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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