The Stacks project

Lemma 31.11.1. Let $X$ be an integral scheme with generic point $\eta $. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U \subset X$ be nonempty open and $s \in \mathcal{F}(U)$. The following are equivalent

  1. for some $x \in U$ the image of $s$ in $\mathcal{F}_ x$ is torsion,

  2. for all $x \in U$ the image of $s$ in $\mathcal{F}_ x$ is torsion,

  3. the image of $s$ in $\mathcal{F}_\eta $ is zero,

  4. the image of $s$ in $j_*\mathcal{F}_\eta $ is zero, where $j : \eta \to X$ is the inclusion morphism.

Proof. Omitted. $\square$

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