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## Tag: 04CC

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Lemma 41.49.1. Let $\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$. Let $f : X \to Y$ be a morphism of schemes. Let $$f_{big} : \mathop{\textit{Sh}}\nolimits((\textit{Sch}/X)_\tau) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/Y)_\tau)$$ be the corresponding morphism of topoi as in Topologies, Lemma 30.3.15, 30.4.15, 30.5.10, 30.6.10, or 30.7.12.
1. The functor $f_{big}^{-1} : \textit{Ab}((\textit{Sch}/Y)_\tau) \to \textit{Ab}((\textit{Sch}/X)_\tau)$ has a left adjoint $$f_{big!} : \textit{Ab}((\textit{Sch}/X)_\tau) \to \textit{Ab}((\textit{Sch}/Y)_\tau)$$ which is exact.
2. The functor $f_{big}^* : \textit{Mod}((\textit{Sch}/Y)_\tau, \mathcal{O}) \to \textit{Mod}((\textit{Sch}/X)_\tau, \mathcal{O})$ has a left adjoint $$f_{big!} : \textit{Mod}((\textit{Sch}/X)_\tau, \mathcal{O}) \to \textit{Mod}((\textit{Sch}/Y)_\tau, \mathcal{O})$$ which is exact.
Moreover, the two functors $f_{big!}$ agree on underlying sheaves of abelian groups.

Proof. Recall that $f_{big}$ is the morphism of topoi associated to the continuous and cocontinuous functor $u : (\textit{Sch}/X)_\tau \to (\textit{Sch}/Y)_\tau$, $U/X \mapsto U/Y$. Moreover, we have $f_{big}^{-1}\mathcal{O} = \mathcal{O}$. Hence the existence of $f_{big!}$ follows from Modules on Sites, Lemma 17.16.2, respectively Modules on Sites, Lemma 17.38.1. Note that if $U$ is an object of $(\textit{Sch}/X)_\tau$ then the functor $u$ induces an equivalence of categories $$u' : (\textit{Sch}/X)_\tau/U \longrightarrow (\textit{Sch}/Y)_\tau/U$$ because both sides of the arrow are equal to $(\textit{Sch}/U)_\tau$. Hence the agreement of $f_{big!}$ on underlying abelian sheaves follows from the discussion in Modules on Sites, Remark 17.38.2. The exactness of $f_{big!}$ follows from Modules on Sites, Lemma 17.16.3 as the functor $u$ above which commutes with fibre products and equalizers. $\square$

\begin{lemma}
\label{lemma-exactness-lower-shriek}
Let $\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$.
Let $f : X \to Y$ be a morphism of schemes. Let
$$f_{big} : \Sh((\Sch/X)_\tau) \longrightarrow \Sh((\Sch/Y)_\tau)$$
be the corresponding morphism of topoi as in
Topologies, Lemma
\ref{topologies-lemma-morphism-big},
\ref{topologies-lemma-morphism-big-etale},
\ref{topologies-lemma-morphism-big-smooth},
\ref{topologies-lemma-morphism-big-syntomic}, or
\ref{topologies-lemma-morphism-big-fppf}.
\begin{enumerate}
\item The functor
$f_{big}^{-1} : \textit{Ab}((\Sch/Y)_\tau) \to \textit{Ab}((\Sch/X)_\tau)$
$$f_{big!} : \textit{Ab}((\Sch/X)_\tau) \to \textit{Ab}((\Sch/Y)_\tau)$$
which is exact.
\item The functor
$f_{big}^* : \textit{Mod}((\Sch/Y)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/X)_\tau, \mathcal{O})$
$$f_{big!} : \textit{Mod}((\Sch/X)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/Y)_\tau, \mathcal{O})$$
which is exact.
\end{enumerate}
Moreover, the two functors $f_{big!}$ agree on underlying sheaves
of abelian groups.
\end{lemma}

\begin{proof}
Recall that $f_{big}$ is the morphism of topoi associated to the
continuous and cocontinuous functor
$u : (\Sch/X)_\tau \to (\Sch/Y)_\tau$, $U/X \mapsto U/Y$.
Moreover, we have $f_{big}^{-1}\mathcal{O} = \mathcal{O}$.
Hence the existence of $f_{big!}$ follows from
respectively
Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}.
Note that if $U$ is an object of $(\Sch/X)_\tau$ then the functor
$u$ induces an equivalence of categories
$$u' : (\Sch/X)_\tau/U \longrightarrow (\Sch/Y)_\tau/U$$
because both sides of the arrow are equal to $(\Sch/U)_\tau$.
Hence the agreement of $f_{big!}$ on underlying abelian sheaves
follows from the discussion in
Modules on Sites, Remark \ref{sites-modules-remark-when-shriek-equal}.
The exactness of $f_{big!}$ follows from
Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek}
as the functor $u$ above which commutes with fibre products and equalizers.
\end{proof}


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