Lemma 58.16.4. Let $f : X \to S$ be a proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization of points of $S$ and let $sp : \pi _1(X_{\overline{s}'}) \to \pi _1(X_{\overline{s}})$ be a specialization map. If $S$ is Noetherian, then there exists a strictly henselian discrete valuation ring $R$ over $S$ such that $sp$ is isomorphic to $sp_ R$ defined above.

**Proof.**
Let $\mathcal{O}_{S, s} \to A$ be the strict henselization constructed using $\kappa (s) \to \kappa (\overline{s})$. Let $A \to \kappa (\overline{s}')$ be the map used to construct $sp$. Let $R \subset \kappa (\overline{s}')$ be a discrete valuation ring dominating the image of $A$, see Algebra, Lemma 10.119.13. Choose a diagram of fields

with $k$ algebraically closed. Let $R^{sh}$ be the strict henselization of $R$ constructed using $R \to k$. Then $R^{sh}$ is a discrete valuation ring by More on Algebra, Lemma 15.45.11. Denote $\eta , o$ the generic and closed point of $\mathop{\mathrm{Spec}}(R^{sh})$. Since the diagram of schemes

commutes, we obtain a commutative diagram

of specialization maps by the construction of these maps. Since the vertical arrows are isomorphisms (Lemma 58.9.3), this proves the lemma. $\square$

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## Comments (1)

Comment #9789 by Haohao Liu on