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Tag 01Y5

Chapter 29: Cohomology of Schemes > Section 29.9: Coherent sheaves on locally Noetherian schemes

Lemma 29.9.7. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Supp}(\mathcal{F})$ is closed, and $\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition 28.5.5.

Proof. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and let $\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$ such that $i_*\mathcal{G} \cong \mathcal{F}$. Since $Z = \text{Supp}(\mathcal{F})$ we see that the support is closed. The scheme $Z$ is locally Noetherian by Morphisms, Lemmas 28.14.5 and 28.14.6. Finally, $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module by Lemma 29.9.1 $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 2198–2205 (see updates for more information).

    \begin{lemma}
    \label{lemma-coherent-support-closed}
    Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent
    $\mathcal{O}_X$-module. Then $\text{Supp}(\mathcal{F})$ is closed, and
    $\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support
    of $\mathcal{F}$, see
    Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}.
    \end{lemma}
    
    \begin{proof}
    Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and
    let $\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$
    such that $i_*\mathcal{G} \cong \mathcal{F}$.
    Since $Z = \text{Supp}(\mathcal{F})$ we see that the support is closed.
    The scheme $Z$ is locally Noetherian by
    Morphisms, Lemmas \ref{morphisms-lemma-immersion-locally-finite-type}
    and \ref{morphisms-lemma-finite-type-noetherian}.
    Finally, $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module by
    Lemma \ref{lemma-coherent-Noetherian}
    \end{proof}

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