Lemma 6.30.12. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{B}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $\mathcal{B}$. Let $\mathcal{O}^{ext}$ be the sheaf of rings on $X$ extending $\mathcal{O}$ and let $\mathcal{F}^{ext}$ be the abelian sheaf on $X$ extending $\mathcal{F}$, see Lemma 6.30.9. There exists a canonical map

which agrees with the given map over elements of $\mathcal{B}$ and which endows $\mathcal{F}^{ext}$ with the structure of an $\mathcal{O}^{ext}$-module.

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