According to this wikipidea link https://en.m.wikipedia.org/wiki/Cohen–Macaulay_ring : Let $R$ be a local ring which is finitely generated as a module over some regular local ring $A$ contained in $R$. Such a subring exists for any localisation $R$ at a prime ideal of a finitely generated algebra over a field by the Noether normalisation lemma.
I want to know why such a subring exists. By Noether normalisation if $S$ is a finitely generated $k$ algebra then $S$ is integral over say $T=k[X_1,\cdots ,X_n]$. Now $T$ is a regular ring. But how $S_p$ will be finitely generated over some regular local ring, where $p$ is a prime ideal in $S$.