Lemma 6.20.1. Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf $\mathcal{O}$-modules. Let $\mathcal{O}^\# $ be the sheafification of $\mathcal{O}$. Let $\mathcal{F}^\# $ be the sheafification of $\mathcal{F}$ as a presheaf of abelian groups. There exists a map of sheaves of sets

which makes the diagram

commute and which makes $\mathcal{F}^\# $ into a sheaf of $\mathcal{O}^\# $-modules. In addition, if $\mathcal{G}$ is a sheaf of $\mathcal{O}^\# $-modules, then any morphism of presheaves of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ (into the restriction of $\mathcal{G}$ to a $\mathcal{O}$-module) factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ where $\mathcal{F}^\# \to \mathcal{G}$ is a morphism of $\mathcal{O}^\# $-modules.

## Comments (0)