Lemma 30.6.1 (07TB)—The Stacks project

Lemma 30.6.1. Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be a filtered colimit of abelian sheaves on $X$ . Then for any $p \geq 0$ we have

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

Proof. We will apply Cohomology, Lemma 20.19.2. Since the affine opens form a basis for the topology of $S$ , it suffices to show that for $U \subset S$ affine open we have $H^ p(f^{-1}U, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(f^{-1}U, \mathcal{F}_ i$ . Since $f^{-1}U$ is quasi-compact and quasi-separated we conclude using Cohomology, Lemma 20.19.1. (Because the basis of affine opens in $f^{-1}U$ satisfies the assumptions of that lemma.) $\square$

Comments (2)

Comment #8261 by Haohao Liu on March 15, 2023 at 08:54

It seems to me that the proof works for every abelian sheaves $F=colim F_i$ .

Comment #8903 by Stacks project on April 09, 2024 at 19:18

Thanks! The proof was a little to sketchy so I added some clarifications as well. See changes.

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