Lemma 30.17.4. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$ be a section. Assume that
$X$ is quasi-compact and quasi-separated, and
$X_ s$ is affine.
Then for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and every $p > 0$ and all $\xi \in H^ p(X, \mathcal{F})$ there exists an $n \geq 0$ such that $s^ n\xi = 0$ in $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$.
Proof. Recall that $H^ p(X_ s, \mathcal{G})$ is zero for every quasi-coherent module $\mathcal{G}$ by Lemma 30.2.2. Hence the lemma follows from Lemma 30.17.3. $\square$
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