Let $A$ be a Cohen-Macaulay local ring, and let $M$ be an $A$-module that is (S$\_2$) and has depth $\ge n$ for some fixed $n$. Let $\mathfrak{p} \subset A$ be a prime of height $\ge n$. Is it true that $\mathrm{depth}\_{A\_{\mathfrak{p}}} M\_{\mathfrak{p}} \ge n$?


I would be happy with a counterexample or a proof even in the special case $n = 3$.



Depth under localization over a Cohen-Macaulay ring
Here is a counter-example when $A$ is regular local of dimension $4$ and $n=3$, the first non-trivial case. Let $A = k[[x,y,z,t]]$, and $P$ be a prime ideal of height $3$, say $P=(x,y,z)$. Let $M=\Omega P$, the syzygy of $P$. Obviously, $M$ is also $\Omega^2 (A/P)$.


The depth of $M$ at various localizations can be computed easily by tracking depth along short exact sequences. In particular, $M$ is $(S\_2)$ since any second syzygy module over $A$ would be. Since $depth(A/P)=1$, it follows that $depth(M)=1+2=3$. On the other hand, when you localize at $P$, $M\_P$ is a second syzygy of the residue field in a $3$-dimensional regular local ring, so $depth(M\_P)=2$. 


PS: as can be seen in the comments below the question, there are different definitions of $(S\_n)$ in the literature, see [What is Serre's condition (S\_n) for sheaves?](https://mathoverflow.net/questions/22228/what-is-serres-condition-s-n-for-sheaves)  However, the notion I assumed for this answer is the most restrictive one ((1) in the one cited), so it applies to others as well. 


