Lemma 35.7.5. 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 flat $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module.

**Proof.**
Omitted. For the affine case, see Algebra, Lemma 10.83.2.
$\square$

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