The Stacks project

Lemma 17.9.5. Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Let $x \in X$. Assume $\mathcal{F}$ of finite type and $\mathcal{F}_ x = 0$. Then there exists an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_ U$ is zero.

Proof. This is a special case of Lemma 17.9.4 applied to the morphism $0 \to \mathcal{F}$. $\square$

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