The Stacks project

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

\[ \ldots \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to \mathcal{Q} \to 0 \]

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

\[ \ldots \to \mathcal{F}_2 \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}_1 \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}_0 \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{Q} \otimes _{p, \mathcal{O}} \mathcal{G} \to 0 \]

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

Proof. Follows from Lemma 18.28.9 by splitting the complex into short exact sequences and using Lemma 18.28.10 to prove inductively that $\mathop{\mathrm{Im}}(\mathcal{F}_{i + 1} \to \mathcal{F}_ i)$ is flat. $\square$

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