Lemma 103.13.4. Let f : \mathcal{X} \to \mathcal{Y} be an affine morphism of algebraic stacks. The functors R^ if_{\mathit{QCoh}, *}, i > 0 vanish and the functor f_{\mathit{QCoh}, *} is exact and commutes with direct sums and all colimits.
Proof. Since we have R^ if_{\mathit{QCoh}, *} = Q \circ R^ if_* we obtain the vanishing from Lemma 103.8.4. The vanishing implies that f_{\mathit{QCoh}, *} is exact as \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} form a \delta -functor, see Proposition 103.11.1. Then f_{\mathit{QCoh}, *} commutes with direct sums for example by Lemma 103.13.3. An exact functor which commutes with direct sums commutes with all colimits. \square
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