The Stacks project

Definition 30.11.1. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $k \geq 0$ be an integer.

1. We say $\mathcal{F}$ has depth $k$ at a point $x$ of $X$ if $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) = k$.

2. We say $X$ has depth $k$ at a point $x$ of $X$ if $\text{depth}(\mathcal{O}_{X, x}) = k$.

3. We say $\mathcal{F}$ has property $(S_ k)$ if

   \[ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) \geq \min (k, \dim (\text{Supp}(\mathcal{F}_ x))) \]

   for all $x \in X$.

4. We say $X$ has property $(S_ k)$ if $\mathcal{O}_ X$ has property $(S_ k)$.