Exercise 5, § 4

Let $A$ be a noetherian local ring of finite homological dimension; deduce from Exercise 16 of $A$, X, p. 208 another proof of the fact that $A$ is regular (observe that one has $\dim_{\kappa_A} m_A/m_A^2 \leqslant \mathrm{dh}(A) \leqslant \operatorname{prof}(A)$).