A Grothendieck topology is called *noetherian* if every object is quasi-compact (which is defined as usual). On such a topology, sheaf cohomology commutes with filtered colimits (and in particular with arbitrary direct sums). A proof can be found in Tamme's *Intoduction to Etale cohomology*, Theorem §3.3.11.1. For the etale topology on the spectrum of a field this shows that Galois cohomology commutes with arbitrary direct sums. For the Zariski topology on a scheme we recover the well-known result cited in Hartshorne. An even more general statement can be found in the Stacks project, [Lemma 19.16.2][1].


  [1]: http://stacks.math.columbia.edu/tag/0739