Lemma 75.7.8 (0CZJ)—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 75.7.8 (cite)

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Lemma 75.7.8. 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. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$ -module. If the support of $\mathcal{F}$ is proper over $Y$ , then the support of $(g')^*\mathcal{F}$ is proper over $Y'$ .

Proof. Observe that the statement makes sense because $(g')*\mathcal{F}$ is of finite type by Modules on Sites, Lemma 18.23.4. We have $\text{Supp}((g')^*\mathcal{F}) = |g'|^{-1}(\text{Supp}(\mathcal{F}))$ by Morphisms of Spaces, Lemma 67.15.2. Thus the lemma follows from Lemma 75.7.4. $\square$

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