The Stacks project

Lemma 17.17.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let

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

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

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

is exact also.

Proof. Follows from Lemma 17.17.7 by splitting the complex into short exact sequences and using Lemma 17.17.8 to prove inductively that $\mathop{\mathrm{Im}}(\mathcal{F}_{i + 1} \to \mathcal{F}_ i)$ is flat. $\square$

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