Lemma 35.12.2. Let $S$ be a scheme. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau $.

If $\mathcal{G}$ is parasitic for the $\tau $-topology, then $H^ p_\tau (U, \mathcal{G}) = 0$ for every $U$ open in $S$, resp. étale over $S$, resp. smooth over $S$, resp. syntomic over $S$, resp. flat and locally of finite presentation over $S$.

If $\mathcal{G}$ is parasitic then $H^ p_\tau (U, \mathcal{G}) = 0$ for every $U$ flat over $S$.

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