced the sheaf ${\mathcal{D}}_{{\mathcal{P}}^{♯},\mathbb{Q}}^{†}:=\underset{\underset{m}{⟶}}{\lim }{\hat{\mathcal{D}}}_{\mathcal{P}}^{(m)}{\otimes }_{\mathbb{Z}}\mathbb{Q}$ of differential operators over $\mathcal{P}$ of finite level. In this paper, we define the notion of overcoherence for complexes of ${\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(\bullet )}$-modules and check that this notion is compatible to that of overcoherence for complexes of ${\mathcal{D}}_{\mathcal{P},\mathbb{Q}}^{†}$-modules.

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