The Stacks project

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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