Higher direct images of qcqs morphisms commute with filtered colimits of sheaves.

Lemma 68.5.2. 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

$R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i.$

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 65.18.8. Recall that the colimit is the sheaf associated to the presheaf colimit. Hence we can apply Lemma 68.5.1 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 65.18.6 to see that there exist enough affine objects. $\square$

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