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.
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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