global sections of higher direct images of étale sheaves

Question

Is there a useful criterion for when $\Gamma(X, R^qf_*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?

Answer 1

The obvious criterion is that this holds whenever $H^k(X,R^{q-k}f_\ast F) = 0$ and $H^{k+1}(X,R^{q-k}f_\ast F) = 0$ for all $k > 0$.

In particular one has $\Gamma(X,R^qf_\ast F) = H^q(X',F)$ for all $q$ whenever $H^k(X,R^qf_\ast F) = 0$ for all $q$ and for all $k>0$.