The Stacks project

Lemma 6.20.3. Let $X$ be a topological space. Let $\mathcal{O} \to \mathcal{O}'$ be a morphism of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf $\mathcal{O}$-modules. Let $x \in X$. We have

\[ \mathcal{F}_ x \otimes _{\mathcal{O}_ x} \mathcal{O}'_ x = (\mathcal{F} \otimes _\mathcal {O} \mathcal{O}')_ x \]

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

Proof. Follows directly from Lemma 6.14.2 and the fact that taking stalks commutes with sheafification. $\square$

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