The Stacks project

Lemma 31.4.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. The set of coherent subsheaves

\[ \{ \mathcal{K} \subset \mathcal{F} \mid \text{Supp}(\mathcal{K})\text{ is nowhere dense in }\text{Supp}(\mathcal{F}) \} \]

has a maximal element $\mathcal{K}$. Setting $\mathcal{F}' = \mathcal{F}/\mathcal{K}$ we have the following

  1. $\text{Supp}(\mathcal{F}') = \text{Supp}(\mathcal{F})$,

  2. $\mathcal{F}'$ has no embedded associated points, and

  3. there exists a dense open $U \subset X$ such that $U \cap \text{Supp}(\mathcal{F})$ is dense in $\text{Supp}(\mathcal{F})$ and $\mathcal{F}'|_ U \cong \mathcal{F}|_ U$.

Proof. This follows from Algebra, Lemmas 10.67.2 and 10.67.3. Note that $U$ can be taken as the complement of the closure of the set of embedded associated points of $\mathcal{F}$. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 31.4: Embedded points

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Your email address will not be published.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 02OL. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 31.4.6 as pdf