Lemma 115.11.4. 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. Let $\mathcal{F}' \subset \mathcal{F}$ be quasi-coherent $\mathcal{O}_ X$-modules. Assume that

$X$ is quasi-compact,

$\mathcal{F}$ is of finite type, and

$\mathcal{F}'|_{X_ s} = \mathcal{F}|_{X_ s}$.

Then there exists an $n \geq 0$ such that multiplication by $s^ n$ on $\mathcal{F}$ factors through $\mathcal{F}'$.

## Comments (0)