# The Stacks Project

## Tag: 054F

This tag has label properties-lemma-locally-Noetherian-specialization-dvr and it points to

The corresponding content:

Lemma 24.5.9. 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 $f : \mathop{\rm Spec}(R) \to X$ such that the generic point $\eta$ of $\mathop{\rm Spec}(R)$ maps to $x'$ and the special point maps to $x$, and
2. given a finitely generated field extension $\kappa(x') \subset K$ we may arrange it so that the extension $\kappa(x') \subset \kappa(\eta)$ induced by $f$ is isomorphic to the given one.

Proof. Let $x' \leadsto x$ be a specialization in $X$, and let $\kappa(x') \subset K$ be a finitely generated extension of fields. By Schemes, Lemma 22.13.2 and the discussion following Schemes, Lemma 22.13.3 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 9.114.11. The ring map $\mathcal{O}_{X, x} \to R$ induces the morphism $f : \mathop{\rm Spec}(R) \to X$, see Schemes, Lemma 22.13.1. This morphism has all the desired properties by construction. $\square$

\begin{lemma}
\label{lemma-locally-Noetherian-specialization-dvr}
Let $X$ be a locally Noetherian scheme.
Let $x' \leadsto x$ be a specialization of points of $X$.
Then
\begin{enumerate}
\item there exists a discrete valuation ring $R$ and a morphism
$f : \Spec(R) \to X$ such that the generic point $\eta$ of
$\Spec(R)$ maps to $x'$ and the special point maps to $x$, and
\item given a finitely generated field extension $\kappa(x') \subset K$
we may arrange it so that the extension $\kappa(x') \subset \kappa(\eta)$
induced by $f$ is isomorphic to the given one.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $x' \leadsto x$ be a specialization in $X$, and let
$\kappa(x') \subset K$ be a finitely generated extension of fields. By
Schemes, Lemma \ref{schemes-lemma-specialize-points}
and the discussion following
Schemes, Lemma \ref{schemes-lemma-characterize-points}
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 \ref{algebra-lemma-exists-dvr}.
The ring map $\mathcal{O}_{X, x} \to R$ induces the morphism
$f : \Spec(R) \to X$, see
Schemes, Lemma \ref{schemes-lemma-morphism-from-spec-local-ring}.
This morphism has all the desired properties by construction.
\end{proof}


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