Lemma 103.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $y : V \to \mathcal{Y}$ in $\mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ with $y$ a flat morphism. Let $\mathcal{F}$ be in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Then $(f_*\mathcal{F})(y) = (f_{\mathit{QCoh}, *}\mathcal{F})(y)$ and $(R^ if_*\mathcal{F})(y) = (R^ if_{\mathit{QCoh}, *}\mathcal{F})(y)$ for all $i \in \mathbf{Z}$.

**Proof.**
This follows from the construction of the functors $R^ if_{\mathit{QCoh}, *}$ in Proposition 103.11.1, the definition of parasitic modules in Definition 103.9.1, and Lemma 103.10.2 part (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Your email address will not be published.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)