Corollary 1

Let $\rho : A \to B$ be a homomorphism of Noetherian rings. If $B$ is a finite $A$-algebra and a Macaulay $A$-module, it is a Macaulay ring. If moreover $\rho$ is injective, one has $\mathrm{ht}(aB) = \mathrm{ht}(a)$ for every ideal $a$ of $A$, and $\mathrm{ht}(b) = \mathrm{ht}(\rho^{-1}(b))$ for every ideal $b$ of $B$.

The first assertion results from Prop. 8. Suppose $\rho$ injective. Let $a$ be an ideal of $A$. We have $\mathrm{ht}(a) = \mathrm{prof}_A(a; B)$ since the $A$-module $B$ is Macaulay, with support equal to $\mathrm{Spec}(A)$ (No. 1, corollary to Prop. 1), $\mathrm{ht}(aB) = \mathrm{prof}_B(aB; B)$ (loc. cit.) and $\mathrm{prof}_A(a; B) = \mathrm{prof}_B(aB; B)$ ($§ 1$, No. 3, Prop. 4), whence $\mathrm{ht}(aB) = \mathrm{ht}(a)$. Let $b$ be an ideal of $B$. From the preceding, we have $\mathrm{ht}(\rho^{-1}(b)) = \mathrm{ht}(\rho^{-1}(b)B)$. But $\rho^{-1}(b)B$ is contained in $b$, hence of height less than $\mathrm{ht}(b)$, and we have $\mathrm{ht}(b) \leq \mathrm{ht}(\rho^{-1}(b))$ by VIII, $§ 2$, No. 3, Theorem 1, b).