General results of this nature can be found in Cohomology, Section 20.19, Sheaves, Lemma 6.29.1, and Modules, Lemma 17.22.8.
Lemma 30.6.1. Let f : X \to S be a quasi-compact and quasi-separated morphism of schemes. Let \mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i be a filtered colimit of abelian sheaves on X. Then for any p \geq 0 we have
Proof. We will apply Cohomology, Lemma 20.19.2. Since the affine opens form a basis for the topology of S, it suffices to show that for U \subset S affine open we have H^ p(f^{-1}U, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(f^{-1}U, \mathcal{F}_ i. Since f^{-1}U is quasi-compact and quasi-separated we conclude using Cohomology, Lemma 20.19.1. (Because the basis of affine opens in f^{-1}U satisfies the assumptions of that lemma.) \square
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Comment #9668 by Jonas Ehrhard on