The Stacks project

Lemma 35.7.7. Let $X$ be a scheme. 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 locally projective $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a locally projective $\mathcal{O}_ X$-module.

Proof. Omitted. For Zariski coverings this is Properties, Lemma 28.21.2. For the affine case this is Algebra, Theorem 10.95.6. $\square$

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