## Tag `01PB`

Chapter 27: Properties of Schemes > Section 27.16: Characterizing modules of finite type and finite presentation

Lemma 27.16.1. 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. $\square$

The code snippet corresponding to this tag is a part of the file `properties.tex` and is located in lines 1871–1877 (see updates for more information).

```
\begin{lemma}
\label{lemma-finite-type-module}
Let $X = \Spec(R)$ 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.
\end{lemma}
\begin{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 \ref{algebra-lemma-cover}.
\end{proof}
```

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