Lemma 6.20.3 (008B)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.20: Sheafification of presheaves of modules
Lemma 6.20.3 (cite)

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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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