Corollary 1

Let $A$ be a noetherian ring. The following conditions are equivalent:

(i) one has $\mathrm{dh}(A) < +\infty$ ; (ii) $A$ is regular and one has $\dim(A) < +\infty$.

If these conditions are satisfied, one has $\dim(A) = \mathrm{dh}(A)$.

If the ring $A$ is regular, one has for every maximal ideal $m$ of $A$ the equality $\dim(A_m) = \mathrm{dh}(A_m)$ (No. 1, prop. 1), and therefore

($§ 3$, No. 2, prop. 3 and VIII, $§ 1$, No. 3, prop. 8). On the other hand, if $\mathrm{dh}(A) < +\infty$, the ring $A$ is regular by prop. 4. The corollary follows.

There exist regular noetherian rings of infinite dimension (VIII, $§ 5$, exerc. 6).