Comparison between admissible and de Jong coverings of rigid analytic spaces in mixed characteristic

TL;DR

This paper demonstrates that in mixed characteristic settings, the inclusion of certain étale covering categories can be strict, leading to non-isomorphic fundamental groups, which contrasts with previous results in equal characteristic zero.

Contribution

It shows that the inclusion of overconvergent coverings into admissible coverings can be strict in mixed characteristic, providing new insights into the structure of étale coverings in this setting.

Findings

The inclusion of categories of coverings can be strict in mixed characteristic.

The natural morphism of Noohi groups is not always an isomorphism.

This contrasts with known results in equal characteristic zero.

Abstract

If $k$ is a complete non-archimedean field and $X$ an adic space locally of finite type over $\mathrm{Spa}(k)$, let $\textbf{Cov}_{X}^{\mathrm{oc}}$ (resp. $\textbf{Cov}_{X}^{\mathrm{adm}}$) be the category of \'etale coverings of $X$ that are locally for the Berkovich overconvergent topology (resp. for the admissible topology) disjoint union of finite \'etale coverings. There is a natural inclusion $\textbf{Cov}_{X}^{\mathrm{oc}}\subseteq \textbf{Cov}_{X}^{\mathrm{adm}}$. Whether or not this inclusion is strict is a question initially asked by de Jong. Some partial answers have been given in the recents works of Achinger, Lara and Youcis in the finite or equal characteristic $0$ cases. The purpose of this note is to show that this inclusion can be strict when $k$ is of mixed characteristic $(0,p)$ and $p$-closed. As a consequence, following the work of Achinger, Lara and Youcis, the natural morphism of Noohi groups $\pi_1^{\mathrm{dJ, \, adm}}(X)\to \pi_1^{\mathrm{dJ, \,oc}}(X)$ is not an isomorphism in general.