On the generalized Krull property in power series rings

Abstract

One open problem in commutative algebra and field arithmetic posed by Jarden is whether the power series ring $R〚X〛$ is a generalized Krull domain if R is a generalized Krull domain. Assuming R is a generalized Krull domain, Paran and Temkin proved that $R〚X〛$ is a generalized Krull domain if and only if $R〚X〛$ is a Krull domain. Hence, if R is a generalized Krull domain that is not a Krull domain, then $R〚X〛$ is never a generalized Krull domain. In this paper, we show that the assumption R is a generalized Krull domain in Paran and Temkin's result can be dropped. In other words, $R〚X〛$ is a generalized Krull domain if and only if $R〚X〛$ is a Krull domain and hence if and only if R is a Krull domain.

Introduction

Let R be an integral domain. An overring of R is a ring lying between R and its quotient field. Consider the following properties on a family ${\{R_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ of valuation overrings of R.

• $R={\cap }_{\alpha \in \mathrm{\Lambda }}{R}_{\alpha }$.

• The family ${\{R_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ has finite character, i.e., if $0\ne a\in R$, then a is a unit in all but finitely many $R_{\alpha }$.

• Each $R_{\alpha }$ is essential for R, i.e., $R_{\alpha }$ is the localization of R at $M_{\alpha }\cap R$, where $M_{\alpha }$ is the maximal ideal of $R_{\alpha }$.

• Each $R_{\alpha }$ is one-dimensional.

• Each $R_{\alpha }$ is one-dimensional and discrete.

If there is a family ${\{R_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ of valuation overrings of R satisfying (a), (b), (c), and (d) (respectively, (d')), then R is called a generalized Krull domain (respectively, a Krull domain). By definition every Krull domain is a generalized Krull domain. However, the converse is not true. A one-dimensional nondiscrete valuation domain is a simple example of a generalized Krull domain that is not a Krull domain.

Krull domains play an important role in commutative algebra. They are a generalization of Dedekind domains. A Krull domain is a Dedekind domain if and only if its Krull dimension is at most one. By Mori-Nagata theorem, the integral closure of a Noetherian domain is a Krull domain. Let R be an integral domain and let $R〚X〛$ be the power series ring over R. Gilmer [7], [8] showed that R is a Krull domain if and only if $R〚X〛$ is a Krull domain.

An application of generalized Krull domains appears in Field Arithmetic and Galois Theory. According to Weissauer's theorem, the quotient field of a generalized Krull domain of dimension exceeding one is Hilbertian [6, Theorem 15.4.6]. Further use of the theorem in Field Arithmetic can be found for example in [15], [17]. Jarden asked: Is the power series ring $R〚X〛$ a generalized Krull domain if R is (see [6, Problem 15.5.9 (a)])? A counterexample was constructed by Paran and Temkin [16]. In fact, they proved that if R is a generalized Krull domain, then $R〚X〛$ is a generalized Krull domain if and only if $R〚X〛$ is a Krull domain [16]. In this paper, we show that the concepts “generalized Krull domain” and “Krull domain” are the same for the power series ring $R〚X〛$ without the assumption that R is a generalized Krull domain. Combining this with Gilmer's result, we get $R〚X〛$ is a generalized Krull domain if and only if $R〚X〛$ is a Krull domain if and only if R is a Krull domain. Therefore, this gives a complete answer to the question: “When is the power series ring $R〚X〛$ a generalized Krull domain?”