Corollary 4

Let $\rho : A \to B$ be a homomorphism of noetherian rings, and $a$ an ideal of $A$. We have the inequality

if the map $^a\rho : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ is surjective. If $B$ is a faithfully flat $A$-module, we have $\text{ht}(\rho(a).B) = \text{ht}(a)$.

If $B$ is faithfully flat over $A$, then $^a\rho$ is surjective (II, $§ 2$, No. 5, Cor. 4 to Prop. 11) and we have $\text{ht}(a) \leqslant \text{ht}(\rho(a).B)$ ($§ 2$, No. 1, corollary of Prop. 2).

It remains therefore to prove inequality (20) under the assumption that $^a\rho$ is surjective. Let $p$ be a prime ideal of $A$ such that $a \subset p$ and $\text{ht}(a) = \text{ht}(p)$ ($§ 1$, no 3, prop. 7). Put $\overline{B} = B \otimes_A \kappa(p)$ and denote by $h$ the canonical homomorphism of $B$ into $\overline{B}$. If $X = {}^a\rho^{-1}(p)$ is the nonempty set of prime ideals of $B$ lying over $p$, it is known ($§ 2$, no 1, lemme 1) that the mapping $^ah$ is a homeomorphism of $\mathrm{Spec}(\overline{B})$ onto the subspace $X$ of $\mathrm{Spec}(B)$. Let $q$ be the image under $^ah$ of a minimal prime ideal of $\overline{B}$; one has $\dim(B_q \otimes_A \kappa(p)) = 0$ and Cor. 1 yields the inequality $\dim(B_q) \leqslant \dim(A_p) = \text{ht}(p) = \text{ht}(a)$, finally

whence the corollary.