The Stacks project

Lemma 31.12.11. Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent reflexive $\mathcal{O}_ X$-module. Let $x \in X$.

  1. If $\text{depth}(\mathcal{O}_{X, x}) \geq 2$, then $\text{depth}(\mathcal{F}_ x) \geq 2$.

  2. If $X$ is $(S_2)$, then $\mathcal{F}$ is $(S_2)$.

Proof. Omitted. See More on Algebra, Lemma 15.23.16. $\square$

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