The Stacks project

Lemma 102.8.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be an affine morphism of algebraic stacks. The functor $f_* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ is exact and commutes with direct sums. The functors $R^ if_*$ for $i > 0$ vanish on $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

Proof. The functors exist by Proposition 102.8.1. By Lemma 102.8.3 this reduces to the case of an affine morphism of algebraic spaces taking higher direct images in the setting of quasi-coherent modules on algebraic spaces. By the discussion in Cohomology of Spaces, Section 68.3 we reduce to the case of an affine morphism of schemes. For affine morphisms of schemes we have the vanishing of higher direct images on quasi-coherent modules by Cohomology of Schemes, Lemma 30.2.3. The vanishing for $R^1f_*$ implies exactness of $f_*$. Commuting with direct sums follows from Morphisms, Lemma 29.11.6 for example. $\square$


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