Remark 59.75.3 (Lifting specializations). Let $S$ be a scheme and let $t \leadsto s$ be a specialization of point on $S$. Choose geometric points $\overline{t}$ and $\overline{s}$ lying over $t$ and $s$. Since $t$ corresponds to a point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ by Schemes, Lemma 26.13.2 and since $\mathcal{O}_{S, s} \to \mathcal{O}^{sh}_{S, \overline{s}}$ is faithfully flat, we can find a point $t' \in \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ mapping to $t$. As $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ is a limit of schemes étale over $S$ we see that $\kappa (t')/\kappa (t)$ is a separable algebraic extension (usually not finite of course). Since $\kappa (\overline{t})$ is algebraically closed, we can choose an embedding $\kappa (t') \to \kappa (\overline{t})$ as extensions of $\kappa (t)$. This choice gives us a commutative diagram

$\xymatrix{ \overline{t} \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \ar[d] & \overline{s} \ar[l] \ar[d] \\ t \ar[r] & S & s \ar[l] }$

of points and geometric points. Thus if $t \leadsto s$ we can always “lift” $\overline{t}$ to a geometric point of the strict henselization of $S$ at $\overline{s}$ and get specialization maps as above.

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