The Stacks project

Lemma 66.4.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent:

  1. there exists a Zariski covering $X = \bigcup X_ i$ and for each $i$ a scheme $U_ i$ and a quasi-compact surjective étale morphism $U_ i \to X_ i$, and

  2. there exist schemes $U_ i$ and étale morphisms $U_ i \to X$ such that the projections $U_ i \times _ X U_ i \to U_ i$ are quasi-compact and $\coprod U_ i \to X$ is surjective.

Proof. If (1) holds then the morphisms $U_ i \to X_ i \to X$ are étale (combine Morphisms, Lemma 29.35.3 and Spaces, Lemmas 63.5.4 and 63.5.3 ). Moreover, as $U_ i \times _ X U_ i = U_ i \times _{X_ i} U_ i$, both projections $U_ i \times _ X U_ i \to U_ i$ are quasi-compact.

If (2) holds then let $X_ i \subset X$ be the open subspace corresponding to the image of the open map $|U_ i| \to |X|$, see Properties of Spaces, Lemma 64.4.10. The morphisms $U_ i \to X_ i$ are surjective. Hence $U_ i \to X_ i$ is surjective étale, and the projections $U_ i \times _{X_ i} U_ i \to U_ i$ are quasi-compact, because $U_ i \times _{X_ i} U_ i = U_ i \times _ X U_ i$. Thus by Spaces, Lemma 63.11.4 the morphisms $U_ i \to X_ i$ are quasi-compact. $\square$


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