The Stacks project

Lemma 69.20.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated and of finite type. The following are equivalent

  1. The morphism $f$ is proper.

  2. For any morphism $Y \to Z$ which is locally of finite presentation the map $|X \times _ Y Z| \to |Z|$ is closed, and

  3. there exists a scheme $V$ and a surjective ├ętale morphism $V \to Y$ such that $|\mathbf{A}^ n \times (X \times _ Y V)| \to |\mathbf{A}^ n \times V|$ is closed for all $n \geq 0$.

Proof. In view of the fact that a proper morphism is the same thing as a separated, finite type, and universally closed morphism, this lemma is a special case of Lemma 69.20.2. $\square$


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