Lemma 10.110.6. A Noetherian local ring $R$ is a regular local ring if and only if it has finite global dimension. In this case $R_{\mathfrak p}$ is a regular local ring for all primes $\mathfrak p$.

**Proof.**
By Propositions 10.110.5 and 10.110.1 we see that a Noetherian local ring is a regular local ring if and only if it has finite global dimension. Furthermore, any localization $R_{\mathfrak p}$ has finite global dimension, see Lemma 10.109.13, and hence is a regular local ring.
$\square$

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