Lemma 18.28.11. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let

be an exact complex of presheaves of $\mathcal{O}$-modules. If $\mathcal{Q}$ and all $\mathcal{F}_ i$ are flat $\mathcal{O}$-modules, then for any presheaf $\mathcal{G}$ of $\mathcal{O}$-modules the complex

is exact also. If $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings then the same result holds $\textit{Mod}(\mathcal{O})$.

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