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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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