The Stacks project

Proof. Choose an equivalence $j : \mathcal{S}_ F \to \mathcal{X}$, see Algebraic Stacks, Sections 94.7 and 94.8. An object of $F_{\acute{e}tale}$ is a scheme $U$ together with an étale morphism $\varphi : U \to F$. Then $\varphi $ is an object of $\mathcal{S}_ F$ over $U$. Hence $j(\varphi )$ is an object of $\mathcal{X}$ over $U$. In this way $j$ induces a functor $u : F_{\acute{e}tale}\to \mathcal{X}$. It is clear that $u$ is continuous and cocontinuous for the étale topology on $\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully faithful. Also, fibre products and equalizers exist in $F_{\acute{e}tale}$ and $u$ commutes with them because these are computed on the level of underlying schemes in $F_{\acute{e}tale}$. Thus Sites, Lemmas 7.21.5, 7.21.6, and 7.21.7 apply. In particular $u$ defines a morphism of topoi $i_ F : \mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ and there exists a left adjoint $i_{F, !}$ of $i_ F^{-1}$ which commutes with fibre products and equalizers.

We claim that $i_{F, !}$ is exact. If this is true, then we can define $\pi _ F$ by the rules $\pi _ F^{-1} = i_{F, !}$ and $\pi _{F, *} = i_ F^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_{F, !}$ is right exact and preserves fibre products. Hence it suffices to show that $i_{F, !}* = *$ where $*$ indicates the final object in the category of sheaves of sets. Let $U$ be a scheme and let $\varphi : U \to F$ be surjective and étale. Set $R = U \times _ F U$. Then

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale})$. Using the right exactness of $i_{F, !}$, using $i_{F, !} = (u_ p\ )^\# $, and using Sites, Lemma 7.5.6 we see that

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale})$. Using that $j$ is an equivalence and that $F = U/R$ it follows that the coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ of the two maps $h_{u(R)} \to h_{u(U)}$ is $*$. We omit the proof that these morphisms are compatible with structure sheaves. $\square$

Proof. This is similar to Topologies, Lemma 34.4.17 (3) but there is a small snag due to the fact that $F \to G$ may not be representable by schemes. In particular we don't get a commutative diagram of ringed sites, but only a commutative diagram of ringed topoi.

Before we start the proof proper, we choose equivalences $j : \mathcal{S}_ F \to \mathcal{X}$ and $j' : \mathcal{S}_ G \to \mathcal{Y}$ which induce functors $u : F_{\acute{e}tale}\to \mathcal{X}$ and $u' : G_{\acute{e}tale}\to \mathcal{Y}$ as in the proof of Lemma 96.10.1. Because of the 2-functoriality of sheaves on categories fibred in groupoids over $\mathit{Sch}_{fppf}$ (see discussion in Section 96.3) we may assume that $\mathcal{X} = \mathcal{S}_ F$ and $\mathcal{Y} = \mathcal{S}_ G$ and that $f : \mathcal{S}_ F \to \mathcal{S}_ G$ is the functor associated to the morphism $f : F \to G$. Correspondingly we will omit $u$ and $u'$ from the notation, i.e., given an object $U \to F$ of $F_{\acute{e}tale}$ we denote $U/F$ the corresponding object of $\mathcal{X}$. Similarly for $G$.

Let $\mathcal{G}$ be a sheaf on $\mathcal{X}_{\acute{e}tale}$. To prove (2) we compute $\pi _{G, *}f_*\mathcal{G}$ and $f_{small, *}\pi _{F, *}\mathcal{G}$. To do this let $V \to G$ be an object of $G_{\acute{e}tale}$. Then

see (96.5.0.1). The fibre product in the formula is

i.e., it is the split category fibred in groupoids associated to the algebraic space $V \times _ G F$. And $\text{pr}^{-1}\mathcal{G}$ is a sheaf on $\mathcal{S}_{V \times _ G F}$ for the étale topology.

In particular, if $V \times _ G F$ is representable, i.e., if it is a scheme, then $\pi _{G, *}f_*\mathcal{G}(V) = \mathcal{G}(V \times _ G F/F)$ and also

which proves the desired equality in this special case.

In general, choose a scheme $U$ and a surjective étale morphism $U \to V \times _ G F$. Set $R = U \times _{V \times _ G F} U$. Then $U/V \times _ G F$ and $R/V \times _ G F$ are objects of the fibre product category above. Since $\text{pr}^{-1}\mathcal{G}$ is a sheaf for the étale topology on $\mathcal{S}_{V \times _ G F}$ the diagram

is an equalizer diagram. Note that $\text{pr}^{-1}\mathcal{G}(U/V \times _ G F) = \mathcal{G}(U/F)$ and $\text{pr}^{-1}\mathcal{G}(R/V \times _ G F) = \mathcal{G}(R/F)$ by the definition of pullbacks. Moreover, by the material in Properties of Spaces, Section 66.18 (especially, Properties of Spaces, Remark 66.18.4 and Lemma 66.18.8) we see that there is an equalizer diagram

Since we also have $\pi _{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ and $\pi _{F, *}\mathcal{G}(U/F) = \mathcal{G}(U/F)$ we obtain a canonical identification $f_{small, *}\pi _{F, *}\mathcal{G}(V) = \pi _{G, *}f_*\mathcal{G}(V)$. We omit the proof that this is compatible with restriction mappings and that it is functorial in $\mathcal{G}$. $\square$