Well-centered overrings of an integral domain

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a simple extension of A, we prove that B is a localization of A if and only if B is flat and well-centered over A. If the integral closure of A is a Krull domain, in particular, if A is Noetherian, we prove that every finitely generated flat well-centered overring of A is a localization of A. We present examples of (non-finitely generated) flat well-centered overrings of a Dedekind domain that are not localizations.Comment: Example 3.11 was replace

Excellent Normal Local Domains and Extensions of Krull Domains

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.Comment: 24 pages to appear in JPA

Generic fiber rings of mixed power series/polynomial rings

Let K be a field, m and n positive integers, and X = {x_1,...,x_n}, and Y = {y_1,..., y_m} sets of independent variables over K. Let A be the polynomial ring K[X] localized at (X). We prove that every prime ideal P in A^ = K[[X]] that is maximal with respect to P\cap A = (0) has height n-1. We consider the mixed power series/polynomial rings B := K[[X]][Y]_{(X,Y)} and C := K[Y]_{(Y)}[[X]]. For each prime ideal P of B^ = C that is maximal with respect to either P \cap B = (0) or P \cap C = (0), we prove that P has height n+m-2. We also prove that each prime ideal P of K[[X, Y]] that is maximal with respect to P \cap K[[X]] = (0) is of height either m or n+m-2.Comment: 28 page

Formal Fibers of Prime Ideals in Polynomial Rings

Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Matsumura has shown that n-1 is the maximal height possible for prime ideals of R^ in the generic formal fiber of R. In this article we prove that every prime ideal of R^ that is maximal in the generic formal fiber of R has height n-1. We also present a related result concerning the generic formal fibers of certain extensions of mixed polynomial-power series rings.Comment: 9 pages to appear in MSRI conference proceeding

Examples of non-Noetherian domains inside power series rings

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice properties from R* such as the Noetherian property. For certain fields L it is possible to approximate A using a localzation B of a nested union of polynomial rings over R associated to A; if B is Noetherian, then B = A. If B is not Noetherian, we can sometimes identify the prime ideals of B that are not finitely generated. We have obtained in this way, for each positive integer s, a 3-dimensional local unique factorization domain B such that the maximal ideal of B is 2-generated, B has precisely s prime ideals of height 2, each prime ideal of B of height 2 is not finitely generated and all the other prime ideals of B are finitely generated. We examine the map Spec A to Spec B for this example. We also present a generalization of this example to dimension 4. We describe a 4-dimensional local non-Noetherian UFD B such that the maximal ideal of B is 3-generated, there exists precisely one prime ideal Q of B of height 3, the prime ideal Q is not finitely generated. We consider the question of whether Q is the only prime ideal of B that is not finitely generated, but have not answered this question.Comment: 32 pages to appear in JC