Lemma 69.21.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian and $f$ is of finite type. Then the following are equivalent

$f$ is universally closed,

$f$ satisfies the existence part of the valuative criterion,

there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $|\mathbf{A}^ n \times X \times _ Y V| \to |\mathbf{A}^ n \times V|$ is closed for all $n \geq 0$,

for all diagrams (69.21.1.1) with $A$ a discrete valuation ring there there exists a finite separable extension $K'/K$ of fields, a discrete valuation ring $A' \subset K'$ dominating $A$, and a morphism $\mathop{\mathrm{Spec}}(A') \to X$ such that the following diagram commutes

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y } \]for all diagrams (69.21.1.1) with $A$ a discrete valuation ring there there exists a field extension $K'/K$, a valuation ring $A' \subset K'$ dominating $A$, and a morphism $\mathop{\mathrm{Spec}}(A') \to X$ such that the following diagram commutes

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

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