Exercise 12, § 2

a) Let $A$ and $K$ be as above, $B$ an integral ring containing $A$ and integral over $A$, $L$ the field of fractions of $B$. Let $k \geq 0$ be an integer. If, for every $m$-tuple $(t_1, ..., t_m)$ of elements of $L$, one has $\dim(A[t_1, ..., t_m]) \leq k$, then, for every $m$-tuple of elements $u_1, ..., u_m$ of $L$, one has $\dim(B[u_1, ..., u_m]) \geq k$. b) Let $p$ be a prime ideal of $A$. Prove that if, for all the $m$-tuples $(t_1, ..., t_m)$ of elements of $K$, one has $\dim(A[t_1, ..., t_m]) \leq k$, then, for all the $m$-tuples $(s_1, ..., s_m)$ of elements of the field of fractions of $A/p$, one has the inequality $\dim((A/p)[s_1, ..., s_m]) \leq k - \mathrm{ht}(p)$.