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 CovXoc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {Cov}}_{X}^{\ extrm{oc}}$$\\end{document} (resp. CovXadm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {Cov}}_{X}^{\ extrm{adm}}$$\\end{document}) 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 CovXoc⊆CovXadm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {Cov}}_{X}^{\ extrm{oc}}\\subseteq {\ extbf {Cov}}_{X}^{\ extrm{adm}}$$\\end{document}. 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 π1dJ,adm(C,x¯)→π1dJ,oc(C,x¯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _1^{\\mathrm {dJ, \\, adm}}(\\mathcal {C}, \\overline{x})\\rightarrow \\pi _1^{\\mathrm {dJ, \\,oc}}(\\mathcal {C},\\overline{x}) $$\\end{document} is not an isomorphism in general.