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

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

2. 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 $q$.

Proof. These are two simple conditions that force the Leray spectral sequence to degenerate at $E_2$. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves. $\square$

Comment #1159 by on

I think you wish to say that the spectral sequence collapses (or degenerates, the latter terminology seems to be the one used in the Stacks project), not converges (which it always does).

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