The Stacks project

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

1. We say $X$ is regular in codimension $k$, or we say $X$ has property $(R_ k)$ if for every $x \in X$ we have

   \[ \dim (\mathcal{O}_{X, x}) \leq k \Rightarrow \mathcal{O}_{X, x}\text{ is regular} \]

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