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.
$\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)