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 $\mathcal{F}$. $\square$

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