The Stacks project

Lemma 70.21.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. 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 point $x \in |X|$ corresponding to $\mathop{\mathrm{Spec}}(K) \to X$ is a codimension $0$ point1 and that $K$ is the residue field of a point on a scheme étale over $X$.

Proof. Choose an affine scheme $V$, a point $v \in V$ and an étale morphism $V \to Y$ mapping $v$ to $y$. The map $|V| \to |Y|$ is open and by Properties of Spaces, Lemma 66.4.3 the image of $|X \times _ Y V| \to |V|$ is the inverse image of the image of $|f|$. We conclude that the point $v$ is in the closure of the image of $|X \times _ Y V| \to |V|$. If we prove the lemma for $X \times _ Y V \to V$ and the point $v$, then the lemma follows for $f$ and $y$. In this way we reduce to the situation described in the next paragraph.

Assume we have $f : X \to Y$ and $y \in |Y|$ as in the lemma where $Y$ is an affine scheme. Since $f$ is quasi-compact, we conclude that $X$ is quasi-compact. Hence we can choose an affine scheme $W$ and a surjective étale morphism $W \to X$. Then the image of $|f|$ is the same as the image of $W \to Y$. In this way we reduce to the case of schemes which is Limits, Lemma 32.15.1. $\square$

[1] See discussion in Properties of Spaces, Section 66.11.

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