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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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