The Stacks project

Lemma 58.30.1. Let $f : X \to S$ be a flat proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization. If $X_ s$ is geometrically reduced, then the specialization map $sp : \pi _1(X_{\overline{s}'}) \to \pi _1(X_{\overline{s}})$ is surjective.

Proof. Since $X_ s$ is geometrically reduced, we may assume all fibres are geometrically reduced after possibly shrinking $S$, see More on Morphisms, Lemma 37.26.7. Let $\mathcal{O}_{S, s} \to A \to \kappa (\overline{s}')$ be as in the construction of the specialization map, see Section 58.16. Thus it suffices to show that

\[ \pi _1(X_{\overline{s}'}) \to \pi _1(X_ A) \]

is surjective. This follows from Proposition 58.15.2 and $\pi _1(\mathop{\mathrm{Spec}}(A)) = \{ 1\} $. $\square$

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