The Stacks project

Lemma 30.9.5. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_ X$-modules. Let $x \in X$.

1. If $\mathcal{F}_ x = 0$ then there exists an open neighbourhood $U \subset X$ of $x$ such that $\mathcal{F}|_ U = 0$.

2. If $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ is injective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi |_ U$ is injective.

3. If $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ is surjective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi |_ U$ is surjective.

4. If $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ is bijective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi |_ U$ is an isomorphism.