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 $R$.



Do height $h$ prime ideals in regular local rings contain regular sequences of length $h$?
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.


