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
30.6 Colimits and higher direct images
General results of this nature can be found in Cohomology, Section 20.19, Sheaves, Lemma 6.29.1, and Modules, Lemma 17.22.8.
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