Proposition 10.110.5. Let $(R, \mathfrak m, \kappa )$ be a Noetherian local ring. The following are equivalent

1. $\kappa$ has finite projective dimension as an $R$-module,

2. $R$ has finite global dimension,

3. $R$ is a regular local ring.

Moreover, in this case the global dimension of $R$ equals $\dim (R) = \dim _\kappa (\mathfrak m/\mathfrak m^2)$.

Proof. We have (3) $\Rightarrow$ (2) by Proposition 10.110.1. The implication (2) $\Rightarrow$ (1) is trivial. Assume (1). By Lemmas 10.110.3 and 10.110.4 we see that $\dim (R) \geq \dim _\kappa (\mathfrak m /\mathfrak m^2)$. Thus $R$ is regular, see Definition 10.60.10 and the discussion preceding it. Assume the equivalent conditions (1) – (3) hold. By Proposition 10.110.1 the global dimension of $R$ is at most $\dim (R)$ and by Lemma 10.110.3 it is at least $\dim _\kappa (\mathfrak m/\mathfrak m^2)$. Thus the stated equality holds. $\square$

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