Discrete valuation overrings of a graded Noetherian domain
type: Item
instance of: scholarly article
title: Discrete valuation overrings of a graded Noetherian domain
zbMATH Open document ID: 1400.13007
DOI: 10.1216/JCA-2018-10-1-45
author: Gyu Whan Chang
author: Dong Yeol Oh
published in: Journal of Commutative Algebra
publication date: 2018-05-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1526608934
review text: In this paper, as usual a valuation ring is an intregral domain \(V\) such that, for every \(x\) in the fraction field of \(V\), \(x\) or \(x^{-1}\) belongs to \(V\). Now, a discrete valuation ring (in short DVR) is a valuation ring such that every primary ideal is a power of its radical. The authors do not make the connection between this definition and the usual definition of a DVR (nor they talk about valuations). This paper is mainly about rings and ideals. The purpose of this paper is to study a graded domain analogue of a result about rings which is called the Krull-Akizuki theorem. In the following, \(R\) is a graded integral domain, \(H\) is the set of nonzero homogeneous elements of \(R\), and the homogeneous quotient field of \(R\) is the graded ring \(R_H\) such that the nonzero homogeneous elements of \(R_H\) are the quotients of the nonzero homogeneous elements of \(R\). A homogeneous graded overring of \(R\) is a graded subring of \(R_H\) (i.e.\ generated by homogeneous elements) which contains \(R\). A homogeneous ideal of \(R\) is an ideal which is generated by homogeneous elements. A homogeneous maximal ideal is an ideal which is maximal among proper homogeneous ideals. If \(R\) satisfies the ascending chain condition on homogeneous ideals, then \(R\) is said to be Noetherian; this equivalent to saying that every homogeneous ideal is finitely generated. The set of homogeneous prime ideals of \(R\) is denoted by h-Spec\((R)\). For \(P\) in h-Spec\((R)\), let h-ht\((P)\) be the supremum of the lengths of chains of homogeneous prime ideals descending from \(P\). The h-dimension h-dim\((R)\) of \(R\) is the supremum of the set \(\{\mathrm{h-ht}(R)\mid P\in \mathrm{h-Spec}(R)\}\). Let \(S(H)=R\backslash \bigcup_{P\in \mathrm{h-Spec}(R)}P\). The first result of this paper states that \(R\) is a graded Noetherian domain with h-dimension \(1\) if, and only if, \(R_{S(H)}\) is a one dimensional Noetherian domain. Furthermore, if this holds, then the set of m…
reviewed by: Gerard Leloup
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13E99
zbMATH DE Number: 6875413
zbMATH Keywords: graded Noetherian domain
zbMATH Keywords: homogeneous prime ideal
zbMATH Keywords: discrete valuation overring
MaRDI profile type: Publication
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OpenAlex ID: W2804693187
Wikidata QID: Q129826019
description: scientific article; zbMATH DE number 6875413