## 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}
```

## Comments (0)

## Add a comment on tag `01Y0`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

There are no comments yet for this tag.