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 $\mathcal{F}'$.
Proof. In other words we claim that $s^ n\mathcal{F} \subset \mathcal{F}' \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ for some $n \geq 0$. In other words, we claim that the quotient map $\mathcal{F} \to \mathcal{F}/\mathcal{F}'$ becomes zero after multiplying by a power of $s$. This follows from Properties, Lemma 28.17.3. $\square$
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