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Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base points $\bar{x}$ and $\bar{y}$ of $X_{\bar{\eta}}$ and $X_{s}$ respectively.

It is a Theorem of Grothendieck that the map $\pi_1(X_s,\bar{y}) \rightarrow \pi_1(X,\bar{y})$ induced by the map $X_s \rightarrow X$ is an isomorphism. Furthermore, if we assume in addition that $\phi$ is smooth with geometrically connected fibers, then the so-called specialization map $\pi_1(X_{\bar{\eta}},\bar{x})^{(p')} \rightarrow \pi_1(X_{\bar{s}}, \bar{y})^{(p')}$ is an isomorphism of maximal prime-to-$p$ quotients, where $p$ is the residue characteristic of $A$.

Question: Are either/both result still true if we relax the hypothesis so that $A$ is henselian, rather than complete?

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  • $\begingroup$ First guess: it should be true. One could use Artin approximation to get a bijection between etale covers of the henselian local ring and etale covers of its completion. I will have to think more if this can be turned into a proof. $\endgroup$
    – Matthias Wendt
    Commented Jul 24, 2014 at 17:08
  • $\begingroup$ You should assume $A$ is strictly henselian (but maybe not complete). The categories of finite etale covers of $X_{\widehat{A}}$ and its special fiber are equivalent due to formal GAGA; that is the only reason completeness enters the proof. So you want the categories of finite etale covers of $X$ and of $X_{\widehat{A}}$ to be equivalent via pullback. Via finite etale Hom-schemes among such $E/X$, full faithfulness holds since $E_{\widehat{A}}$ is connected if $E$ is (by formal geometry). Essential surjectivity is Artin approx (as Wendt says); doesn't Artin mention this in his paper? $\endgroup$
    – user27920
    Commented Jul 25, 2014 at 0:53
  • $\begingroup$ Is strictness really necessary? I seem to have found a source which suggests henselianity suffices (see Theorem 8.1 of mi.fu-berlin.de/users/kindler/documents/basechange.pdf). Of course I should have looked for this before asking the question...but it seems believable to me. He works over an algebraically closed field but comments that it's not necessary...I guess that's the only possible hitch? $\endgroup$
    – KristianJS
    Commented Jul 25, 2014 at 10:23
  • $\begingroup$ The entire setup only depends on $A$ through its strict henselization (with respect to $\overline{s}$), and defining the specialization map without conjugation ambiguity rests on going up to the strict henselization, so there is truly no loss of generality for the purpose of your question in considering only strictly henselian $A$. By the way, applying Artin approximation does require using standard limit arguments to reduce to $A$ that is the henselization of a local ring on a finitely generated $\mathbf{Z}$-algebra (and likewise the conclusion then applies without noetherian hypotheses). $\endgroup$
    – user27920
    Commented Jul 25, 2014 at 16:44

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