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