The Stacks project

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

1. $X$ is quasi-compact and quasi-separated, and

2. $X_ s$ 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 $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$.