The Stacks project

Lemma 66.16.5. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact morphism of algebraic spaces over $S$. Let $Z$ be the scheme theoretic image of $f$. Let $z \in |Z|$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Z \ar[r] & Y } \]

such that the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $z$.

Proof. Choose an affine scheme $V$ with a point $z' \in V$ and an ├ętale morphism $V \to Y$ mapping $z'$ to $z$. Let $Z' \subset V$ be the scheme theoretic image of $X \times _ Y V \to V$. By Lemma 66.16.3 we have $Z' = Z \times _ Y V$. Thus $z' \in Z'$. Since $f$ is quasi-compact and $V$ is affine we see that $X \times _ Y V$ is quasi-compact. Hence there exists an affine scheme $W$ and a surjective ├ętale morphism $W \to X \times _ Y V$. Then $Z' \subset V$ is also the scheme theoretic image of $W \to V$. By Morphisms, Lemma 29.6.5 we can choose a diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & W \ar[r] \ar[d] & X \times _ Y V \ar[d] \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Z' \ar[r] & V \ar[r] & Y } \]

such that the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $z'$. Composing with $Z' \to Z$ and $W \to X \times _ Y V \to X$ we obtain a solution. $\square$


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