The Stacks project

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 $.

  1. 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 $S$.

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

Proof. Proof in case $\tau = fppf$; the other cases are proved in the exact same way. The assumption means that $\mathcal{G}(U) = 0$ for any $U \to S$ flat and locally of finite presentation. Apply Cohomology on Sites, Lemma 21.10.9 to the subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ consisting of $U \to S$ flat and locally of finite presentation and the collection $\text{Cov}$ of all fppf coverings of elements of $\mathcal{B}$. $\square$

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