Corollary

For every ideal a of A, one has $\mathrm{ht}(a) \leq \mathrm{ht}(\rho(a)B)$.

Let q be a prime ideal of B containing $\rho(a)B$, and $p = \rho^{-1}(q)$. According to prop. 1, the local homomorphism $\rho_q : A_p \to B_q$ deduced from $\rho$ satisfies (PM), and we therefore have $\dim(A_p) \leq \dim(B_q)$ according to prop. 2. According to prop. 7 of § 7, no 3, we have $\mathrm{ht}(a) \leq \dim(A_p)$ since p contains a, whence $\mathrm{ht}(a) \leq \dim(B_q)$ for every prime ideal q of B containing $\rho(a)B$. The corollary then follows from loc. cit.