The Stacks project

Lemma 28.5.10. Let $X$ be a locally Noetherian scheme. Let $x' \leadsto x$ be a specialization of points of $X$. Then

  1. there exists a discrete valuation ring $R$ and a morphism $f : \mathop{\mathrm{Spec}}(R) \to X$ such that the generic point $\eta $ of $\mathop{\mathrm{Spec}}(R)$ maps to $x'$ and the special point maps to $x$, and

  2. given a finitely generated field extension $K/\kappa (x')$ we may arrange it so that the extension $\kappa (\eta )/\kappa (x')$ induced by $f$ is isomorphic to the given one.

Proof. Let $x' \leadsto x$ be a specialization in $X$, and let $K/\kappa (x')$ be a finitely generated extension of fields. By Schemes, Lemma 26.13.2 and the discussion following Schemes, Lemma 26.13.3 this leads to ring maps $\mathcal{O}_{X, x} \to \kappa (x') \to K$. Let $R \subset K$ be any discrete valuation ring whose field of fractions is $K$ and which dominates the image of $\mathcal{O}_{X, x} \to K$, see Algebra, Lemma 10.119.13. The ring map $\mathcal{O}_{X, x} \to R$ induces the morphism $f : \mathop{\mathrm{Spec}}(R) \to X$, see Schemes, Lemma 26.13.1. This morphism has all the desired properties by construction. $\square$

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