I have a question about definition of $depth$ of a ring in view of the Serre's criterion for normality:
https://en.wikipedia.org/wiki/Serre%27s_criterion_for_normality
Let $A$ be a commutative ring. $A$ fulfills the condition $S_k$ iff
$$ S_{k}:\operatorname {depth}A_{{{\mathfrak {p}}}}\geq \inf\{k,\operatorname {ht}({\mathfrak {p}})\} $$
for every prime ideal $\mathfrak {p}$.
My question is quite simple: the $\operatorname {depth}A_{{{\mathfrak {p}}}}$ of the localized ring $A_{\mathfrak p}$ is considered with respect to which ideal? Also $\mathfrak {p}$?
I'm only familiar with with term "depth" as introduced here
https://stacks.math.columbia.edu/tag/00LE
Here we consider an ideal $I \subset A$ of a ring and a $A$-module $M$. Then $depth_I(M)$ with respect to $I$ is defined as the supremum of all lengths of $M$-regular sequences in $I$ (see def. in https://stacks.math.columbia.edu/tag/00LF).
The problem is with respect which ideal $I$ and a $A$-module $M$ is defined $$\operatorname {depth}A_{{{\mathfrak {p}}}}\ ?$$
Intuitively I guess $I = \mathfrak{p}$ and $M = A_p$ as $A$-module? Is it true?