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Let $R$ be a regular local ring and let $P$ be a prime ideal of height $h$ in $R$. Is it always the case that $P$ contains a regular sequence of lenght $h$?

This is clear if $h$ is $0,1$ or $\dim R$.

Since $R$ is regular, the localization $R_P$ is regular local of dimension $h$, and thus contains an $R_P$-regular sequence $f_1,\dots,f_h$, which can be chosen to live in $R$. However, it is a priori not clear that $f_1,\dots,f_h$ is a regular sequence in $R$.

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The answer is yes. In fact, more generally, if $R$ is a Cohen-Macaulay ring, then the height of any prime ideal in it is equal to its depth, which is just the length of a maximal regular sequence contained in P. See for example Theorem 2.1.2 of the book Cohen-Macaulay rings by Bruns and Herzog.

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  • $\begingroup$ Thank you very much! Did you mean to write "grade on $R$" instead of "depth"? $\endgroup$
    – Uriya First
    Commented Jun 8, 2020 at 7:08
  • $\begingroup$ The grade of an ideal is often called the depth of an ideal. $\endgroup$
    – Liran Shaul
    Commented Jun 8, 2020 at 7:42

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