# The Stacks Project

## Tag 01Y0

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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