Excellent normal local domains and extensions of Krull domains

• William Heinzer ^[1] ; Christel Rotthaus ^[2] ; Sylvia Wiegand ^[3]
  1. [1] Purdue University

     Purdue University

     Township of Wabash, Estados Unidos

     
  2. [2] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
  3. [3] University of Nebraska (USA)

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• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 3 ((March 2015)), 2015 (Ejemplar dedicado a: Special Issue in honor of Prof. Hans-Bjørn Foxby), págs. 510-529
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2014.05.010
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• Resumen

  • We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let (R,m)(R,m) be an excellent normal local domain with field of fractions K, let y be a nonzero element of m and let R⁎R⁎ denote the (y)-adic completion of R . For elements τ1,…,τsτ1,…,τs of yR⁎yR⁎ that are algebraically independent over R , we construct two associated Krull domains: an intersection domain A:=K(τ1,…τs)∩R⁎A:=K(τ1,…τs)∩R⁎ and its approximation domain B; see Setting 2.2.

    If in addition R is countable with dimR≥2, we prove that there exist elements τ1,…,τs,…τ1,…,τs,… as above such that, for each s∈Ns∈N, the extension R[τ1,…,τs]↪R⁎[1/y]R[τ1,…,τs]↪R⁎[1/y] is flat; equivalently, B=AB=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⁎R⁎; and B contains a height-one prime ideal p such that R⁎/pR⁎R⁎/pR⁎ 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⊊AB⊊A and A is Noetherian while B is not Noetherian; and other examples where B=AB=A and A is not Noetherian