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.

## Comments (0)