Exercise 6, § 6

Let $\rho : A \to B$ be an injective homomorphism of noetherian rings, making $B$ an $A$-algebra of finite type. Let $q$ be a prime ideal of $B$ distinct from $B$, and put $p = \rho^{-1}(q)$. a) Now prove that one has, with the notations of the preceding exercise, the inequality

(One may proceed by induction on $d_A(B)$.) b) Now prove that if the $A$-algebra $B$ is generated by $d$ elements and $d_A(B) = d$, there is equality in (). c) Now prove that if $A$ is integral and such that the polynomial ring $A[T_1, ..., T_n]$ is catenary for every $n$, there is equality in () for every integral $A$-algebra of finite type $B$ and every prime ideal $q$ of $B$ distinct from $B$.