Lemma 27.20.1. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a (finite) locally free $\mathcal{O}_ X$-module of if and only if $M$ is a (finite) locally free $R$-module.

Proof. Follows from the definitions, see Modules, Definition 17.14.1 and Algebra, Definition 10.77.1. $\square$

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