Let $R$ be a regular local ring of dimension $d$ and let $x_1,x_2,...,x_d$ be a regular system of parameters. Now, for any $y\in R$, the colon ideal $(x_1,x_2,...,x_h):y$ where $h\leq d$ is a prime ideal or the whole ring. I was wondering, if given a prime ideal $P$ in a regular local ring $R$, does there exist a subset of a regular system of parameters, such that $P$ is a colon ideal of the above form?

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    $\begingroup$ Since each ideal generated by a partial system of parameters $(x_1, \dots, x_h)$ is prime, your colon ideal $(x_1, \dots, x_h):y$ is just $(x_1, \dots, x_h)$ again as long as $y \notin (x_1, \dots, x_h)$, and is the ring otherwise. $\endgroup$ – Graham Leuschke Sep 9 '11 at 10:24

The answer is no. It is easy enough to construct counter-examples, but to convince yourself that such a statement is hopeless, here is a salient point.

Any ideal $P$ such that $P$ is a "colon ideal" of a regular system of parameters is such that $R/P$ is itself a regular ring (indeed $R/P$ is of dimension $d-h$ and its maximal ideal is generated by $x_{h+1},\cdots,x_{d}$). On the other hand, Cohen's structure theorem tells you that any equicharacteristic complete noetherian local domain is a quotient of a regular ring by a prime ideal (and quite a bit more than that, of course). So any equicharacteristic complete noetherian local domain which is not regular would provide a counterexample.


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