Exercise 26, § 1

Let $\mathbf{A}$ be an integral domain and $\mathbf{K}$ its field of fractions. A valuation $v$ on $\mathbf{K}$ is called essential for $\mathbf{A}$ if the ring of $v$ is a local ring $\mathbf{A}_p$ at a prime ideal $p$ of $\mathbf{A}$ (the intersection of $\mathbf{A}$ and the ideal of $v$).

(a) Let $v$ be an essential valuation for $\mathbf{A}$ of height 1 and let $p$ be the prime ideal $\mathbf{A}$ consisting of those $x$ such that $v(x) > 0$. Show that $p$ is of height 1. If $x \in \mathbf{K}$ does not belong to $\mathbf{A}_p$, then $\mathbf{A} \cap x^{-1}\mathbf{A} \subset p$. If $w$ is a valuation on $\mathbf{K}$ whose ring $B$ contains $\mathbf{A}$ and whose ideal $q$ satisfies $q \cap \mathbf{A} \subset p$, show that $w$ is equivalent to $v$.

(b) An integrally closed domain $\mathbf{A}$ is said to be of finite character and real type if there exists a family $(v_i)_{i \in I}$ of valuations on $\mathbf{K}$ of height 1 satisfying properties (AK$_{II}$) and (AK$_{III}$). Show that under these conditions, if $z \in \mathbf{K}$ does not belong to $\mathbf{A}$ and $p$ is a prime ideal of $\mathbf{A}$ such that $\mathbf{A} \cap z^{-1}\mathbf{A} \subset p$, there exists $i \in I$ such that $v_i(z) < 0$ and that the prime ideal $q_i$ of $x \in \mathbf{A}$ such that $v_i(x) > 0$ is contained in $p$. (Argue by reductio ad absurdum by considering the finite number of $v_{i_k}$ such that $v_{i_k}(z) < 0$; prove the existence of an $a \in \mathbf{A}$ such that $a \notin p$ and $v_{i_k}(a) > 0$ for all $k$; deduce that $a^n z \notin \mathbf{A}$ for every integer $n > 0$ and show that this implies a contradiction.) Conclude from this that every essential valuation for $\mathbf{A}$ of height 1 is equivalent to one of the $v_i$ (use (a)). Every finite intersection of subdomains of $\mathbf{K}$ which are of finite character and real type is also a domain of finite character and real type.

(c) Suppose that the hypotheses of (b) are satisfied and moreover that all the $v_i$ are essential. Show that, for every prime ideal $p$ of $\mathbf{A}$ of height 1, $\mathbf{A}$, is the ring of one of the valuations $v_i$ (use (b) taking $z^{-1} \in p$). For all $x \in \mathbf{A}$, the principal ideal $Ax$ then admits a unique reduced primary decomposition (Chapter IV, § 2, Exercise 20), the prime ideals corresponding to this decomposition being the prime ideals of height 1 containing $x$.

(d) Suppose that the hypotheses of (b) are satisfied. Let $S$ be a multiplicative subset of $\mathbf{A}$ not containing 0 and $J \subset I$ the set of $t \in I$ such that $v_t(x) = 0$ on $S$. Show that the family $(v_i)_{i \in I - J}$ satisfies properties (AK$_{II}$) and (AK$_{III}$) for the ring $S^{-1}\mathbf{A}$; this family consists of essential valuations for $S^{-1}\mathbf{A}$ if $(v_i)_{i \in I}$ consists of essential valuations for $\mathbf{A}$.

(e) Generalize Proposition 9 of no. 5 to the case where the hypotheses of (c) are satisfied.

(f) Suppose the hypotheses of (b) are satisfied. Let $K'$ be a finite extension of $K$ and $A'$ the integral closure of $A$ in $K'$. Show that the valuations (no two of which are equivalent) on $K'$ which extend the $v_i$ satisfy properties (AK$_{II}$) and (AK$_{III}$) for $A'$ and the latter is therefore a domain of finite character and real type; if the $v_i$ are all essential for $\mathbf{A}$, so are their extensions for $A'$ (argue as in no. 8, Proposition 12 and use also Chapter VI, § 8, no. 3, Remark).

(g) If $\mathbf{A}$ is a domain of finite character and real type, so is $\mathbf{A}[X]$; if the conditionsof (c) are fulfilled, determine the essential valuations for $A[X]$ (argue as in no. 9). Generalize Exercise 8 similarly.