Lemma 103.8.4. 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 103.8.1. By Lemma 103.8.3 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 69.3 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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)