Lemma 30.9.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Let $x \in X$. Suppose $\psi : \mathcal{G}_ x \to \mathcal{F}_ x$ is a map of $\mathcal{O}_{X, x}$-modules. Then there exists an open neighbourhood $U \subset X$ of $x$ and a map $\varphi : \mathcal{G}|_ U \to \mathcal{F}|_ U$ such that $\varphi _ x = \psi $.

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