## 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 17.11.6.

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 quasi-coherent 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. Recall that $R^ pf_*\mathcal{F}$ is the sheaf associated to $U \mapsto H^ p(f^{-1}U, \mathcal{F})$, see Cohomology, Lemma 20.7.3. Recall that the colimit is the sheaf associated to the presheaf colimit (taking colimits over opens). Hence we can apply Cohomology, Lemma 20.19.1 to $H^ p(f^{-1}U, -)$ where $U$ is affine to conclude. (Because the basis of affine opens in $f^{-1}U$ satisfies the assumptions of that lemma.) $\square$

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