Lemma 6.30.13. 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 $X$. Denote $\textit{Mod}(\mathcal{O}|_\mathcal {B})$ the category of sheaves of $\mathcal{O}|_\mathcal {B}$-modules on $\mathcal{B}$. There is an equivalence of categories

which assigns to a sheaf of $\mathcal{O}$-modules on $X$ its restriction to the members of $\mathcal{B}$.

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