## 28.16 Characterizing modules of finite type and finite presentation

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following lemma implies that $\mathcal{F}$ is of finite type (see Modules, Definition 17.9.1) if and only if $\mathcal{F}$ is on each open affine $\mathop{\mathrm{Spec}}(A) = U \subset X$ of the form $\widetilde M$ for some finite type $A$-module $M$. Similarly, $\mathcal{F}$ is of finite presentation (see Modules, Definition 17.11.1) if and only if $\mathcal{F}$ is on each open affine $\mathop{\mathrm{Spec}}(A) = U \subset X$ of the form $\widetilde M$ for some finitely presented $A$-module $M$.

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. $\square$

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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