The Stacks project

Definition 18.21.2. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

  1. The ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$ is called the localization of the ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ at $\mathcal{F}$.

  2. The morphism of ringed topoi $(j_\mathcal {F}, j_\mathcal {F}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of Lemma 18.21.1 is called the localization morphism.

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View Definition 18.21.2 as pdf