Theorem 1 (Auslander-Buchsbaum)

Let $A$ be a noetherian local ring and $M$ a finitely generated $A$-module of finite projective dimension. Then one has the equality

Let us reason by induction on $\mathrm{dp}_A(M)$.

a) If $\mathrm{dp}_A(M)$ is zero, $M$ is a nonzero finitely generated free $A$-module; its depth is equal to $\mathrm{prof}(A)$ (§ 1, No. 1, remark 4).

b) Suppose $\mathrm{dp}_A(M) = 1$ and choose a minimal projective resolution

of $M$ (No. 3, prop. 4, (iv)). The $A$-modules $L_0$ and $L_1$ are nonzero finitely generated free modules, hence of depth $\mathrm{prof}(A)$ (§ 1, No. 1, remark 4). The mapping $1_{\kappa_A} \otimes d_1 : \kappa_A \otimes_A L_1 \to \kappa_A \otimes_A L_0$ is zero, which implies that $d_1$ belongs to $m_A \mathrm{Hom}_A(L_1, L_0)$. By remark 5 of § 1, No. 1, one has $\mathrm{prof}_A(M) = \mathrm{prof}(A) - 1$.

c) Suppose $\mathrm{dp}_A(M) > 1$. Choose an exact sequence

where $L$ is a finitely generated free $A$-module. One then has $\mathrm{prof}_A(L) = \mathrm{prof}(A)$ (§ 1, No. 1, remark 4), $\mathrm{dp}_A(N) = \mathrm{dp}_A(M) - 1$ (A, X, p. 135, cor. 2 c)), whence $\mathrm{prof}_A(N) = \mathrm{prof}(A) - \mathrm{dp}_A(N)$ (induction hypothesis), and in particular $\mathrm{prof}_A(N) < \mathrm{prof}_A(L)$. By prop. 1 of § 1, No. 1, one then has $\mathrm{prof}_A(M) = \mathrm{prof}_A(N) - 1$, which completes the proof.