SUR LA PRÉSERVATION DE LA COHÉRENCE PAR IMAGE INVERSE EXTRAORDINAIRE D’UNE IMMERSION FERMÉE

Abstract

Let ${\mathcal{V}}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $u:{\mathcal{Z}}{\hookrightarrow}\mathfrak{X}$ be a closed immersion of smooth, quasi-compact, separated formal schemes over ${\mathcal{V}}$, $T$ be a divisor of $X$ such that $U:=T\cap Z$ is a divisor of $Z$, and $\mathfrak{D}$ a strict normal crossing divisor of $\mathfrak{X}$ such that $u^{-1}(\mathfrak{D})$ is a strict normal crossing divisor of ${\mathcal{Z}}$. We pose $\mathfrak{X}^{\sharp }:=(\mathfrak{X},\mathfrak{D})$, ${\mathcal{Z}}^{\sharp }:=({\mathcal{Z}},u^{-1}\mathfrak{D})$ and $u^{\sharp }:{\mathcal{Z}}^{\sharp }{\hookrightarrow}\mathfrak{X}^{\sharp }$ the exact closed immersion of smooth logarithmic formal schemes over ${\mathcal{V}}$. In Berthelot’s theory of arithmetic ${\mathcal{D}}$-modules, we work with the inductive system of sheaves of rings $\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T):=(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T))_{m\in \mathbb{N}}$, where $\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}$ is the $p$-adic completion of the ring of differential operators of level $m$ over $\mathfrak{X}^{\sharp }$ and where $T$ means that we add overconvergent singularities along the divisor $T$. Moreover, Berthelot introduced the sheaf ${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}:=\underset{\underset{m}{\longrightarrow }}{\lim }\,\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T)\otimes _{\mathbb{Z}}\mathbb{Q}$ of differential operators over $\mathfrak{X}^{\sharp }$ of finite level with overconvergent singularities along $T$. Let ${\mathcal{E}}^{(\bullet )}\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T))$ and ${\mathcal{E}}:=\varinjlim ~({\mathcal{E}}^{(\bullet )})$ be the corresponding object of $D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}})$. In this paper, we study sufficient conditions on ${\mathcal{E}}$ so that if $u^{\sharp !}({\mathcal{E}})\in D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{{\mathcal{Z}}^{\sharp }}^{\dagger }(\text{}^{\dagger }U)_{\mathbb{Q}})$ then $u^{\sharp (\bullet )!}({\mathcal{E}}^{(\bullet )})\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{{\mathcal{Z}}^{\sharp }}^{(\bullet )}(U))$. For instance, we check that this is the case when ${\mathcal{E}}$ is a coherent ${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}$-module such that the cohomological spaces of $u^{\sharp !}({\mathcal{E}})$ are isocrystals on ${\mathcal{Z}}^{\sharp }$ overconvergent along $U$.