The Stacks project

Lemma 35.12.3. Let $f : T \to S$ be a morphism of schemes. For any parasitic $\mathcal{O}$-module on $(\mathit{Sch}/T)_\tau $ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^ if_*\mathcal{F}$ are parasitic $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau $.

Proof. Recall that $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf

\[ U \mapsto H^ i((\mathit{Sch}/U \times _ S T)_\tau , \mathcal{F}) \]

see Cohomology on Sites, Lemma 21.7.4. If $U \to S$ is flat, then $U \times _ S T \to T$ is flat as a base change. Hence the displayed group is zero by Lemma 35.12.2. If $\{ U_ i \to U\} $ is a $\tau $-covering then $U_ i \times _ S T \to T$ is also flat. Hence it is clear that the sheafification of the displayed presheaf is zero on schemes $U$ flat over $S$. $\square$

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