Exercise 7, § 1

Let $K$ be a field and $(v_i)_{i \in I}$ a family of discrete valuations on $K$ satisfying conditions (AK$_I$) and (AK$_{III}$) of no. 3 and such that, for every integer $r$, every family $(n_h)_{1 \leq h \leq r}$ of rational integers and every family $(t_h)_{1 \leq h \leq r}$ of distinct elements of $I$, there exists $x \in K$ such that $v_{t_h}(x) = n_h$ for $1 \leq h \leq r$ and $v_i(x) \geq 0$ for $i$ distinct from the $t_h$. Show that the intersection of the valuation rings of the $v_i$ is a Krull domain $A$ whose field of fractions is $K$ and that $(v_i)_{i \in I}$ is the family of essential valuations of $A$. (Show first that $K$ is the field of fractions of $A$ and hence that $A$ is a Krull domain; then prove that for all $i \in I$ the prime ideal of $A$ consisting of those $x$ such that $v_i(x) > 0$ is necessarily of height 1, arguing by reductio ad absurdum and using Corollary 2 to Proposition 6 of no. 4.)