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

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.