18.22 Localization of morphisms of ringed topoi
This section is the analogue of Sites, Section 7.31.
Lemma 18.22.1. Let
\[ f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \]
be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then there exists a commutative diagram of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[rr]_{(j_\mathcal {F}, j_\mathcal {F}^\sharp )} \ar[d]_{(f', (f')^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G}) \ar[rr]^{(j_\mathcal {G}, j_\mathcal {G}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') } \]
We have $f'_*j_\mathcal {F}^{-1} = j_\mathcal {G}^{-1}f_*$ and $f'_*j_\mathcal {F}^* = j_\mathcal {G}^*f_*$. Moreover, the morphism $f'$ is characterized by the rule
\[ (f')^{-1}(\mathcal{H} \xrightarrow {\varphi } \mathcal{G}) = (f^{-1}\mathcal{H} \xrightarrow {f^{-1}\varphi } \mathcal{F}). \]
Proof.
By Sites, Lemma 7.31.1 we have the diagram of underlying topoi, the equality $f'_*j_\mathcal {F}^{-1} = j_\mathcal {G}^{-1}f_*$, and the description of $(f')^{-1}$. To define $(f')^\sharp $ we use the map
\[ (f')^\sharp : \mathcal{O}'_\mathcal {G} = j_\mathcal {G}^{-1} \mathcal{O}' \xrightarrow {j_\mathcal {G}^{-1}f^\sharp } j_\mathcal {G}^{-1} f_*\mathcal{O} = f'_* j_\mathcal {F}^{-1}\mathcal{O} = f'_* \mathcal{O}_\mathcal {F} \]
or equivalently the map
\[ (f')^\sharp : (f')^{-1}\mathcal{O}'_\mathcal {G} = (f')^{-1}j_\mathcal {G}^{-1} \mathcal{O}' = j_\mathcal {F}^{-1}f^{-1}\mathcal{O}' \xrightarrow {j_\mathcal {F}^{-1}f^\sharp } j_\mathcal {F}^{-1} \mathcal{O} = \mathcal{O}_\mathcal {F}. \]
We omit the verification that these two maps are indeed adjoint to each other. The second construction of $(f')^\sharp $ shows that the diagram commutes in the $2$-category of ringed topoi (as the maps $j_\mathcal {F}^\sharp $ and $j_\mathcal {G}^\sharp $ are identities). Finally, the equality $f'_*j_\mathcal {F}^* = j_\mathcal {G}^*f_*$ follows from the equality $f'_*j_\mathcal {F}^{-1} = j_\mathcal {G}^{-1}f_*$ and the fact that pullbacks of sheaves of modules and sheaves of sets agree, see Lemma 18.21.1.
$\square$
Lemma 18.22.2. Let
\[ f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \]
be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. If $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ and $\mathcal{G} = h_ V^\# $, then the commutative diagrams of Lemma 18.20.1 and Lemma 18.22.1 agree via the identifications of Lemma 18.21.3.
Proof.
At the level of morphisms of topoi this is Sites, Lemma 7.31.2. This works also on the level of morphisms of ringed topoi since the formulas defining $(f')^\sharp $ in the proofs of Lemma 18.20.1 and Lemma 18.22.1 agree.
$\square$
Lemma 18.22.3. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[rr]_{(j_\mathcal {F}, j_\mathcal {F}^\sharp )} \ar[d]_{(f_ c, f_ c^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_\mathcal {G}) \ar[rr]^{(j_\mathcal {G}, j_\mathcal {G}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}'). } \]
The morphism $(f_ s, f_ s^\sharp )$ is equal to the composition of the morphism
\[ (f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/{\mathcal{G}}, \mathcal{O}'_\mathcal {G}) \]
of Lemma 18.22.1 and the morphism
\[ (j, j^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/f^{-1}\mathcal{G}, \mathcal{O}_{f^{-1}\mathcal{G}}) \]
of Lemma 18.21.4. Given any morphisms $b : \mathcal{G}' \to \mathcal{G}$, $a : \mathcal{F}' \to \mathcal{F}$, and $s' : \mathcal{F}' \to f^{-1}\mathcal{G}'$ such that
\[ \xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_ a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} } \]
commutes, then the following diagram of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}', \mathcal{O}_{\mathcal{F}'}) \ar[rr]_{(j_{\mathcal{F}'/\mathcal{F}}, j_{\mathcal{F}'/\mathcal{F}}^\sharp )} \ar[d]_{(f_{s'}, f_{s'}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[d]^{(f_ s, f_ s^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}', \mathcal{O}'_{\mathcal{G}'}) \ar[rr]^{(j_{\mathcal{G}'/\mathcal{G}}, j_{\mathcal{G}'/\mathcal{G}}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G}, \mathcal{O}'_{\mathcal{G}'}) } \]
commutes.
Proof.
On the level of morphisms of topoi this is Sites, Lemma 7.31.3. To check that the diagrams commute as morphisms of ringed topoi use the commutative diagrams of Lemmas 18.21.4 and 18.22.1.
$\square$
Lemma 18.22.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$, $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be as in Lemma 18.22.3. If $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ and $\mathcal{G} = h_ V^\# $, $\mathcal{F} = h_ U^\# $ and $s$ comes from a morphism $c : U \to u(V)$, then the commutative diagrams of Lemma 18.20.2 and Lemma 18.22.3 agree via the identifications of Lemma 18.21.3.
Proof.
This is formal using Lemmas 18.21.5 and 18.22.2.
$\square$
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