The Stacks project

Lemma 65.6.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a Zariski covering $X = \bigcup X_ i$ such that each algebraic space $X_ i$ has a surjective étale covering by an affine scheme. We may in addition assume each $X_ i$ maps into an affine open of $S$.

Proof. By Lemma 65.6.1 we can find a surjective étale morphism $U = \coprod U_ i \to X$, with $U_ i$ affine and mapping into an affine open of $S$. Let $X_ i \subset X$ be the open subspace of $X$ such that $U_ i \to X$ factors through an étale surjective morphism $U_ i \to X_ i$, see Lemma 65.4.10. Since $U = \bigcup U_ i$ we see that $X = \bigcup X_ i$. As $U_ i \to X_ i$ is surjective it follows that $X_ i \to S$ maps into an affine open of $S$. $\square$

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