Characterization of prime ideals in regular local rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:36:13Z http://mathoverflow.net/feeds/question/74974 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74974/characterization-of-prime-ideals-in-regular-local-rings Characterization of prime ideals in regular local rings Satyajit Sahu 2011-09-09T05:28:39Z 2011-09-09T07:47:21Z <p>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?</p> http://mathoverflow.net/questions/74974/characterization-of-prime-ideals-in-regular-local-rings/74981#74981 Answer by Olivier for Characterization of prime ideals in regular local rings Olivier 2011-09-09T07:47:21Z 2011-09-09T07:47:21Z <p>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. </p> <p>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. </p>