A characterization of the overcoherence

Abstract

Let $\mathcal{P}$ be a proper smooth formal $\mathcal{V}$-scheme, $X$ a closed subscheme of the special fiber of $\mathcal{P}$, $\mathcal{E} \in F\text{-}D ^\mathrm{b}_\mathrm{coh} (\D ^\dag_{\mathcal{P},\mathbb{Q}})$ with support in $X$. We check that $\mathcal{E}$ is $\D ^\dag _{\mathcal{P},\mathbb{Q}}$-overcoherent if and only if, for any morphism $f : \mathcal{P}' \to \mathcal{P}$ of smooth formal $\mathcal{V}$-schemes, $f ^! (\mathcal{E}) $ is $\D ^\dag_{\mathcal{P}', \mathbb{Q}}$-coherent.