The Stacks project

Lemma 30.9.3. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Any quasi-coherent submodule of $\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\mathcal{F}$ is coherent.

Proof. We may assume that $X$ is affine, say $X = \mathop{\mathrm{Spec}}(A)$. Properties, Lemma 28.5.2 implies that $A$ is Noetherian. Lemma 30.9.1 turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite $A$-module is a finite $A$-module, see for example Algebra, Lemma 10.51.1. $\square$

Comments (3)

Comment #2650 by Student on

Isn't the quasi-coherence assumption superfluous? That is, any submodule of is coherent, as the notions of finite type and finite presentations coincide over locally Noetherian schemes.

Comment #2651 by on

Nope because you can have things like where is an open. Such modules aren't even locally generated by sections, and a fortiori not quasi-coherent and a fortiori not coherent.

Comment #2652 by on

In comment #2651 I forgot to say: is the open immersion and is extension by zero.

There are also:

  • 2 comment(s) on Section 30.9: Coherent sheaves on locally Noetherian schemes

Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01Y1. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 30.9.3 as pdf