Lemma 70.16.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$ such that $|X|$ has finitely many irreducible components.

There exists a surjective finite morphism $f : Y \to X$ of finite presentation where $Y$ is a scheme such that $f$ is finite étale over a quasi-compact dense open $U \subset X$,

given a surjective étale morphism $V \to X$ we may choose $Y \to X$ such that for every $y \in Y$ there is an open neighbourhood $W \subset Y$ such that $W \to X$ factors through $V$.

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