Corollary 2

Let $A$ be an integrally closed noetherian ring and $B$ a ring containing $A$. Suppose that $B$ is a torsion-free $A$-module, of finite type. If $B$ is a Macaulay ring, the $A$-module $B$ is Macaulay.

In fact, two prime ideals of $B$ lying over the same ideal of $A$ have the same height (VIII, $§ 2$, No. 3, Theorem 2). One can therefore apply Prop. 8 with $N = B$.