Lemma 77.6.2 (0CW2)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 77: Flatness on Algebraic Spaces
Section 77.6: A criterion for purity
Lemma 77.6.2 (cite)

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Lemma 77.6.2. Let $Y$ be an algebraic space over a scheme $S$ . Let $g : X' \to X$ be a morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$ , then for any morphism $Z \to Y$ we have $\text{Ass}_{X_ Z/Z}(\mathcal{F}_ Z) \subset g_ Z(|X'_ Z|)$ .

Proof. By Properties of Spaces, Lemma 66.4.3 the map $|X'_ Z| \to |X_ Z| \times _{|X|} |X'|$ is surjective as $X'_ Z$ is equal to $X_ Z \times _ X X'$ . By Divisors on Spaces, Lemma 71.4.7 the map $|X_ Z| \to |X|$ sends $\text{Ass}_{X_ Z/Z}(\mathcal{F}_ Z)$ into $\text{Ass}_{X/Y}(\mathcal{F})$ . The lemma follows. $\square$

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