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Specialization for the pro-étale fundamental group
- Part of
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- Journal:
- Compositio Mathematica / Volume 158 / Issue 8 / August 2022
- Published online by Cambridge University Press:
- 27 September 2022, pp. 1713-1745
- Print publication:
- August 2022
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- Article
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For a formal scheme
$\mathfrak {X}$ of finite type over a complete rank-one valuation ring, we construct a specialization morphism
from the de Jong fundamental group of the rigid generic fiber to the Bhatt–Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of
\[ \pi^{\mathrm{dJ}}_1(\mathfrak{X}_\eta) \to \pi^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}_1(\mathfrak{X}_k) \]
$\mathfrak {X}$ and normalizations of the irreducible components of 
$\mathfrak {X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent.
