The Stacks project

Lemma 17.14.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $r \geq 0$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of finite locally free $\mathcal{O}_ X$-modules of rank $r$. Then $\varphi $ is an isomorphism if and only if $\varphi $ is surjective.

Proof. Assume $\varphi $ is surjective. Pick $x \in X$. There exists an open neighbourhood $U$ of $x$ such that both $\mathcal{F}|_ U$ and $\mathcal{G}|_ U$ are isomorphic to $\mathcal{O}_ U^{\oplus r}$. Pick lifts of the free generators of $\mathcal{G}|_ U$ to obtain a map $\psi : \mathcal{G}|_ U \to \mathcal{F}|_ U$ such that $\varphi |_ U \circ \psi = \text{id}$. Hence we conclude that the map $\Gamma (U, \mathcal{F}) \to \Gamma (U, \mathcal{G})$ induced by $\varphi $ is surjective. Since both $\Gamma (U, \mathcal{F})$ and $\Gamma (U, \mathcal{G})$ are isomorphic to $\Gamma (U, \mathcal{O}_ U)^{\oplus r}$ as an $\Gamma (U, \mathcal{O}_ U)$-module we may apply Algebra, Lemma 10.16.4 to see that $\Gamma (U, \mathcal{F}) \to \Gamma (U, \mathcal{G})$ is injective. This finishes the proof. $\square$

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