The Stacks project

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 10.60.13 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. $\square$

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