The Stacks project

Proposition 103.11.7. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable morphisms of algebraic stacks. Assume that

  1. $f$ is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and

  2. $g$ is quasi-compact and quasi-separated.

If $\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ then there is a spectral sequence

\[ E_2^{p, q} = R^ q(g \circ f_ p)_{\mathit{QCoh}, *}f_ p^*\mathcal{F} \Rightarrow R^{p + q}g_{\mathit{QCoh}, *}\mathcal{F} \]

in $\mathit{QCoh}(\mathcal{O}_\mathcal {Y})$.

Proof. Note that each of the morphisms $f_ p : \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$ is quasi-compact and quasi-separated, hence $g \circ f_ p$ is quasi-compact and quasi-separated, hence the assertion makes sense (i.e., the functors $R^ q(g \circ f_ p)_{\mathit{QCoh}, *}$ are defined). There is a spectral sequence

\[ E_2^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]

by Sheaves on Stacks, Proposition 96.21.1. Applying the exact functor $Q_\mathcal {Y} : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {Y})$ gives the desired spectral sequence in $\mathit{QCoh}(\mathcal{O}_\mathcal {Y})$. $\square$


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Your email address will not be published.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 0785. Beware of the difference between the letter 'O' and the digit '0'.