18.21 Localization of ringed topoi
This section is the analogue of Sites, Section 7.30 in the setting of ringed topoi.
Lemma 18.21.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a sheaf. For a sheaf $\mathcal{H}$ on $\mathcal{C}$ denote $\mathcal{H}_\mathcal {F}$ the sheaf $\mathcal{H} \times \mathcal{F}$ seen as an object of the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$. The pair $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$ is a ringed topos and there is a canonical morphism of ringed topoi
\[ (j_\mathcal {F}, j_\mathcal {F}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \]
which is a localization as in Section 18.19 such that
the functor $j_\mathcal {F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal {F}$,
the functor $j_\mathcal {F}^*$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal {F}$,
the functor $j_{\mathcal{F}!}$ on sheaves of sets is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$,
the functor $j_{\mathcal{F}!}$ on sheaves of modules associates to the $\mathcal{O}_\mathcal {F}$-module $\varphi : \mathcal{G} \to \mathcal{F}$ the $\mathcal{O}$-module which is the sheafification of the presheaf
\[ V \longmapsto \bigoplus \nolimits _{s \in \mathcal{F}(V)} \{ \sigma \in \mathcal{G}(V) \mid \varphi (\sigma ) = s \} \]
for $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.
Proof.
By Sites, Lemma 7.30.1 we see that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is a topos and that (1) and (3) are true. In particular this shows that $j_\mathcal {F}^{-1}\mathcal{O} = \mathcal{O}_\mathcal {F}$ and shows that $\mathcal{O}_\mathcal {F}$ is a sheaf of rings. Thus we may choose the map $j_\mathcal {F}^\sharp $ to be the identity, in particular we see that (2) is true. Moreover, the proof of Sites, Lemma 7.30.1 shows that we may assume $\mathcal{C}$ is a site with all finite limits and a subcanonical topology and that $\mathcal{F} = h_ U$ for some object $U$ of $\mathcal{C}$. Then (4) follows from the description of $j_{\mathcal{F}!}$ in Lemma 18.19.2. Alternatively one could show directly that the functor described in (4) is a left adjoint to $j_\mathcal {F}^*$.
$\square$
Definition 18.21.2. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.
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 $\mathcal{F}$.
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.
We continue the tradition, established in the chapter on sites, that we check the localization constructions on topoi are compatible with the constructions of localization on sites, whenever this makes sense.
Lemma 18.21.3. With $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ as in Lemma 18.21.1. If $\mathcal{F} = h_ U^\# $ for some object $U$ of $\mathcal{C}$ then via the identification $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ of Sites, Lemma 7.25.4 we have
canonically $\mathcal{O}_ U = \mathcal{O}_\mathcal {F}$, and
with these identifications we have $(j_\mathcal {F}, j_\mathcal {F}^\sharp ) = (j_ U, j_ U^\sharp )$.
Proof.
The assertion for underlying topoi is Sites, Lemma 7.30.5. Note that $\mathcal{O}_ U$ is the restriction of $\mathcal{O}$ which by Sites, Lemma 7.25.7 corresponds to $\mathcal{O} \times h_ U^\# $ under the equivalence of Sites, Lemma 7.25.4. By definition of $\mathcal{O}_\mathcal {F}$ we get (1). What's left is to prove that $j_\mathcal {F}^\sharp = j_ U^\sharp $ under this identification. We omit the verification.
$\square$
Localization is functorial in the following two ways: We can “relocalize” a localization (see Lemma 18.21.4) or we can given a morphism of ringed topoi, localize upstairs at the inverse image of a sheaf downstairs and get a commutative diagram of ringed topoi (see Lemma 18.22.1).
Lemma 18.21.4. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. If $s : \mathcal{G} \to \mathcal{F}$ is a morphism of sheaves on $\mathcal{C}$ then there exists a natural commutative diagram of morphisms of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{G}, \mathcal{O}_\mathcal {G}) \ar[rd]_{(j_\mathcal {G}, j_\mathcal {G}^\sharp )} \ar[rr]_{(j, j^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F}) \ar[ld]^{(j_\mathcal {F}, j_\mathcal {F}^\sharp )} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) & } \]
where $(j, j^\sharp )$ is the localization morphism of the ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}, \mathcal{O}_\mathcal {F})$ at the object $\mathcal{G}/\mathcal{F}$.
Proof.
All assertions follow from Sites, Lemma 7.30.6 except the assertion that $j_\mathcal {G}^\sharp = j^\sharp \circ j^{-1}(j_\mathcal {F}^\sharp )$. We omit the verification.
$\square$
Lemma 18.21.5. With $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $s : \mathcal{G} \to \mathcal{F}$ as in Lemma 18.21.4. If there exist a morphism $f : V \to U$ of $\mathcal{C}$ such that $\mathcal{G} = h_ V^\# $ and $\mathcal{F} = h_ U^\# $ and $s$ is induced by $f$, then the diagrams of Lemma 18.19.5 and Lemma 18.21.4 agree via the identifications $(j_\mathcal {F}, j_\mathcal {F}^\sharp ) = (j_ U, j_ U^\sharp )$ and $(j_\mathcal {G}, j_\mathcal {G}^\sharp ) = (j_ V, j_ V^\sharp )$ of Lemma 18.21.3.
Proof.
All assertions follow from Sites, Lemma 7.30.7 except for the assertion that the two maps $j^\sharp $ agree. This holds since in both cases the map $j^\sharp $ is simply the identity. Some details omitted.
$\square$
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