Lemma 66.31.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent

for some scheme $U$ and surjective étale morphism $U \to X$ the restriction $\mathcal{F}|_ U$ is locally projective on $U$, and

for any scheme $U$ and any étale morphism $U \to X$ the restriction $\mathcal{F}|_ U$ is locally projective on $U$.

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