The Stacks project

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

1. $\mathcal{F}$ is torsion free,

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

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

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