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

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

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.

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