Let X be a quasi-compact and separated algebraic space. Let U be an affine scheme, and let f : U —> X be a surjective étale morphism. Let d be an upper bound for the size of the fibres of |U| —> |X|. Then for any quasi-coherent O_X-module F we have H^q(X,F)=0 for q ≥ d. See Tag 072B.

Note: This is interesting even when X is a scheme.