Corollary

A noetherian local integrally closed ring of dimension $\geqslant 2$ has depth $\geqslant 2$.

Let $A$ be a normal ring (No. 8) and $\mathfrak{p}$ a prime ideal of $A$. Then: a) if $\operatorname{ht}(\mathfrak{p}) = 0$, $A_{\mathfrak{p}}$ is a field; b) if $\operatorname{ht}(\mathfrak{p}) = 1$, $A_{\mathfrak{p}}$ is a discrete valuation ring (VII, § 1, No. 7, Proposition 11); c) if $\operatorname{ht}(\mathfrak{p}) \geqslant 2$, one has $\operatorname{prof}(A_{\mathfrak{p}}) \geqslant 2$ (corollary to Proposition 16).

Conversely: