Lemma 10.110.8. Let $R$ be a Noetherian ring. The following are equivalent:

1. $R$ has finite global dimension $n$,

2. $R$ is a regular ring of dimension $n$,

3. there exists an integer $n$ such that all the localizations $R_{\mathfrak m}$ at maximal ideals are regular of dimension $\leq n$ with equality for at least one $\mathfrak m$, and

4. there exists an integer $n$ such that all the localizations $R_{\mathfrak p}$ at prime ideals are regular of dimension $\leq n$ with equality for at least one $\mathfrak p$.

Proof. This follows from the discussion above. More precisely, it follows by combining Definition 10.110.7 with Lemma 10.110.2 and Proposition 10.110.5. $\square$

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