Lemma 34.4.16 (021H)—The Stacks project

Lemma 34.4.16. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$. The functor

\[ u : (\mathit{Sch}/T)_{\acute{e}tale}\longrightarrow (\mathit{Sch}/S)_{\acute{e}tale}, \quad V/T \longmapsto V/S \]

is cocontinuous, and has a continuous right adjoint

\[ v : (\mathit{Sch}/S)_{\acute{e}tale}\longrightarrow (\mathit{Sch}/T)_{\acute{e}tale}, \quad (U \to S) \longmapsto (U \times _ S T \to T). \]

They induce the same morphism of topoi

\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale}) \]

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 (details omitted; compare with the proof of Lemma 34.4.13). 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 $f_{big, *}$. $\square$

The Stacks project

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