Exercise 3, § 5

Let $A$ be a local noetherian ring. We propose to prove that $A$ is regular if and only if $\mathrm{dh}(A)$ is finite. a) Suppose $\mathrm{dh}(A)$ is finite. To prove that $A$ is regular, use A, X, p. 208, exerc. 13. b) Suppose $A$ is regular. Let $x$ be a completely secant family generating $m_A$. Then, the Koszul complex corresponding to $x$ is a finite resolution of $A/m_A$ by free $A$-modules. Apply then A, X, p. 203, exerc. 16.