Let $R$ be a CM ring and $P$ be a prime ideal. Let $P^{(n)}$ denote the $n$th symbolic power of $P$. Is the sequence $\operatorname {depth} (R/P^{(n)})$ eventually constant?
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$\begingroup$ Remark:1. If instead of symbolic power one considers the ordinary powers then the answer is affirmative by a theorem of Brodmann, for instance if for any prime $Q$ strictly containing $P$, $\ell(P_Q)<ht(Q)$ (a Theorem of Huneke-Huckaba). Remark2. Brodmann's result follows from the fact that for any f.g module $M$, Ass$(M/P^nM)$ is eventually the same for any $n$. So that a negative answer to the question is tantamount to the existence of a f.g. module $M$ such that Ass$(M/P^{(n)}M)$ is NOT eventually the same for any $n$. $\endgroup$– S.Hamid HassanzadehCommented Sep 27, 2016 at 21:22
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$\begingroup$ Remark 3. The answer of the question is negative if one does not assume that $P$ is a prime. A.K.Singh (TAMS 2003) presents an example of a CM ring whose canonical cover is Noetherian but not CM, which means that the depth of the symbolic powers of the canonical ideal(=module) is periodic and not the same for all. As far as I checked in Macaualay2, Singh's ideal is not primary. $\endgroup$– S.Hamid HassanzadehCommented Sep 27, 2016 at 21:22
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