Lemma 18.28.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. If $\mathcal{F}$ is a flat $\mathcal{O}$-module, then $\mathcal{F}|_ U$ is a flat $\mathcal{O}_ U$-module.

Proof. Let $\mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3$ be an exact complex of $\mathcal{O}_ U$-modules. Since $j_{U!}$ is exact (Lemma 18.19.3) and $\mathcal{F}$ is flat as an $\mathcal{O}$-modules then we see that the complex made up of the modules

$j_{U!}(\mathcal{G}_ i \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U) = j_{U!}\mathcal{G}_ i \otimes _\mathcal {O} \mathcal{F}$

(Lemma 18.27.9) is exact. We conclude that $\mathcal{G}_1 \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U \to \mathcal{G}_2 \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U \to \mathcal{G}_3 \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U$ is exact by Lemma 18.19.4. $\square$

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