The Stacks project

Lemma 32.15.1. Let $f : X \to Y$ be a morphism of schemes. Assume $f$ finite type and $Y$ locally Noetherian. Let $y \in Y$ be a point in the closure of the image of $f$. Then there exists a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y } \]

where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\mathop{\mathrm{Spec}}(A)$ to $y$. Moreover, we can assume that the image point of $\mathop{\mathrm{Spec}}(K) \to X$ is a generic point $\eta $ of an irreducible component of $X$ and that $K = \kappa (\eta )$.

Proof. By the non-Noetherian version of this lemma (Morphisms, Lemma 29.6.5) there exists a point $x \in X$ such that $f(x)$ specializes to $y$. We may replace $x$ by any point specializing to $x$, hence we may assume that $x$ is a generic point of an irreducible component of $X$. This produces a ring map $\mathcal{O}_{Y, y} \to \kappa (x)$ (see Schemes, Section 26.13). Let $R \subset \kappa (x)$ be the image. Then $R$ is Noetherian as a quotient of the Noetherian local ring $\mathcal{O}_{Y, y}$. On the other hand, the extension $\kappa (x)$ is a finitely generated extension of the fraction field of $R$ as $f$ is of finite type. Thus there exists a discrete valuation ring $A \subset \kappa (x)$ with fraction field $\kappa (x)$ dominating $R$ by Algebra, Lemma 10.119.13. Then

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\kappa (x)) \ar[d] \ar[rrr] & & & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & \mathop{\mathrm{Spec}}(R) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \ar[r] & Y } \]

gives the desired diagram. $\square$

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