The Stacks project

Lemma 31.15.5. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor. If $X$ is $(S_ k)$, then $D$ is $(S_{k - 1})$.

Proof. Let $x \in D$. Then $\mathcal{O}_{D, x} = \mathcal{O}_{X, x}/(f)$ where $f \in \mathcal{O}_{X, x}$ is a nonzerodivisor. By assumption we have $\text{depth}(\mathcal{O}_{X, x}) \geq \min (\dim (\mathcal{O}_{X, x}), k)$. By Algebra, Lemma 10.72.7 we have $\text{depth}(\mathcal{O}_{D, x}) = \text{depth}(\mathcal{O}_{X, x}) - 1$ and by Algebra, Lemma 10.60.13 $\dim (\mathcal{O}_{D, x}) = \dim (\mathcal{O}_{X, x}) - 1$. It follows that $\text{depth}(\mathcal{O}_{D, x}) \geq \min (\dim (\mathcal{O}_{D, x}), k - 1)$ as desired. $\square$

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