The Stacks project

Lemma 74.6.4. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite locally free $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module.

Proof. This follows from the case of schemes, see Descent, Lemma 35.7.6, by étale localization. $\square$

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