Lemma 58.16.1. Consider a commutative diagram
\xymatrix{ Y \ar[d]_ g \ar[r] & X \ar[d]^ f \\ T \ar[r] & S }
of schemes where f and g are proper with geometrically connected fibres. Let t' \leadsto t be a specialization of points in T and consider a specialization map sp : \pi _1(Y_{\overline{t}'}) \to \pi _1(Y_{\overline{t}}) as above. Then there is a commutative diagram
\xymatrix{ \pi _1(Y_{\overline{t}'}) \ar[r]_{sp} \ar[d] & \pi _1(Y_{\overline{t}}) \ar[d] \\ \pi _1(X_{\overline{s}'}) \ar[r]^{sp} & \pi _1(X_{\overline{s}}) }
of specialization maps where \overline{s} and \overline{s}' are the images of \overline{t} and \overline{t}'.
Proof.
Let B be the strict henselization of \mathcal{O}_{T, t} with respect to \kappa (t) \subset \kappa (t)^{sep} \subset \kappa (\overline{t}). Pick \psi : \overline{t}' \to \mathop{\mathrm{Spec}}(B) lifting \overline{t}' \to T as in the construction of the specialization map. Let s and s' denote the images of t and t' in S. Let A be the strict henselization of \mathcal{O}_{S, s} with respect to \kappa (s) \subset \kappa (s)^{sep} \subset \kappa (\overline{s}). Since \kappa (\overline{s}) = \kappa (\overline{t}), by the functoriality of strict henselization (Algebra, Lemma 10.155.10) we obtain a ring map A \to B fitting into the commutative diagram
\xymatrix{ \overline{t}' \ar[r]_-\psi \ar[d] & \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & T \ar[d] \\ \overline{s}' \ar[r]^-\varphi & \mathop{\mathrm{Spec}}(A) \ar[r] & S }
Here the morphism \varphi : \overline{s}' \to \mathop{\mathrm{Spec}}(A) is simply taken to be the composition \overline{t}' \to \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A). Applying base change we obtain a commutative diagram
\xymatrix{ Y_{\overline{t}'} \ar[r] \ar[d] & Y_ B \ar[d] \\ X_{\overline{s}'} \ar[r] & X_ A }
and from the construction of the specialization map the commutativity of this diagram implies the commutativity of the diagram of the lemma.
\square
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