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The Stacks project

Lemma 34.5.10. Let \mathit{Sch}_{smooth} be a big smooth site. Let f : T \to S be a morphism in \mathit{Sch}_{smooth}. The functor

u : (\mathit{Sch}/T)_{smooth} \longrightarrow (\mathit{Sch}/S)_{smooth}, \quad V/T \longmapsto V/S

is cocontinuous, and has a continuous right adjoint

v : (\mathit{Sch}/S)_{smooth} \longrightarrow (\mathit{Sch}/T)_{smooth}, \quad (U \to S) \longmapsto (U \times _ S T \to T).

They induce the same morphism of topoi

f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_{smooth}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{smooth})

We have f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S). We have f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T). Also, f_{big}^{-1} has a left adjoint f_{big!} which commutes with fibre products and equalizers.

Proof. The functor u is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for f_{big}^{-1} and the existence of f_{big!}. Moreover, the functor v is a right adjoint because given U/T and V/S we have \mathop{\mathrm{Mor}}\nolimits _ S(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ T(U, V \times _ S T) as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for f_{big, *}. \square


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