The Stacks project

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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