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
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
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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