Lemma 31.11.10. Let $X$ be a locally Noetherian integral scheme with generic point $\eta $. Let $\mathcal{F}$ be a nonzero coherent $\mathcal{O}_ X$-module. The following are equivalent

$\mathcal{F}$ is torsion free,

$\eta $ is the only associated prime of $\mathcal{F}$,

$\eta $ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has property $(S_1)$, and

$\eta $ is in the support of $\mathcal{F}$ and $\mathcal{F}$ has no embedded associated prime.

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