The Stacks project

18.44 Localizing sheaves of rings

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{S} \subset \mathcal{O}$ be a sub-presheaf of sets such that for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the set $\mathcal{S}(U) \subset \mathcal{O}(U)$ is a multiplicative subset, see Algebra, Definition 10.9.1. In this case we can consider the presheaf of rings

The restriction mapping sends the section $f/s$, $f \in \mathcal{O}(U)$, $s \in \mathcal{S}(U)$ to $(f|_ V)/(s|_ V)$ for $V \to U$ in $\mathcal{C}$.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{S} \subset \mathcal{O}$ be a sub-presheaf of sets such that for all $U \in \mathcal{C}$ the set $\mathcal{S}(U) \subset \mathcal{O}(U)$ is a multiplicative subset. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. In this case we can consider the presheaf of $\mathcal{S}^{-1}\mathcal{O}$-modules

The restriction mapping sends the section $t/s$, $t \in \mathcal{F}(U)$, $s \in \mathcal{S}(U)$ to $(t|_ V)/(s|_ V)$ if $V \to U$ is a morphism of $\mathcal{C}$. Then $\mathcal{S}^{-1}\mathcal{F}$ is a presheaf of $\mathcal{S}^{-1}\mathcal{O}$-modules.