Comparison between admissible and de Jong coverings in mixed characteristic

Pub Date : 2024-07-05 DOI:10.1007/s00229-024-01578-8
Sylvain Gaulhiac
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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.

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混合特征中可容许覆盖与德容覆盖的比较
让 X 是一个局部有限类型的、在完全非拱顶域 k 上的 adic 空间,并表示 \({\textbf {Cov}}_{X}^{\textrm{oc}}\) (respect.\({/textbf{Cov}}_{X}^{/textrm{adm}}/))是 X 的 étale 覆盖的范畴,这些覆盖对于伯克维奇超收敛拓扑学(或者对于可容许拓扑学)来说是有限 étale 覆盖的局部不相交的联合。有一个自然包含 \({\textbf {Cov}}_{X}^{\textrm{oc}}}subseteq {\textbf {Cov}}_{X}^{\textrm{adm}}\).这个包含是否严格是德容最初提出的问题。Achinger, Lara 和 Youcis 最近的著作给出了有限或等特征 0 情况下的部分答案。本注释表明,当 k 为混合特征(0,p)且 p 封闭时,这种包含是严格的。因此,Noohi 群的自然变形(\pi _1^{\mathrm {dJ,\, adm}}(\mathcal {C}, \overline{x})\rightarrow \pi _1^{mathrm {dJ, \,oc}}(\mathcal {C},\overline{x}) \)在一般情况下不是同构的。
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