Excellent normal local domains and extensions of Krull domains

Abstract We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let ( R , m ) be an excellent normal local domain with field of fractions K, let y be a nonzero element of m and let R ⁎ denote the (y)-adic completion of R. For elements τ 1 , … , τ s of y R ⁎ that are algebraically independent over R, we construct two associated Krull domains: an intersection domain A : = K ( τ 1 , … τ s ) ∩ R ⁎ and its approximation domain B; see Setting 2.2 . If in addition R is countable with dim R ≥ 2 , we prove that there exist elements τ 1 , … , τ s , … as above such that, for each s ∈ N , the extension R [ τ 1 , … , τ s ] ↪ R ⁎ [ 1 / y ] is flat; equivalently, B = A and A is Noetherian. Using this result we establish the existence of a normal Noetherian local domain B such that: B dominates R; B has (y)-adic completion R ⁎ ; and B contains a height-one prime ideal p such that R ⁎ / p R ⁎ is not reduced. Thus B is not a Nagata domain and hence is not excellent. We present several theorems involving the construction. These theorems yield examples where B ⊊ A and A is Noetherian while B is not Noetherian; and other examples where B = A and A is not Noetherian.