Exercise 9, § 2

Let $A$ be an integral ring and $B$ an integral ring containing $A$ and integral over $A$. Let $p$ be a prime ideal of $A$ such that the integral closure of the ring of fractions $A_p$ is a local ring. For every prime ideal $q$ of $B$ lying over $p$, one has $\mathrm{ht}(q) = \mathrm{ht}(p)$ (use V, § 2, No. 1, Prop. 2).