Purity for overconvergence

Abstract

Let $X\hookrightarrow \overline{X}$ be an open immersion of smooth varieties over a field of characteristic $p>0$ such that the complement is a simple normal crossing divisor and let $\overline{Z}\subseteq Z\subseteq \overline{X}$ be closed subschemes of codimension at least $2$. In this paper, we prove that the canonical restriction functor between the category of overconvergent $F$-isocrystals $F\text{-}{\mathrm{Isoc}}^{†}(X,\overline{X})⟶F\text{-}{\mathrm{Isoc}}^{†}(X\setminus Z,\overline{X}\setminus \overline{Z})$ is an equivalence of categories. We also prove an application to the category of $p$-adic representations of the fundamental group of $X$, which is a higher-dimensional version of a result of Tsuzuki.

22 pages