The Stacks project

Lemma 17.12.6. Let $X$ be a ringed space. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_ X$-modules. Let $x \in X$. Assume $\mathcal{G}$ of finite type, $\mathcal{F}$ coherent and the map on stalks $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ injective. Then there exists an open neighbourhood $x \in U \subset X$ such that $\varphi |_ U$ is injective.

Proof. Denote $\mathcal{K} \subset \mathcal{G}$ the kernel of $\varphi $. By Lemma 17.12.4 we see that $\mathcal{K}$ is a finite type $\mathcal{O}_ X$-module. Our assumption is that $\mathcal{K}_ x = 0$. By Lemma 17.9.5 there exists an open neighbourhood $U$ of $x$ such that $\mathcal{K}|_ U = 0$. Then $U$ works. $\square$

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