The Stacks project

Lemma 35.7.4. 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 locally generated by $r$ sections as an $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_ X$-module.

Proof. By Lemma 35.7.1 we see that $\mathcal{F}$ is of finite type. Hence Nakayama's lemma (Algebra, Lemma 10.20.1) implies that $\mathcal{F}$ is generated by $r$ sections in the neighbourhood of a point $x \in X$ if and only if $\dim _{\kappa (x)} \mathcal{F}_ x \otimes \kappa (x) \leq r$. Choose an $i$ and a point $x_ i \in X_ i$ mapping to $x$. Then $\dim _{\kappa (x)} \mathcal{F}_ x \otimes \kappa (x) = \dim _{\kappa (x_ i)} (f_ i^*\mathcal{F})_{x_ i} \otimes \kappa (x_ i)$ which is $\leq r$ as $f_ i^*\mathcal{F}$ is locally generated by $r$ sections. $\square$

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