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

1. 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

2. 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.

Proof. Let $s' \leadsto s$ be a specialization in $S$, and let $\kappa (s') \subset K$ be an extension of fields. By Lemma 25.13.2 and the discussion following Lemma 25.13.3 this leads to ring maps $\mathcal{O}_{S, s} \to \kappa (s') \to K$. Let $A \subset K$ be any valuation ring whose field of fractions is $K$ and which dominates the image of $\mathcal{O}_{S, s} \to K$, see Algebra, Lemma 10.49.2. The ring map $\mathcal{O}_{S, s} \to A$ induces the morphism $f : \mathop{\mathrm{Spec}}(A) \to S$, see Lemma 25.13.1. This morphism has all the desired properties by construction. $\square$

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