6.14 Stalks of presheaves of modules

Lemma 6.14.1. Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $x \in X$. The canonical map $\mathcal{O}_ x \times \mathcal{F}_ x \to \mathcal{F}_ x$ coming from the multiplication map $\mathcal{O} \times \mathcal{F} \to \mathcal{F}$ defines a $\mathcal{O}_ x$-module structure on the abelian group $\mathcal{F}_ x$.

Proof. Omitted. $\square$

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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