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Tag 01Y0

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

Lemma 29.9.2. Let $X$ be a locally Noetherian scheme. The category of coherent $\mathcal{O}_X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\mathcal{O}_X$-modules are coherent. Any extension of coherent sheaves is coherent.

Proof. This is a restatement of Modules, Lemma 17.12.4 in a particular case. $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 2110–2117 (see updates for more information).

    \begin{lemma}
    \label{lemma-coherent-abelian-Noetherian}
    Let $X$ be a locally Noetherian scheme.
    The category of coherent $\mathcal{O}_X$-modules is abelian.
    More precisely, the kernel and cokernel of a map of coherent
    $\mathcal{O}_X$-modules are coherent. Any extension
    of coherent sheaves is coherent.
    \end{lemma}
    
    \begin{proof}
    This is a restatement of
    Modules, Lemma \ref{modules-lemma-coherent-abelian}
    in a particular case.
    \end{proof}

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