Lemma 31.4.6 (02OL)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.4: Embedded points
Lemma 31.4.6 (cite)

numberstags

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

$\text{Supp}(\mathcal{F}') = \text{Supp}(\mathcal{F})$ ,
$\mathcal{F}'$ has no embedded associated points, and
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

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).

Comments (0)

Post a comment