## Tag `01PC`

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

Lemma 27.16.2. 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 27.16.1 we see that $M$ is a finite $R$-module. Choose a surjection $R^n \to M$ with kernel $K$. By Schemes, Lemma 25.5.4 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 27.16.1 again we see that $K$ is a finite $R$-module. Hence $M$ is an $R$-module of finite presentation. $\square$

The code snippet corresponding to this tag is a part of the file `properties.tex` and is located in lines 1891–1896 (see updates for more information).

```
\begin{lemma}
\label{lemma-finite-presentation-module}
Let $X = \Spec(R)$ 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.
\end{lemma}
\begin{proof}
Assume $\widetilde M$ is an $\mathcal{O}_X$-module of finite presentation.
By Lemma \ref{lemma-finite-type-module} we see that $M$ is a finite $R$-module.
Choose a surjection $R^n \to M$ with kernel $K$. By
Schemes, Lemma \ref{schemes-lemma-spec-sheaves}
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
\ref{modules-lemma-kernel-surjection-finite-free-onto-finite-presentation}
we see that $\widetilde{K}$ is a finite type $\mathcal{O}_X$-module.
Hence by Lemma \ref{lemma-finite-type-module}
again we see that $K$ is a finite $R$-module.
Hence $M$ is an $R$-module of finite presentation.
\end{proof}
```

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