The Stacks project

Lemma 28.16.1. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_ X$-modules $\widetilde M$ is a finite type $\mathcal{O}_ X$-module if and only if $M$ is a finite $R$-module.

Proof. Assume $\widetilde M$ is a finite type $\mathcal{O}_ X$-module. This means there exists an open covering of $X$ such that $\widetilde M$ restricted to the members of this covering is globally generated by finitely many sections. Thus there also exists a standard open covering $X = \bigcup _{i = 1, \ldots , n} D(f_ i)$ such that $\widetilde M|_{D(f_ i)}$ is generated by finitely many sections. Thus $M_{f_ i}$ is finitely generated for each $i$. Hence we conclude by Algebra, Lemma 10.23.2. $\square$

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