Lemma 10.106.3. Let $R$ be a regular local ring and let $x_1, \ldots , x_ d$ be a minimal set of generators for the maximal ideal $\mathfrak m$. Then $x_1, \ldots , x_ d$ is a regular sequence, and each $R/(x_1, \ldots , x_ c)$ is a regular local ring of dimension $d - c$. In particular $R$ is Cohen-Macaulay.
Proof. Note that $R/x_1R$ is a Noetherian local ring of dimension $\geq d - 1$ by Lemma 10.60.12 with $x_2, \ldots , x_ d$ generating the maximal ideal. Hence it is a regular local ring by definition. Since $R$ is a domain by Lemma 10.106.2 $x_1$ is a nonzerodivisor. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.