Proposition 10.110.5. Let (R, \mathfrak m, \kappa ) be a Noetherian local ring. The following are equivalent
\kappa has finite projective dimension as an R-module,
R has finite global dimension,
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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