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Tag 01Y5

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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