The Stacks project

Proposition 102.5.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $Rf_*$ induces a commutative diagram

\[ \xymatrix{ D^{+}_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X}) \ar[r] \ar[d]^{Rf_*} & D^{+}_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X}) \ar[r] \ar[d]^{Rf_*} & D(\mathcal{O}_\mathcal {X}) \ar[d]^{Rf_*} \\ D^{+}_{\mathcal{P}_\mathcal {Y}}(\mathcal{O}_\mathcal {Y}) \ar[r] & D^{+}_{\mathcal{M}_\mathcal {Y}}(\mathcal{O}_\mathcal {Y}) \ar[r] & D(\mathcal{O}_\mathcal {Y}) } \]

and hence induces a functor

\[ Rf_{\mathit{QCoh}, *} : D^{+}_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow D^{+}_\mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \]

on quotient categories. Moreover, the functor $R^ if_\mathit{QCoh}$ of Cohomology of Stacks, Proposition 101.10.1 are equal to $H^ i \circ Rf_{\mathit{QCoh}, *}$ with $H^ i$ as in (

Proof. We have to show that $Rf_*E$ is an object of $D^{+}_{\mathcal{M}_\mathcal {Y}}(\mathcal{O}_\mathcal {Y})$ for $E$ in $D^{+}_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$. This follows from Cohomology of Stacks, Proposition 101.7.4 and the spectral sequence $R^ if_*H^ j(E) \Rightarrow R^{i + j}f_*E$. The case of parasitic modules works the same way using Cohomology of Stacks, Lemma 101.8.3. The final statement is clear from the definition of $H^ i$ in ( $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07BC. Beware of the difference between the letter 'O' and the digit '0'.