Exercise 5, § 6

Let $\rho : A \to B$ be an injective homomorphism of noetherian rings making $B$ a finitely generated $A$-algebra. We call degree of transcendence of $B$ over $A$, and denote it by $d_A(B)$, the largest integer $d$ such that there exists an injective homomorphism of $A$-algebras $b : A[X_1, ..., X_d] \to B$. (a) Prove that if $\sigma : B \to C$ satisfies the same hypotheses as $\rho$, one has the inequality

(b) Prove that if $A$ and $B$ are integral, $d_A(B)$ is equal to the degree of transcendence of the field of fractions of $B$ over that of $A$. Moreover if in $a$ one supposes $C$ integral, one has the equality $d_A(C) = d_A(B) + d_B(C)$. (c) Prove that if $B$ is a subalgebra of $A[v, v^{-1}]$ containing $A[v]$, where $v$ is an indeterminate, one has $d_A(B) = 1$.