The Stacks project

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

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

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

Proof. Let $U \to X$ be as in (1) and let $V \to X$ be étale where $V$ is a scheme. Then $\{ U \times _ X V \to V\} $ is an fppf covering of schemes. Hence if $\mathcal{F}|_ U$ is locally projective, then $\mathcal{F}|_{U \times _ X V}$ is locally projective (see Properties, Lemma 28.21.3) and hence $\mathcal{F}|_ V$ is locally projective, see Descent, Lemma 35.7.7. $\square$

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