The Stacks project

Lemma 18.40.12. Localization of locally ringed sites and topoi.

1. Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. Let $U$ be an object of $\mathcal{C}$. Then the localization $(\mathcal{C}/U, \mathcal{O}_ U)$ is a locally ringed site, and the localization morphism

   $$(j_ U, j_ U^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$$

   is a morphism of locally ringed topoi.

2. Let $(\mathcal{C}, \mathcal{O})$ be a locally ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then the morphism

   $$(j, j^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V), \mathcal{O}_ V) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U)$$

   of Lemma 18.19.5 is a morphism of locally ringed topoi.

3. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$ and let $U = u(V)$. Then the morphism

   $$(f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V)$$

   of Lemma 18.20.1 is a morphism of locally ringed sites.

4. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$, $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $c : U \to u(V)$. Then the morphism

   $$(f_ c, (f_ c)^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V)$$

   of Lemma 18.20.2 is a morphism of locally ringed topoi.

5. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a locally ringed topos. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then the localization $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$ is a locally ringed topos and the localization morphism

   $$(j_\mathcal {F}, j_\mathcal {F}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$$

   is a morphism of locally ringed topoi.

6. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a locally ringed topos. Let $s : \mathcal{G} \to \mathcal{F}$ be a map of sheaves on $\mathcal{C}$. Then the morphism

   $$(j, j^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal {G}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$$

   of Lemma 18.21.4 is a morphism of locally ringed topoi.

7. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of locally ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then the morphism

   $$(f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G})$$

   of Lemma 18.22.1 is a morphism of locally ringed topoi.

8. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of locally ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be a morphism of sheaves. Then the morphism

   $$(f_ s, (f_ s)^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G})$$

   of Lemma 18.22.3 is a morphism of locally ringed topoi.