{"title":"Comparison between admissible and de Jong coverings in mixed characteristic","authors":"Sylvain Gaulhiac","doi":"10.1007/s00229-024-01578-8","DOIUrl":null,"url":null,"abstract":"<p>Let <i>X</i> be an adic space locally of finite type over a complete non-archimedean field <i>k</i>, and denote <span>\\({\\textbf {Cov}}_{X}^{\\textrm{oc}}\\)</span> (resp. <span>\\({\\textbf {Cov}}_{X}^{\\textrm{adm}}\\)</span>) the category of étale coverings of <i>X</i> that are locally for the Berkovich overconvergent topology (resp. for the admissible topology) disjoint union of finite étale coverings. There is a natural inclusion <span>\\({\\textbf {Cov}}_{X}^{\\textrm{oc}}\\subseteq {\\textbf {Cov}}_{X}^{\\textrm{adm}}\\)</span>. 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 <i>k</i> is of mixed characteristic (0, <i>p</i>) and <i>p</i>-closed. As a consequence, the natural morphism of Noohi groups <span>\\(\\pi _1^{\\mathrm {dJ, \\, adm}}(\\mathcal {C}, \\overline{x})\\rightarrow \\pi _1^{\\mathrm {dJ, \\,oc}}(\\mathcal {C},\\overline{x}) \\)</span> is not an isomorphism in general.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01578-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.