Higher direct images of qcqs morphisms commute with filtered colimits of sheaves.
Lemma 69.5.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be a filtered colimit of abelian sheaves on $X_{\acute{e}tale}$. Then for any $p \geq 0$ we have
Proof.
We will use that the morphism of topoi $f_{small} : X_{small} \to Y_{small}$ comes from the morphism of sites $f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}$ corresponding to the continuous functor $V \longmapsto X \times _ Y V$, see Properties of Spaces, Lemma 66.18.8. We will apply Cohomology on Sites, Lemma 21.16.4 to this morphism of sites. Since every object of $Y_{spaces, {\acute{e}tale}}$ has a covering by affine objects, it suffices to show that for $V$ affine and étale over $Y$ we have $H^ p(X \times _ Y V, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X \times _ Y V, \mathcal{F}_ i)$. Since $V$ is affine, the algebraic space $X \times _ Y V$ is quasi-compact and quasi-separated. Hence we can apply Lemma 69.5.1 to conclude.
$\square$
\[ R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i. \]
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Comment #1286 by Johan Commelin on