Excellent normal local domains and extensions of Krull domains
/ Authors
/ Abstract
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.
Journal: Journal of Pure and Applied Algebra