Proposition 4

Let $A$ be a noetherian ring. The following conditions are equivalent: (i) $A$ is regular; (ii) every finitely generated $A$-module has projective dimension $< +\infty$; (iii) for every maximal ideal $m$ of $A$, the projective dimension of $A/m$ is finite; (iv) for every prime ideal $p$ of $A$, the local ring $A_p$ is regular.

Let us prove that (i) implies (ii). Let $M$ be a finitely generated $A$-module. Under hypothesis (i), one has $\mathrm{dp}_{A_m}(M_m) \leq \mathrm{dh}(A_m) < +\infty$ for every maximal ideal $m$ of $A$ (No. 1, prop. 1); therefore $M$ has projective dimension $< +\infty$ ($§ 3$, No. 2, cor. 2 of prop. 3), whence (ii).