Lemma 25.20.4. Let $S$ be a scheme. Let $s' \leadsto s$ be a specialization of points of $S$. Then

there exists a valuation ring $A$ and a morphism $\mathop{\mathrm{Spec}}(A) \to S$ such that the generic point $\eta $ of $\mathop{\mathrm{Spec}}(A)$ maps to $s'$ and the special point maps to $s$, and

given a field extension $\kappa (s') \subset K$ we may arrange it so that the extension $\kappa (s') \subset \kappa (\eta )$ induced by $f$ is isomorphic to the given extension.

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