Proposition 10

Let A be an integrallyclosed Noetherian domain and a an integral ideal of A. The following conditions are equivalent: (a) a is divisorial; (b) the prime ideals associated with A/a are of height 1.

Recall that, if a = $\bigcap^n q_i$ is a reduced primary decomposition of a and p_i denotes the prime ideal corresponding to q_i, the prime ideals associated with A/a are just the p_i (Chapter IV, § 2, no. 3, Proposition 4). The fact that (a) implies (b) then follows from Proposition 8 of no. 4. Conversely, if, in the above notation, the p_i are of height 1, A_{p_i} is a discrete valuation ring (Theorem 4); now, q_i = q_i A_{p_i} ∩ A (Chapter IV, § 2, no. 1, Proposition 3); denoting by v_i the essential valuation corresponding to p_i, there therefore exists an integer n_i such that q_i is the set of x ∈ A such that v_i(x) ≥ n_i; this shows that the q_i are divisorial (no. 4, Proposition 5), hence also is a.