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The Stacks project

Lemma 6.14.2. Let X be a topological space. Let \mathcal{O} \to \mathcal{O}' be a morphism of presheaves of rings on X. Let \mathcal{F} be a presheaf of \mathcal{O}-modules. Let x \in X. We have

\mathcal{F}_ x \otimes _{\mathcal{O}_ x} \mathcal{O}'_ x = (\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{O}')_ x

as \mathcal{O}'_ x-modules.

Proof. Omitted. \square

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