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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

\[ \pi^{\mathrm{dJ}}_1(\mathfrak{X}_\eta) \to \pi^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}_1(\mathfrak{X}_k) \]

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

$\mathfrak {X}$ 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.

We investigate the action of the Weil group on the compactly supported $\ell$-adic étale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of $\ell$ when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on $\ell$-independence for nearby cycle cohomology, which leads to our $\ell$-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of Huber, which requires the restriction on the characteristic of the base field.

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Specialization for the pro-étale fundamental group

On $\ell$-independence for the étale cohomology of rigid spaces over local fields

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Specialization for the pro-étale fundamental group

On $\ell$-independence for the étale cohomology of rigid spaces over local fields