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
\[ \xymatrix{ \kappa (\overline{s}) \ar[r] & k \\ A/\mathfrak m_ A \ar[r] \ar[u] & R/\mathfrak m_ R \ar[u] } \]
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
\[ \xymatrix{ \overline{\eta } \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(R^{sh}) \ar[d] & \mathop{\mathrm{Spec}}(k) \ar[d] \ar[l] \\ \overline{s}' \ar[r] & \mathop{\mathrm{Spec}}(A) & \overline{s} \ar[l] } \]
commutes, we obtain a commutative diagram
\[ \xymatrix{ \pi _1(X_{\overline{\eta }}) \ar[d] \ar[r]_{sp_{R^{sh}}} & \pi _1(X_ o) \ar[d] \\ \pi _1(X_{\overline{s}'}) \ar[r]^{sp} & X_{\overline{s}} } \]
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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