2014/04/08 by William Heinzer, Heinzer, William, Christel Rotthaus +3
Computer Science · Mathematics · #13A15 #13B35 #13J10 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1404.2307
openalex publication_date 2014/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 in m, and let R* denote the (y)-adic completion of R. For a finite set w of elements of yR* that are algebraically independent over R, we construct two Krull domains: an intersection domain A that is the intersection of R* with the field of fractions of K[w], and an approximation domain B to A. If R is countable with dim R at least 2, we prove that there exist sets w as above such that the extension R[w] to R*[1/y] is flat. In this case B = A is Noetherian, but may fail to be excellent as we demonstrate with examples. We present several theorems involving the construction. These theorems yield examples where B is properly contained in A and A is Noetherian while B is not Noetherian, and other examples where B = A is not Noetherian.