Proposition 10.109.5. A Noetherian local ring whose residue field has finite projective dimension is a regular local ring. In particular a Noetherian local ring of finite global dimension is a regular local ring.
By Lemmas 10.109.3 and 10.109.4 we see that $\dim (R) \geq \dim _\kappa (\mathfrak m /\mathfrak m^2)$. Thus the result follows immediately from Definition 10.59.9.
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