Some technical lemmas concerning exactness properties of pushforward.
Lemma 18.15.1. Let f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) be a morphism of topoi. The following are equivalent:
f^{-1}f_*\mathcal{F} \to \mathcal{F} is surjective for all \mathcal{F} in \textit{Ab}(\mathcal{C}), and
f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D}) reflects surjections.
In this case the functor f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D}) is faithful.
Proof. Assume (1). Suppose that a : \mathcal{F} \to \mathcal{F}' is a map of abelian sheaves on \mathcal{C} such that f_*a is surjective. As f^{-1} is exact this implies that f^{-1}f_*a : f^{-1}f_*\mathcal{F} \to f^{-1}f_*\mathcal{F}' is surjective. Combined with (1) this implies that a is surjective. This means that (2) holds.
Assume (2). Let \mathcal{F} be an abelian sheaf on \mathcal{C}. We have to show that the map f^{-1}f_*\mathcal{F} \to \mathcal{F} is surjective. By (2) it suffices to show that f_*f^{-1}f_*\mathcal{F} \to f_*\mathcal{F} is surjective. And this is true because there is a canonical map f_*\mathcal{F} \to f_*f^{-1}f_*\mathcal{F} which is a one-sided inverse.
We omit the proof of the final assertion. \square
Lemma 18.15.2. Let f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) be a morphism of topoi. Assume at least one of the following properties holds
f_* transforms surjections of sheaves of sets into surjections,
f_* transforms surjections of abelian sheaves into surjections,
f_* commutes with coequalizers on sheaves of sets,
f_* commutes with pushouts on sheaves of sets,
Then f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D}) is exact.
Proof. Since f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D}) is a right adjoint we already know that it transforms a short exact sequence 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 of abelian sheaves on \mathcal{C} into an exact sequence
see Categories, Sections 4.23 and 4.24 and Homology, Section 12.7. Hence it suffices to prove that the map f_*\mathcal{F}_2 \to f_*\mathcal{F}_3 is surjective. If (1), (2) holds, then this is clear from the definitions. By Sites, Lemma 7.41.1 we see that either (3) or (4) formally implies (1), hence in these cases we are done also. \square
Lemma 18.15.3. Let f : \mathcal{D} \to \mathcal{C} be a morphism of sites associated to the continuous functor u : \mathcal{C} \to \mathcal{D}. Assume u is almost cocontinuous. Then
f_* : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C}) is exact.
if f^\sharp : f^{-1}\mathcal{O}_\mathcal {C} \to \mathcal{O}_\mathcal {D} is given so that f becomes a morphism of ringed sites, then f_* : \textit{Mod}(\mathcal{O}_\mathcal {D}) \to \textit{Mod}(\mathcal{O}_\mathcal {C}) is exact.
Proof. Part (2) follows from part (1) by Lemma 18.14.2. Part (1) follows from Sites, Lemmas 7.42.6 and 7.41.1. \square
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