Lemma 66.29.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. The following are equivalent

$\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module,

there exists an étale morphism $f : Y \to X$ of algebraic spaces over $S$ with $|f| : |Y| \to |X|$ surjective such that $f^*\mathcal{F}$ is quasi-coherent on $Y$,

there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that $\varphi ^*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module, and

for every affine scheme $U$ and étale morphism $\varphi : U \to X$ the restriction $\varphi ^*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module.

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