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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: