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Variants of the de Jong fundamental group
Algebraic Geometry · Submitted 22 Mar 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ 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.
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The abstract, sentence by sentence
For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ 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 $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ 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. | result 0.78 | 0.95 | 0.66 | asserted 1.00 |
| s1 | In addition, we prove that the objects of $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze. | result 0.98 | 0.92 | 0.20 | asserted 1.00 |
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The paper
Original abstract
For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ 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 $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.
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How the literature uses it
3 arXiv papers cite this in 7 sentences
Papers citing it · 6 sentences without a confident label
Cites it as support 1 paper
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Claim (4) follows from [ref] ), and claim (3) then follows from claim (4), as each projection $π_0(S_i)→π_0(S_j)$ is a surjection of compact Hausdorff spaces (see 0906), and so $π_0(S_i)→π_0(S_i)$ is surjective by Tychonoff's theorem.
Canonical Integral Models of Shimura Varieties of Abelian Type · Step 3: Global construction relative to a very good Hodge type datum · 0.96
What it cites · 6 sentences without a confident label
Cites as support 1 paper
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By [ref] we know that $[f^-1(U)]$ is homeomorphic to $[f]^-1(_X(U))$.
Geometric arcs and fundamental groups of rigid spaces · Extending finite étale components · 0.96
Cites as background 1 paper
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In [ref] we introduced the notion of a geometric covering of $X$.
Geometric arcs and fundamental groups of rigid spaces · The categories CovX and Cov tau X · 0.89
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