Lemma 66.4.9 (03IE)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.4: Points of algebraic spaces
Lemma 66.4.9 (cite)

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Lemma 66.4.9. Let $S$ be a scheme. Let $X$ , $Y$ be algebraic spaces over $S$ . Let $X' \subset X$ be an open subspace. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$ . Then $f$ factors through $X'$ if and only if $|f| : |Y| \to |X|$ factors through $|X'| \subset |X|$ .

Proof. By Spaces, Lemma 65.12.3 we see that $Y' = Y \times _ X X' \to Y$ is an open immersion. If $|f|(|Y|) \subset |X'|$ , then clearly $|Y'| = |Y|$ . Hence $Y' = Y$ by Lemma 66.4.8. $\square$

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