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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