Lemma 35.7.7 (05JZ)—The Stacks project

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Part 2: Schemes
Chapter 35: Descent
Section 35.7: Descent of finiteness properties of modules
Lemma 35.7.7 (cite)

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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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