Remark

In view of cor. 2 of prop. 4 (No. 3), theorem 1 applied to the $A$-module $\kappa_A$ implies that one is in one or the other of the following cases:

(i) one has $\mathrm{dp}_A(\kappa_A) = \mathrm{dh}(A) = +\infty$ ; (ii) one has $\mathrm{dp}_A(\kappa_A) = \mathrm{dh}(A) = \mathrm{prof}(A) < +\infty$.

We shall see later (§ 4, No. 2) that (ii) characterizes regular local rings.