Lemma 18.28.13. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on a site $\mathcal{C}$. If $\mathcal{G}$ is a flat $\mathcal{O}_1$-module, then $\mathcal{G} \otimes _{\mathcal{O}_1} \mathcal{O}_2$ is a flat $\mathcal{O}_2$-module.

**Proof.**
This is true because

\[ (\mathcal{G} \otimes _{\mathcal{O}_1} \mathcal{O}_2) \otimes _{\mathcal{O}_2} \mathcal{H} = \mathcal{G} \otimes _{\mathcal{O}_1} \mathcal{F} \]

(as sheaves of abelian groups for example). $\square$

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