The Stacks project

Lemma 17.16.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ X$-modules. Let $x \in X$. There is a canonical isomorphism of $\mathcal{O}_{X, x}$-modules

\[ (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G})_ x = \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{G}_ x \]

functorial in $\mathcal{F}$ and $\mathcal{G}$.

Proof. Omitted. $\square$

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