Lemma 46.6.3 (06ZP)—The Stacks project

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Lemma 46.6.3. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. For any parasitic adequate $\mathcal{O}_ T$ -module on $(\mathit{Sch}/T)_\tau$ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^ if_*\mathcal{F}$ are parasitic adequate $\mathcal{O}_ S$ -modules on $(\mathit{Sch}/S)_\tau$ .

Proof. We have already seen in Lemma 46.5.12 that these higher direct images are adequate. Hence it suffices to show that $(R^ if_*\mathcal{F})(U_ i) = 0$ for any $\tau$ -covering $\{ U_ i \to S\}$ open. And $R^ if_*\mathcal{F}$ is parasitic by Descent, Lemma 35.12.3. $\square$

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