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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 $n = 3$.

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    $\begingroup$ I do not immediately know the answer, but the numerical invariant that definitely behaves well with respect to localization is the "codepth" rather than the depth, cf. EGA IV_2, Proposition 6.11.2 of Auslander. $\endgroup$ – Jason Starr Dec 27 '17 at 16:42
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    $\begingroup$ The result is not true without further hypotheses. Here is a counterexample. Let $A$ be $k[x_1,\dots,x_n,y_1,\dots,y_n]_{\mathfrak{m}}$, for the maximal ideal $\mathfrak{m}=\langle x_1,\dots,x_n,y_1,\dots,y_n\rangle$. Let $\mathfrak{p}$ be $\langle x_1,\dots,x_n\rangle$. Let $M$ be $A/\mathfrak{p}.$ Then $M_{\mathfrak{p}}$ is the residue field of $A_\mathfrak{p}$. This has depth $0$. $\endgroup$ – Jason Starr Dec 27 '17 at 16:49
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    $\begingroup$ @HailongDao. What definition of $S2$ are you using? I am using the definition from EGA IV_2, Definition 5.7.2, p. 103. According to the definition in EGA, the module in my comment is $S2$. $\endgroup$ – Jason Starr Dec 27 '17 at 17:00
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    $\begingroup$ @JasonStarr: yes, indeed there are different notions of $(S_n)$ in the literature, which are radically different. Basically, whether a module should be $(S_n)$ on the whole spectrum of $R$ or just on it's support. I tried to alert people's attention to that issue here: mathoverflow.net/questions/22228/… $\endgroup$ – Hailong Dao Dec 27 '17 at 17:06
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    $\begingroup$ In the words of Shawn Spencer, "I've heard it both ways." :) I agree that Lisa S. should clarify the intended definition. $\endgroup$ – Jason Starr Dec 27 '17 at 17:08
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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? 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.

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