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 $S$.

Thank you.

  • $\begingroup$ Maybe it helps to know that regular is stable under localization. You can just look at $S_\mathfrak{p}$ over $T_{T\cap\mathfrak{p}}$. The latter ring is local and regular, and the first is finite over the latter. (Note that finitely generated and integral means finite.) $\endgroup$ – user213008 Dec 3 '17 at 15:41
  • $\begingroup$ @user213008 Can you please explain why $S_p$ is finitely generated as $T_ {T\cap p}$ module? $\endgroup$ – user411792 Dec 3 '17 at 15:52
  • $\begingroup$ Sorry, I don't know the details for sure. I can't explain it. $\endgroup$ – user213008 Dec 3 '17 at 16:31

You can see a very partial result here, but probably it will answer your doubts about the difficulty of the question.


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