The Stacks project

Lemma 35.25.5. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $Z$ be a closed subset of $X$. If there exists an fpqc covering $\{ Y_ i \to Y\} $ such that the inverse image $Z_ i \subset Y_ i \times _ Y X$ is proper over $Y_ i$ (Cohomology of Schemes, Definition 30.26.2) then $Z$ is proper over $Y$.

Proof. Endow $Z$ with the reduced induced closed subscheme structure, see Schemes, Definition 26.12.5. For every $i$ the base change $Y_ i \times _ Y Z$ is a closed subscheme of $Y_ i \times _ Y X$ whose underlying closed subset is $Z_ i$. By definition (via Cohomology of Schemes, Lemma 30.26.1) we conclude that the projections $Y_ i \times _ Y Z \to Y_ i$ are proper morphisms. Hence $Z \to Y$ is a proper morphism by Lemma 35.23.14. Thus $Z$ is proper over $Y$ by definition. $\square$

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