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
Comments (1)
Comment #9789 by Haohao Liu on