The Stacks project

Lemma 28.16.2. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. The quasi-coherent sheaf of $\mathcal{O}_ X$-modules $\widetilde M$ is an $\mathcal{O}_ X$-module of finite presentation if and only if $M$ is an $R$-module of finite presentation.

Proof. Assume $\widetilde M$ is an $\mathcal{O}_ X$-module of finite presentation. By Lemma 28.16.1 we see that $M$ is a finite $R$-module. Choose a surjection $R^ n \to M$ with kernel $K$. By Schemes, Lemma 26.5.4 there is a short exact sequence

\[ 0 \to \widetilde{K} \to \bigoplus \mathcal{O}_ X^{\oplus n} \to \widetilde{M} \to 0 \]

By Modules, Lemma 17.11.3 we see that $\widetilde{K}$ is a finite type $\mathcal{O}_ X$-module. Hence by Lemma 28.16.1 again we see that $K$ is a finite $R$-module. Hence $M$ is an $R$-module of finite presentation. $\square$

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