Lemma 20.13.6. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module.

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

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

## Comments (2)

Comment #1159 by Pieter Belmans on

Comment #1176 by Johan on