Comparison between admissible and de Jong coverings in mixed characteristic

Abstract

Let X be an adic space locally of finite type over a complete non-archimedean field k, and denote \({\textbf {Cov}}_{X}^{\textrm{oc}}\) (resp. \({\textbf {Cov}}_{X}^{\textrm{adm}}\)) the category of étale coverings of X that are locally for the Berkovich overconvergent topology (resp. for the admissible topology) disjoint union of finite étale coverings. There is a natural inclusion \({\textbf {Cov}}_{X}^{\textrm{oc}}\subseteq {\textbf {Cov}}_{X}^{\textrm{adm}}\). 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 present note shows that this inclusion can be strict when k is of mixed characteristic (0, p) and p-closed. As a consequence, the natural morphism of Noohi groups \(\pi _1^{\mathrm {dJ, \, adm}}(\mathcal {C}, \overline{x})\rightarrow \pi _1^{\mathrm {dJ, \,oc}}(\mathcal {C},\overline{x}) \) is not an isomorphism in general.