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Suppose $R$ is a regular local ring. If $I$ is ideal of height strictly larger than $h$, does $I$ contain a height $h$ prime ideal?

I’m particularly interested in the case of mixed characteristic complete rings.

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  • $\begingroup$ Do you have an example where this fails if R is not regular? $\endgroup$ Oct 12, 2020 at 22:42
  • $\begingroup$ And am I right to think that if I were prime, this would follow from the definition of height? $\endgroup$ Oct 12, 2020 at 22:44
  • $\begingroup$ Might not need regular for all I know, and yes, for primes this would be just from the definition. $\endgroup$
    – Danny
    Oct 12, 2020 at 23:26
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    $\begingroup$ For a counterexample when $R$ is not regular, I guess you could take $R = k[x,y,z]/xy$ and $I = (z)$ which has height 1 but isn't prime. Then the height-0 / minimal primes are $(x)$ and $(y)$, neither of which contains $I$. $\endgroup$ Oct 13, 2020 at 2:34

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