Variants of the de Jong fundamental group

Abstract

For a rigid space $X$, we answer two questions of de Jong about the category ${\mathbf{Cov}}_X^{\mathrm{adm}}$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite etale coverings: we show that this class is different from the one used by de Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of ${\mathbf{Cov}}_X^{\mathrm{et}}$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.

19 pages. Comments welcome! This is the second part of the paper arXiv:2105.05184v3 which was split off the original article "Geometric arcs and fundamental groups of rigid spaces" (arXiv:2105.05184v2) in the review process and then revised