Definition 10.157.1. Let $R$ be a Noetherian ring. Let $k \geq 0$ be an integer.

1. We say $R$ has property $(R_ k)$ if for every prime $\mathfrak p$ of height $\leq k$ the local ring $R_{\mathfrak p}$ is regular. We also say that $R$ is regular in codimension $\leq k$.

2. We say $R$ has property $(S_ k)$ if for every prime $\mathfrak p$ the local ring $R_{\mathfrak p}$ has depth at least $\min \{ k, \dim (R_{\mathfrak p})\}$.

3. Let $M$ be a finite $R$-module. We say $M$ has property $(S_ k)$ if for every prime $\mathfrak p$ the module $M_{\mathfrak p}$ has depth at least $\min \{ k, \dim (\text{Supp}(M_{\mathfrak p}))\}$.

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