The Stacks project

Lemma 66.6.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if there exists an étale surjective morphism $U \to X$ with $U$ an affine scheme.

Proof. If there exists an étale surjective morphism $U \to X$ with $U$ affine then $X$ is quasi-compact by Definition 66.5.1. Conversely, if $X$ is quasi-compact, then $|X|$ is quasi-compact. Let $U = \coprod _{i \in I} U_ i$ be a disjoint union of affine schemes with an étale and surjective map $\varphi : U \to X$ (Lemma 66.6.1). Then $|X| = \bigcup \varphi (|U_ i|)$ and by quasi-compactness there is a finite subset $i_1, \ldots , i_ n$ such that $|X| = \bigcup \varphi (|U_{i_ j}|)$. Hence $U_{i_1} \cup \ldots \cup U_{i_ n}$ is an affine scheme with a finite surjective morphism towards $X$. $\square$

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