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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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