The Stacks project

Lemma 101.6.2. 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 101.5.1 to prove this. We will check its assumptions (1) – (4). Parts (1) and (2) follows from Sheaves on Stacks, Lemma 94.11.8. Part (3) follows from Lemma 101.6.1. 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 94.21.3 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 67.3.1. 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!

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