Definition 18.19.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

The ringed site $(\mathcal{C}/U, \mathcal{O}_ U)$ is called the

*localization of the ringed site $(\mathcal{C}, \mathcal{O})$ at the object $U$*.The morphism of ringed topoi $(j_ U, j_ U^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is called the

*localization morphism*.The functor $j_{U*} : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ is called the

*direct image functor*.For a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ the sheaf $j_ U^*\mathcal{F}$ is called the

*restriction of $\mathcal{F}$ to $\mathcal{C}/U$*. We will sometimes denote it by $\mathcal{F}|_{\mathcal{C}/U}$ or even $\mathcal{F}|_ U$. It is described by the simple rule $j_ U^*(\mathcal{F})(X/U) = \mathcal{F}(X)$.The left adjoint $j_{U!} : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ of restriction is called

*extension by zero*. It exists and is exact by Lemmas 18.19.2 and 18.19.3.

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