Lemma 66.5.2 (03E4)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.5: Quasi-compact spaces
Lemma 66.5.2 (cite)

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Lemma 66.5.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Then $X$ is quasi-compact if and only if $|X|$ is quasi-compact.

Proof. Choose a scheme $U$ and an étale surjective morphism $U \to X$ . We will use Lemma 66.4.4. If $U$ is quasi-compact, then since $|U| \to |X|$ is surjective we conclude that $|X|$ is quasi-compact. If $|X|$ is quasi-compact, then since $|U| \to |X|$ is open we see that there exists a quasi-compact open $U' \subset U$ such that $|U'| \to |X|$ is surjective (and still étale). Hence we win. $\square$

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