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30.6 Colimits and higher direct images

General results of this nature can be found in Cohomology, Section 20.19, Sheaves, Lemma 6.29.1, and Modules, Lemma 17.22.8.

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 (1)

Comment #9668 by Jonas Ehrhard on

The second sentence of the proof is missing a bracket at the end.


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