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.

We say $\mathcal{F}$ has

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

*depth $k$ at a point*$x$ of $X$ if $\text{depth}(\mathcal{O}_{X, x}) = k$.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 $x \in X$.

We say $X$ has property

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

## Comments (1)

Comment #948 by correction_bot on