Definition 28.12.1. Let $X$ be a locally Noetherian scheme. Let $k \geq 0$.

We say $X$ is

*regular in codimension $k$*, or we say $X$ has property*$(R_ k)$*if for every $x \in X$ we have\[ \dim (\mathcal{O}_{X, x}) \leq k \Rightarrow \mathcal{O}_{X, x}\text{ is regular} \]We say $X$ has property

*$(S_ k)$*if for every $x \in X$ we have $\text{depth}(\mathcal{O}_{X, x}) \geq \min (k, \dim (\mathcal{O}_{X, x}))$.

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