Lemma 75.7.4 (0CZF)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 75: Derived Categories of Spaces
Section 75.7: Being proper over a base
Lemma 75.7.4 (cite)

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Lemma 75.7.4. Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces over $S$

$\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

with $f$ locally of finite type. If $T$ is a closed subset of $|X|$ proper over $Y$ , then $|g'|^{-1}(T)$ is a closed subset of $|X'|$ proper over $Y'$ .

Proof. Observe that the statement makes sense as $f'$ is locally of finite type by Morphisms of Spaces, Lemma 67.23.3. Let $Z \subset X$ be the reduced induced closed subspace structure on $T$ . Denote $Z' = (g')^{-1}(Z)$ the scheme theoretic inverse image. Then $Z' = X' \times _ X Z = (Y' \times _ Y X) \times _ X Z = Y' \times _ Y Z$ is proper over $Y'$ as a base change of $Z$ over $Y$ (Morphisms of Spaces, Lemma 67.40.3). On the other hand, we have $T' = |Z'|$ . Hence the lemma holds. $\square$

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