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Let $R$ be a henselian dvr, $s,\eta\in\text{Spec}(R)$ the closed and generic points, and $f : X\to \text{Spec}(R)$ a proper smooth scheme.

For a prime $\ell$ invertible on $R$, is there a specialization map

$$sp^i_{\eta,s} : R^if_*(\mu_{\ell^n})_{s}\to R^if_*(\mu_{\ell^n})_{\eta}$$

(with $\eta$ and $s$, not the geometric points over them)?

Is it an isomorphism, injective, surjective?

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    $\begingroup$ ... it means forming the (filtered) colimit of evaluation of your étale sheaf at all those $U\to X$ étale through which $s\to X$ factors. This makes perfectly good sense if $s$ is non geometric, with the difference that the stalk at $s$ cannot be evaluated as evaluation at $\text{Spec}(\mathcal{O}_{S,s})$. $\endgroup$ – user95222 Feb 17 '18 at 9:32
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    $\begingroup$ Your question contains as a subquestion “does étale cohomology of varieties defined over a non-separably closed field exist?” Of course it does. Only, typically the finiteness and base change theorems for geometric étale cohomology break down, hence my question. $\endgroup$ – user95222 Feb 17 '18 at 9:35
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    $\begingroup$ @Ben Lim on the étale site, stalks at geometric points are enough to detect monomorphisms and epimorphisms (evaluation at geometric points gives a conservative family of fiber functors). Nothing forbids you to evaluate stalks at non-geometric points. $\endgroup$ – user92332 Feb 17 '18 at 9:39
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If $\ell$ is invertible in $R$, then $R^i f_* (\mu_{\ell^n}) $ is a locally constant sheaf on $R$ by smooth and proper base change. Hence it is a represenation of the fundamental group of $R$, which is equal to the Galois group of the residue field, because $R$ is Henselian.

This fundamental group is easily seen to be a quotient of the Galois groups of both $s$ and $\eta$, and hence the "stalks" at $s$ and $\eta$, which are the Galois-invariants of the geometric stalks, are both equal to the $\pi_1$-invariants, hence naturally isomorphic.

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    $\begingroup$ For a general sheaf, the natural specialization map actually goes the other way. Given a section at $s$, you obtain a section over an etale neighborhood of $s$, which wlog is $R$ because $R$ is henselian, and hence a section at $\eta$. $\endgroup$ – Will Sawin Feb 17 '18 at 12:10
  • $\begingroup$ Can one relax the henselianity assumption on $R$ in any way? Probably not $\endgroup$ – user95222 Feb 18 '18 at 2:27
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    $\begingroup$ @AG2073951378 For lisse sheaves on non-henselian schemes the specialization map does go from $\eta$ to $s$. More precisely, there is a map from global sections to the stalk at $\eta$ and a map from global sections to the stalk at $s$. The first map is an isomorphism in the lisse case (as long as the scheme is normal, otherwise just an injection). The second map is an isomorphism in the Henselian case. $\endgroup$ – Will Sawin Feb 18 '18 at 6:40

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