Sur la préservation de la cohérence par image inverse extraordinaire par 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 $Z$, $\mathfrak{D}$ 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 ${\mathfrak{X}}^{♯}:=(\mathfrak{X},\mathfrak{D})$, ${\mathcal{Z}}^{♯}:=(\mathcal{Z},u^{-1}\mathfrak{D})$ and $u^{♯}:{\mathcal{Z}}^{♯}\hookrightarrow {\mathfrak{X}}^{♯}$ the exact closed immersion of smooth logarithmic formal schemes over $\V$. Let ${\mathcal{E}}^{(\bullet )}\in {\underrightarrow{LD}}_{\mathbb{Q},\mathrm{coh}}^{\mathrm{b}}({\hat{\mathcal{D}}}_{{\mathfrak{X}}^{♯}}^{(\bullet )}(T))$ and $\mathcal{E}:=\underrightarrow{\lim }({\mathcal{E}}^{(\bullet )})$ the corresponding objet of $D ^{\mathrm{b}}_{\mathrm{coh}}(\smash{\mathcal{D}} ^†_{\mathfrak{X} ^{\sharp}}(\hdag T)_{\mathbb{Q}})$. In this paper, we study sufficient conditions on $\mathcal{E}$ so that if $u ^{\sharp !} (\mathcal{E}) \in D ^{\mathrm{b}}_{\mathrm{coh}}(\smash{\mathcal{D}} ^†_{\mathcal{Z} ^{\sharp}}(\hdag U)_{\mathbb{Q}})$ then $u^{♯(\bullet )!}({\mathcal{E}}^{(\bullet )})\in {\underrightarrow{LD}}_{\mathbb{Q},\mathrm{coh}}^{\mathrm{b}}({\hat{\mathcal{D}}}_{{\mathcal{Z}}^{♯}}^{(\bullet )}(U))$.