The Stacks project

Lemma 17.16.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{G}$ be an $\mathcal{O}_ X$-module. If $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is an exact sequence of $\mathcal{O}_ X$-modules then the induced sequence

\[ \mathcal{F}_1 \otimes _{\mathcal{O}_ X} \mathcal{G} \to \mathcal{F}_2 \otimes _{\mathcal{O}_ X} \mathcal{G} \to \mathcal{F}_3 \otimes _{\mathcal{O}_ X} \mathcal{G} \to 0 \]

is exact.

Proof. This follows from the fact that exactness may be checked at stalks (Lemma 17.3.1), the description of stalks (Lemma 17.16.1) and the corresponding result for tensor products of modules (Algebra, Lemma 10.12.10). $\square$

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