Sur la pr\'eservation de la coh\'erence par image inverse extraordinaire par une immersion ferm\'ee

Abstract

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $u\colon \mathcal{Z} \hookrightarrow \mathfrak{X}$ 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 $\mathfrak{X} ^{\sharp}:= (\mathfrak{X}, \mathfrak{D})$, $\mathcal{Z} ^{\sharp}:= (\mathcal{Z}, u ^{-1}\mathfrak{D})$ and $u ^{\sharp}\colon \mathcal{Z} ^{\sharp} \hookrightarrow \mathfrak{X} ^{\sharp}$ the exact closed immersion of smooth logarithmic formal schemes over $\V$. Let $\mathcal{E} ^{(\bullet)} \in \smash{\underrightarrow{LD}} ^{\mathrm{b}}_{\mathbb{Q}, \mathrm{coh}} (\smash{\hat{\mathcal{D}}}_{\mathfrak{X} ^{\sharp}} ^{(\bullet)} (T))$ and $\mathcal{E} := \underrightarrow{\lim} ~ (\mathcal{E} ^{(\bullet)}) $ the corresponding objet of $D ^{\mathrm{b}}_{\mathrm{coh}}(\smash{\mathcal{D}} ^\dag_{\mathfrak{X} ^{\sharp}}(\hdag T)_{\mathbb{Q}})$. In this paper, we study sufficient conditions on $\mathcal{E}$ so that if $u ^{\sharp !} (\mathcal{E}) \in D ^{\mathrm{b}}_{\mathrm{coh}}(\smash{\mathcal{D}} ^\dag_{\mathcal{Z} ^{\sharp}}(\hdag U)_{\mathbb{Q}})$ then $u ^{\sharp (\bullet) !} (\mathcal{E} ^{(\bullet)}) \in \smash{\underrightarrow{LD}} ^{\mathrm{b}}_{\mathbb{Q}, \mathrm{coh}} (\smash{\hat{\mathcal{D}}}_{\mathcal{Z} ^{\sharp}} ^{(\bullet)} (U))$.