Lemma 103.6.2. 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 103.5.1 to prove this. We will check its assumptions (1) – (4). Parts (1) and (2) follows from Sheaves on Stacks, Lemma 96.12.4. Part (3) follows from Lemma 103.6.1. 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

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 96.22.3 we have

The right hand side is a quasi-coherent sheaf on $V_{\acute{e}tale}$ by Cohomology of Spaces, Lemma 69.3.1. This implies the left hand side is quasi-coherent which is what we had to prove. $\square$

## 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Your email address will not be published.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 (2)

Comment #33 by David Zureick-Brown on

Comment #36 by Johan on