Purity for overconvergence

TL;DR

This paper proves a purity result for overconvergent $F$-isocrystals, showing that removing certain closed subschemes of codimension at least 2 does not change the category, with applications to $p$-adic fundamental group representations.

Contribution

It establishes an equivalence of categories for overconvergent $F$-isocrystals under removal of codimension 2 subschemes and extends Tsuzuki's results to higher dimensions.

Findings

The restriction functor is an equivalence of categories.

Application to $p$-adic representations of the fundamental group.

Extension of known results to higher-dimensional varieties.

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 $F\text{-}{\rm Isoc}^{\dagger}(X,\overline{X}) \longrightarrow F\text{-}{\rm Isoc}^{\dagger}(X \setminus Z, \overline{X} \setminus \overline{Z})$ 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.