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## Tag: 021F

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Lemma 30.4.12. Let $\textit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\textit{Sch}_{\acute{e}tale}$. The functor $T_{\acute{e}tale} \to (\textit{Sch}/S)_{\acute{e}tale}$ is cocontinuous and induces a morphism of topoi $$i_f : \mathop{\textit{Sh}}\nolimits(T_{\acute{e}tale}) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{\acute{e}tale})$$ For a sheaf $\mathcal{G}$ on $(\textit{Sch}/S)_{\acute{e}tale}$ we have the formula $(i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.

Proof. Denote the functor $u : T_{\acute{e}tale} \to (\textit{Sch}/S)_{\acute{e}tale}$. In other words, given an étale morphism $j : U \to T$ corresponding to an object of $T_{\acute{e}tale}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 30.4.10. Let $a, b : U \to V$ be two morphisms in $T_{\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is $$V \times_{\Delta_{V/T}, V \times_T V, (a, b)} U \times_T U$$ which is a fibre product of schemes étale over $T$, hence étale over $T$. Thus $T_{\acute{e}tale}$ has equalizers and $u$ commutes with them. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas 7.20.5 and 7.20.6. $\square$

\begin{lemma}
\label{lemma-put-in-T-etale}
Let $\Sch_\etale$ be a big \'etale site.
Let $f : T \to S$ be a morphism in $\Sch_\etale$.
The functor $T_\etale \to (\Sch/S)_\etale$
is cocontinuous and induces a morphism of topoi
$$i_f : \Sh(T_\etale) \longrightarrow \Sh((\Sch/S)_\etale)$$
For a sheaf $\mathcal{G}$ on $(\Sch/S)_\etale$
we have the formula $(i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$.
The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes
with fibre products and equalizers.
\end{lemma}

\begin{proof}
Denote the functor $u : T_\etale \to (\Sch/S)_\etale$.
In other words, given an \'etale morphism $j : U \to T$ corresponding
to an object of $T_\etale$ we set $u(U \to T) = (f \circ j : U \to S)$.
This functor commutes with fibre products, see
Lemma \ref{lemma-fibre-products-etale}.
Let $a, b : U \to V$ be two morphisms in $T_\etale$.
In this case the equalizer of $a$ and $b$ (in the category of schemes) is
$$V \times_{\Delta_{V/T}, V \times_T V, (a, b)} U \times_T U$$
which is a fibre product of schemes \'etale over $T$, hence \'etale
over $T$. Thus $T_\etale$ has equalizers and $u$ commutes with them.
It is clearly cocontinuous.
It is also continuous as $u$ transforms coverings to coverings and
commutes with fibre products. Hence the Lemma follows from
Sites, Lemmas \ref{sites-lemma-when-shriek}
and \ref{sites-lemma-preserve-equalizers}.
\end{proof}


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