The Stacks project

Lemma 21.14.6. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {C}$-module.

1. If $R^ qf_*\mathcal{F} = 0$ for $q > 0$, then $H^ p(\mathcal{C}, \mathcal{F}) = H^ p(\mathcal{D}, f_*\mathcal{F})$ for all $p$.

2. If $H^ p(\mathcal{D}, R^ qf_*\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^ q(\mathcal{C}, \mathcal{F}) = H^0(\mathcal{D}, R^ qf_*\mathcal{F})$ for all $q$.