Lemma 103.6.2 (075Z)—The Stacks project

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

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

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 96.22.3 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 69.3.1. This implies the left hand side is quasi-coherent which is what we had to prove. $\square$

The Stacks project

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