Lemma 10.106.7. Suppose $R$ is a Noetherian local ring. Let $x \in \mathfrak m$ be a nonzerodivisor such that $R/xR$ is a regular local ring. Then $R$ is a regular local ring. More generally, if $x_1, \ldots , x_ r$ is a regular sequence in $R$ such that $R/(x_1, \ldots , x_ r)$ is a regular local ring, then $R$ is a regular local ring.

Proof. This is true because $x$ together with the lifts of a system of minimal generators of the maximal ideal of $R/xR$ will give $\dim (R)$ generators of $\mathfrak m$. Use Lemma 10.60.13. The last statement follows from the first and induction. $\square$

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