The Stacks project

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

  1. the functor $j_\mathcal {F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal {F}$,

  2. the functor $j_\mathcal {F}^*$ is the functor $\mathcal{H} \mapsto \mathcal{H}_\mathcal {F}$,

  3. the functor $j_{\mathcal{F}!}$ on sheaves of sets is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$,

  4. 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 $j_\mathcal {F}^*$. $\square$

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