The Stacks project

Lemma 18.28.12. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $\mathcal{G}$ and $\mathcal{F}$ are flat $\mathcal{O}$-modules, then $\mathcal{G} \otimes _\mathcal {O} \mathcal{F}$ is a flat $\mathcal{O}$-module.

Proof. This is true because

\[ (\mathcal{G} \otimes _\mathcal {O} \mathcal{F}) \otimes _\mathcal {O} \mathcal{H} = \mathcal{G} \otimes _\mathcal {O} (\mathcal{F} \otimes _\mathcal {O} \mathcal{H}) \]

and a composition of exact functors is exact. $\square$

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