Exercise 1, § 3

Let $A$ be a ring, $\mathfrak{p}$ a prime ideal of $A$, $B$ a finitely generated $A$-algebra and let $r_1 \subset \ldots \subset r_m$ be a non-empty increasing sequence of ideals of $B$ lying above $\mathfrak{p}$. Show that there exist an element $a \in A - \mathfrak{p}$ and a finite family $(y_k)_{1 \leq k \leq q}$ of elements of $B$ with the following properties:

(1) If we write $B' = A[y_1, \ldots, y_q]$ and if $S$ is the set of powers of $a$, the ring $S^{-1}B$ is integral over $S^{-1}B'$.

(2) For all $i$ such that $1 \leq i \leq m$ there exists an integer $h(i) \geq 0$ such that $r_i$ contains $y_1, \ldots, y_{h(i)}$ and for every polynomial $P \in A[X_{h(i)+1}, \ldots, X_q]$ the relation $P(y_{h(i)+1}, \ldots, y_r) \in r_i$ implies that all the coefficients of $P$ belong to $\mathfrak{p}$.

The ideal $r_i \cap B'$ is then generated by $\mathfrak{p}$ and the $y_k$ such that $k \leq h(i)$. (Argue by induction on $m$, thus reducing it to the case $m = 1$. Then let $c = r_1a$; argue by induction on the number $n$ of generators of the $A$-algebra $B$. Let $(x_i)_{1 \leq i \leq n}$ be such a system of generators and let $s$ be the ideal of $A[X_1, \ldots, X_n]$ consisting of those $P$ such that $P(x_1, \ldots, x_i) \in c$; then $s \supseteq \mathfrak{p}A[X_1, \ldots, X_n]$; distinguish two cases according to whether $s$ is equal or not to $\mathfrak{p}A[X_1, \ldots, X_n]$. In the second case, there is a polynomial $Q \in A[X_1, \ldots, X_n]$ none of whose coefficients is in $\mathfrak{p}$ and which belongs to $B$. Show that it is possible to determine integers $m_i$ ($2 \leq i \leq n$) such that, if we write $x'_i = Q(x_1, \ldots, x_n), x'_i = x_i - x_1^{m_i}$ ($2 \leq i \leq n$), there exists an element $b \in A - \mathfrak{p}$ for which $bx_1$ is integral over $C = A[x'_2, \ldots, x'_n]$; apply the induction hypothesis to the $A$-algebra

and the ideal $c \cap B_1$.