The Stacks project

Lemma 103.13.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $f_{\mathit{QCoh}, *}$ and the functors $R^ if_{\mathit{QCoh}, *}$ commute with direct sums and filtered colimits.

Proof. The functors $f_*$ and $R^ if_*$ commute with direct sums and filtered colimits on all modules by Lemma 103.13.2. The lemma follows as $f_{\mathit{QCoh}, *} = Q \circ f_*$ and $R^ if_{\mathit{QCoh}, *} = Q \circ R^ if_*$ and $Q$ commutes with all colimits, see Lemma 103.10.2. $\square$

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