The Stacks project

Lemma 64.6.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a surjective étale morphism $U \to X$ where $U$ is a disjoint union of affine schemes. We may in addition assume each of these affines maps into an affine open of $S$.

Proof. Let $V \to X$ be a surjective étale morphism. Let $V = \bigcup _{i \in I} V_ i$ be a Zariski open covering such that each $V_ i$ maps into an affine open of $S$. Then set $U = \coprod _{i \in I} V_ i$ with induced morphism $U \to V \to X$. This is étale and surjective as a composition of étale and surjective representable transformations of functors (via the general principle Spaces, Lemma 63.5.4 and Morphisms, Lemmas 29.9.2 and 29.35.3). $\square$


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