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Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$. Let $P$ be a prime ideal of height $d-1$. I want to know if $P^2$ is always a $P$ primary ideal ie if $P/P^2$ is torsion free as $R/P$ module. Thanks.

Let me explain few of my observations. Assume that $R$ is a complete local ring. Then by Cohen's structure theorem $R$ is a power series ring over a DVR. In eqicharacteristic case it is actually a power series over a field. So can we prove the statement in the following special cases.

  1. $R=k[[X_1, X_2, \ldots, X_n]]$.
  2. $R=k[X_1, X_2, \ldots, X_n]_{(X_1, X_2, \ldots, X_n)}$.

Note that if $R/P$ is regular then $P$ is generated by a sequence. If $dimR=2$ then $P$ is principal . So in either cases $P^2$ is $P$ primary. In this problem we have to prove or disprove that $\mathfrak{m} \notin Ass R /P^2$ or $R/P^2$ is one dimensional cohen macaulay ring.

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This is false in general. If $R$ is a $3$ dimensional regular local ring and $P$ is a height $2$ prime ideal which is not a complete intersection then $P^n \neq P^{(n)}$ for $n\geq 2$ which is equivalent to saying none of the powers of $P$ other than $P$ itself are $P$-primary.

First paragraph in the following article for reference

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