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.

Proof. Part (1) is clear since $\mathcal{O}_ U$ is just the restriction of $\mathcal{O}$, so Lemmas 18.40.5 and 18.40.11 apply. Part (2) is clear as the morphism $(j, j^\sharp )$ is actually a localization of a locally ringed site so (1) applies. Part (3) is clear also since $(f')^\sharp $ is just the restriction of $f^\sharp $ to the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$, see proof of Lemma 18.22.1 (hence the diagram of Definition 18.40.9 for the morphism $f'$ is just the restriction of the corresponding diagram for $f$, and restriction is an exact functor). Part (4) follows formally on combining (2) and (3). Parts (5), (6), (7), and (8) follow from their counterparts (1), (2), (3), and (4) by enlarging the sites as in Lemma 18.7.2 and translating everything in terms of sites and morphisms of sites using the comparisons of Lemmas 18.21.3, 18.21.5, 18.22.2, and 18.22.4. (Alternatively one could use the same arguments as in the proofs of (1), (2), (3), and (4) to prove (5), (6), (7), and (8) directly.) $\square$

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