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 $U$.

中文翻译：

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SUR LA PRÉSERVATION DE LA COHÉRENCE PAR 图像 INVERSE EXTRAORDINAIRE D'UNE IMMERSION FERMÉE

让${\mathcal{V}}$是一个完全离散的不等特征的估值环，具有完美的剩余场，$u:{\mathcal{Z}}{\hookrightarrow}\mathfrak{X}$是平滑的、准紧凑的、分离的形式方案的封闭沉浸${\mathcal{V}}$,$T$成为除数$X$这样$U:=T\cap Z$是一个除数$Z$， 和$\mathfrak{D}$的严格正态交叉除数$\mathfrak{X}$这样$u^{-1}(\mathfrak{D})$是一个严格的正态交叉除数${\mathcal{Z}}$. 我们摆姿势$\mathfrak{X}^{\sharp }:=(\mathfrak{X},\mathfrak{D})$,${\mathcal{Z}}^{\sharp }:=({\mathcal{Z}},u^{-1}\mathfrak{D})$和$u^{\sharp }:{\mathcal{Z}}^{\sharp }{\hookrightarrow}\mathfrak{X}^{\sharp }$平滑对数形式方案的精确封闭浸入${\mathcal{V}}$. 在贝特洛的算术理论中${\mathcal{D}}$-模块，我们使用环轮的感应系统$\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T):=(\widehat{{\mathcal{D}}}_ {\mathfrak{X}^{\sharp }}^{(m)}(T))_{m\in \mathbb{N}}$， 在哪里$\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}$是个$p$-水平微分算子环的进阶完成$m$超过$\mathfrak{X}^{\sharp }$和在哪里$T$意味着我们沿除数添加过收敛奇点$T$. 此外，Berthelot 还介绍了捆${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}:=\underset {\underset{m}{\longrightarrow }}{\lim }\,\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T) \otimes _{\mathbb{Z}}\mathbb{Q}$微分算子$\mathfrak{X}^{\sharp }$具有过收敛奇点的有限水平$T$. 让${\mathcal{E}}^{(\bullet )}\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text {b}}(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T))$和${\mathcal{E}}:=\varinjlim ~({\mathcal{E}}^{(\bullet )})$成为对应的对象$D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{ \dagger }T)_{\mathbb{Q}})$. 在本文中，我们研究了${\mathcal{E}}$所以如果$u^{\sharp !}({\mathcal{E}})\in D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{{\mathcal{Z }}^{\sharp }}^{\dagger }(\text{}^{\dagger }U)_{\mathbb{Q}})$然后$u^{\sharp (\bullet )!}({\mathcal{E}}^{(\bullet )})\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{ Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{{\mathcal{Z}}^{\sharp }}^{(\bullet ) }(U))$. 例如，我们检查是否是这种情况${\mathcal{E}}$是一个连贯的${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}$-模使得上同调空间$u^{\sharp !}({\mathcal{E}})$是等晶在${\mathcal{Z}}^{\sharp }$过度收敛美元.