The Stacks project

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

\[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}|_\mathcal {B}) \]

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

Proof. The inverse functor in given in Lemma 6.30.12 above. Checking the obvious functorialities is left to the reader. $\square$

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