Theorem 1 (Serre)

A noetherian local ring is regular if and only if its homological dimension is finite.

We have seen that a regular noetherian local ring is of finite homological dimension (prop. 1).

Conversely, let $A$ be a noetherian local ring of finite homological dimension $n$; by § 3, No. 3, Corollary 2 of Proposition 4 and No. 5, Theorem 1, one has

If $n = 0$, the $A$-module $\kappa_A$ is free, hence $m_A = 0$ and $A$ is a field. Suppose $n > 0$ and argue by induction on $n$. Since $\mathrm{prof}(A) > 0$, the ideal $m_A$ is not associated to $A$ (§ 1, No. 1, Remark 2), hence is not contained in the union of $m_A^2$ and the ideals associated to $A$ (II, $§ 1$, No. 1, Proposition 2). Consequently (IV, $§ 1$, No. 1, Corollary 2 of Proposition 2), one can find an element $x$ of $m_A - m_A^2$ such that the homothety $x_A$ is injective. Let $B$ denote the noetherian local ring $A/xA$ and consider the sequence of $A$-modules where the mapping $i$ is deduced by passing to quotients from the mapping $a \mapsto ax$ of $A$ into $m_A$, and where $p$ is the canonical surjection; it is exact. Since the class of $x$ in the $\kappa_A$-vector space $m_A/m_A^2$ is non-zero, there exists an $A$-linear mapping $\phi : m_A \to \kappa_A$ with $\phi(x) = 1$; by passing to the quotient, one deduces from $\phi$ a retraction of $i$, so that the preceding exact sequence is split. This implies the relations (Corollary 2 of Proposition 7 of § 3, No. 4 and $A$, X, p. 135, Corollary 1). Corollary 2 of loc. cit. applied to the exact sequence of $B$-modules $0 \to m_B \to B \to \kappa_B \to 0$ implies $\mathrm{dp}_B(\kappa_B) < +\infty$. The ring $B$ is therefore of finite homological dimension (§ 3, No. 3, Corollary 2 of Proposition 4), and of depth $n - 1$ (§ 1, No. 4, Proposition 7 and No. 3, Corollary of Proposition 4). It follows from the induction hypothesis that $B$ is regular, hence that $A$ is regular (VIII, $§ 5$, No. 3, Corollary 1 of Proposition 2).

Consequently, if $A$ is a noetherian local ring, there is an equivalence between the following three properties: (i) $A$ is regular; (ii) the $A$-module $\kappa_A$ is of finite projective dimension; (iii) every finitely generated $A$-module is of projective dimension $< +\infty$.