Proposition 3

Let $\rho : A \to B$ be a homomorphism of rings. Put $n = \sup_{p \in \mathrm{Spec}(A)} \dim(B \otimes_A \kappa(p))$. One has the inequality

We may suppose $\dim(A) \neq -\infty$ and $n < +\infty$. Let $q_0 \subset ... \subset q_m$ be a chain of prime ideals of B; put $p_i = \rho^{-1}(q_i)$. The sequence of the $p_i$ is increasing, hence the set of its values has cardinal $\leq \dim(A) + 1$. For each $p \in \mathrm{Spec}(A)$, the set of the $q_j$ such that $p_j = p$ is a chain of the subset $^a\rho^{-1}(p)$ of $\mathrm{Spec}(B)$, hence has cardinal less than $\dim(B \otimes_A \kappa(p)) + 1$ (no 1, Lemma 1), and consequently less than or equal to $(n + 1)$. It follows that $m + 1 \leq (\dim(A) + 1)(n + 1)$, whence the proposition.