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.

Proof. This is a translation of More on Algebra, Lemma 15.22.8 into the language of schemes. We omit the translation. $\square$

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