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.
Recall that $R^ pf_*\mathcal{F}$ is the sheaf on $Y_{spaces, {\acute{e}tale}}$ associated to $V \mapsto H^ p(V \times _ Y X, \mathcal{F})$, see Cohomology on Sites, Lemma 21.7.4 and Properties of Spaces, Lemma 66.18.8. Recall that the colimit is the sheaf associated to the presheaf colimit. Hence we can apply Lemma 69.5.1 to $H^ p(V \times _ Y X, -)$ where $V$ is affine to conclude (because when $V$ is affine, then $V \times _ Y X$ is quasi-compact and quasi-separated). Strictly speaking this also uses Properties of Spaces, Lemma 66.18.6 to see that there exist enough affine objects.
$\square$
\[ R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i. \]
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Comment #1286 by Johan Commelin on