## Tag `01BW`

Chapter 17: Sheaves of Modules > Section 17.12: Coherent modules

Lemma 17.12.2. Let $(X, \mathcal{O}_X)$ be a ringed space. Any coherent $\mathcal{O}_X$-module is of finite presentation and hence quasi-coherent.

Proof.Let $\mathcal{F}$ be a coherent sheaf on $X$. Pick a point $x \in X$. By (1) of the definition of coherent, we may find an open neighbourhood $U$ and sections $s_i$, $i = 1, \ldots, n$ of $\mathcal{F}$ over $U$ such that $\Psi : \bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{F}$ is surjective. By (2) of the definition of coherent, we may find an open neighbourhood $V$, $x \in V \subset U$ and sections $t_1, \ldots, t_m$ of $\bigoplus_{i = 1, \ldots, n} \mathcal{O}_V$ which generate the kernel of $\Psi|_V$. Then over $V$ we get the presentation $$ \bigoplus\nolimits_{j = 1, \ldots, m} \mathcal{O}_V \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_V \to \mathcal{F}|_V \to 0 $$ as desired. $\square$

The code snippet corresponding to this tag is a part of the file `modules.tex` and is located in lines 1646–1651 (see updates for more information).

```
\begin{lemma}
\label{lemma-coherent-finite-presentation}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Any coherent $\mathcal{O}_X$-module is of finite presentation
and hence quasi-coherent.
\end{lemma}
\begin{proof}
Let $\mathcal{F}$ be a coherent sheaf on $X$.
Pick a point $x \in X$.
By (1) of the definition of coherent, we may find an open neighbourhood $U$
and sections $s_i$, $i = 1, \ldots, n$ of $\mathcal{F}$ over $U$
such that $\Psi : \bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{F}$
is surjective. By (2) of the definition of coherent, we may find
an open neighbourhood $V$, $x \in V \subset U$ and sections
$t_1, \ldots, t_m$ of $\bigoplus_{i = 1, \ldots, n} \mathcal{O}_V$
which generate the kernel of $\Psi|_V$. Then over $V$ we get the
presentation
$$
\bigoplus\nolimits_{j = 1, \ldots, m}
\mathcal{O}_V
\longrightarrow
\bigoplus\nolimits_{i = 1, \ldots, n}
\mathcal{O}_V
\to
\mathcal{F}|_V
\to
0
$$
as desired.
\end{proof}
```

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