Lemma 103.13.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be a filtered colimit of abelian sheaves on $\mathcal{X}$. Then for any $p \geq 0$ we have

$R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i.$

The same is true for abelian sheaves on $\mathcal{X}_{\acute{e}tale}$ taking higher direct images in the étale topology.

Proof. We will prove this for the fppf topology; the proof for the étale topology is the same. Recall that $R^ if_*\mathcal{F}$ is the sheaf on $\mathcal{Y}_{fppf}$ associated to the presheaf

$(y : V \to \mathcal{Y}) \longmapsto H^ i(V \times _{y, \mathcal{Y}} \mathcal{X}, \text{pr}^{-1}\mathcal{F})$

See Sheaves on Stacks, Lemma 96.21.2. Recall that the colimit is the sheaf associated to the presheaf colimit. When $V$ is affine, the fibre product $V \times _\mathcal {Y} \mathcal{X}$ is quasi-compact and quasi-separated. Hence we can apply Lemma 103.13.1 to $H^ p(V \times _\mathcal {Y} \mathcal{X}, -)$ where $V$ is affine. Since every $V$ has an fppf covering by affine objects this proves the lemma. Some details omitted. $\square$

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