The Stacks project

Lemma 115.16.7. Let $S$ be a scheme. Let $X$ be a very reasonable algebraic space over $S$. There exists a set of schemes $U_ i$ and morphisms $U_ i \to X$ such that

  1. each $U_ i$ is a quasi-compact scheme,

  2. each $U_ i \to X$ is étale,

  3. both projections $U_ i \times _ X U_ i \to U_ i$ are quasi-compact, and

  4. the morphism $\coprod U_ i \to X$ is surjective (and étale).

Proof. Decent Spaces, Definition 68.6.1 says that there exist $U_ i \to X$ such that (2), (3) and (4) hold. Fix $i$, and set $R_ i = U_ i \times _ X U_ i$, and denote $s, t : R_ i \to U_ i$ the projections. For any affine open $W \subset U_ i$ the open $W' = t(s^{-1}(W)) \subset U_ i$ is a quasi-compact $R_ i$-invariant open (see Groupoids, Lemma 39.19.2). Hence $W'$ is a quasi-compact scheme, $W' \to X$ is étale, and $W' \times _ X W' = s^{-1}(W') = t^{-1}(W')$ so both projections $W' \times _ X W' \to W'$ are quasi-compact. This means the family of $W' \to X$, where $W \subset U_ i$ runs through the members of affine open coverings of the $U_ i$ gives what we want. $\square$

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