Exercise 8, § 2

Now prove that the polynomial ring $A[X]$ is a faithfully flat $A$-module, that for every maximal ideal $m$ of $A$ one has $\dim(A[X]/mA[X]) = 1$, and deduce from the preceding exercise the existence of ring homomorphisms $\rho : A \to B$ satisfying condition (PM) and such that $\dim(B) > \dim(A) + \inf_{m \in S} \dim(B/mB)$ (where $S$ is the set of maximal ideals of $A$).