Theorem 58.30.3. Let $f : X \to S$ be a smooth proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization. If the characteristic of $\kappa (s)$ is $p$, then the specialization map
is surjective and induces an isomorphism
of the maximal prime-to-p quotients
Proof. This is proved in exactly the same manner as Proposition 58.30.2 with the following differences
Given $X/A$ we no longer show that the functor $\textit{FÉt}_ X \to \textit{FÉt}_{X_{\overline{\eta }}}$ is essentially surjective. We show only that Galois objects whose Galois group has order prime to $p$ are in the essential image. This will be enough to conclude the injectivity of $\pi '_1(X_{\overline{s}'}) \to \pi '_1(X_{\overline{s}})$ by exactly the same argument.
The extensions $\mathcal{O}_{X_ B, \xi _ B} \subset \mathcal{O}_{Z, \xi _ i}$ are tamely ramified as the associated extension of fraction fields is Galois with group of order prime to $p$. See More on Algebra, Lemma 15.112.2.
The extension $\kappa _ B/\kappa _ A$ is no longer necessarily trivial, but it is purely inseparable. Hence the morphism $X_{\kappa _ B} \to X_{\kappa _ A}$ is a universal homeomorphism and induces an isomorphism of fundamental groups by Proposition 58.8.4.
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