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18.15 Exactness of pushforward

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:

  1. $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is surjective for all $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, and

  2. $f_* : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ 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

  1. $f_*$ transforms surjections of sheaves of sets into surjections,

  2. $f_*$ transforms surjections of abelian sheaves into surjections,

  3. $f_*$ commutes with coequalizers on sheaves of sets,

  4. $f_*$ 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

\[ 0 \to f_*\mathcal{F}_1 \to f_*\mathcal{F}_2 \to f_*\mathcal{F}_3 \]

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

  1. $f_* : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})$ is exact.

  2. 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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