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The Stacks project

Lemma 17.11.6. Let (X, \mathcal{O}_ X) be a ringed space. Let \mathcal{F} be a finitely presented \mathcal{O}_ X-module. Let x \in X such that \mathcal{F}_ x \cong \mathcal{O}_{X, x}^{\oplus r}. Then there exists an open neighbourhood U of x such that \mathcal{F}|_ U \cong \mathcal{O}_ U^{\oplus r}.

Proof. Choose s_1, \ldots , s_ r \in \mathcal{F}_ x mapping to a basis of \mathcal{O}_{X, x}^{\oplus r} by the isomorphism. Choose an open neighbourhood U of x such that s_ i lifts to s_ i \in \mathcal{F}(U). After shrinking U we see that the induced map \psi : \mathcal{O}_ U^{\oplus r} \to \mathcal{F}|_ U is surjective (Lemma 17.9.4). By Lemma 17.11.3 we see that \mathop{\mathrm{Ker}}(\psi ) is of finite type. Then \mathop{\mathrm{Ker}}(\psi )_ x = 0 implies that \mathop{\mathrm{Ker}}(\psi ) becomes zero after shrinking U once more (Lemma 17.9.5). \square


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