The Stacks project

Lemma 100.6.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_\mathcal {X}$-module on $\mathcal{X}_{\acute{e}tale}$. Then $R^ if_*\mathcal{F}$ (computed in the étale topology) is locally quasi-coherent on $\mathcal{Y}_{\acute{e}tale}$.

Proof. We will use Lemma 100.5.1 to prove this. We will check its assumptions (1) – (4). Parts (1) and (2) follows from Sheaves on Stacks, Lemma 93.11.8. Part (3) follows from Lemma 100.6.1. Thus it suffices to show (4).

Suppose $f : \mathcal{X} \to \mathcal{Y}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes $X$ and $Y$. Choose any object $y$ of $\mathcal{Y}$ lying over a scheme $V$. For clarity, denote $\mathcal{V} = (\mathit{Sch}/V)_{fppf}$ the algebraic stack corresponding to $V$. Consider the cartesian diagram

\[ \xymatrix{ \mathcal{Z} \ar[d] \ar[r]_ g \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{V} \ar[r]^ y & \mathcal{Y} } \]

Thus $\mathcal{Z}$ is representable by the scheme $Z = V \times _ Y X$ and $f'$ is quasi-compact and separated (even affine). By Sheaves on Stacks, Lemma 93.21.3 we have

\[ R^ if_*\mathcal{F}|_{V_{\acute{e}tale}} = R^ if'_{small, *}\big (g^*\mathcal{F}|_{Z_{\acute{e}tale}}\big ) \]

The right hand side is a quasi-coherent sheaf on $V_{\acute{e}tale}$ by Cohomology of Spaces, Lemma 66.3.1. This implies the left hand side is quasi-coherent which is what we had to prove. $\square$


Comments (2)

Comment #33 by David Zureick-Brown on

Typo -- stacs

Comment #36 by Johan on

Fixed. Thanks!


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Your email address will not be published.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 075Z. Beware of the difference between the letter 'O' and the digit '0'.