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

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 $\mathcal{F}'|_ U \cong \mathcal{F}|_ U$.

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