First neighborhood complete ideals in two-dimensional Muhly local domains
type: Item
instance of: scholarly article
title: First neighborhood complete ideals in two-dimensional Muhly local domains
zbMATH Open document ID: 1163.13004
author: Raymond Debremaeker
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-30T00:00:00Z
review text: Let \((R, \mathfrak m)\) be a two-dimensional Muhly local domain [\textit{H. T. Muhly} and \textit{M. Sakuma}, Trans. Am. Math. Soc. 106, 210--221 (1963; Zbl 0123.03601); \textit{H. T. Muhly}, J. Lond. Math. Soc. 40, 99--107 (1965; Zbl 0123.03701)]. The author studies these rings from the point of view of first neighborhood complete ideals, also known as complete ideals adjacent to the maximal ideal. The study extends the ideas in \textit{S.~Noh} [Commun. Algebra 23, No. 8, 3127--3143 (1995; Zbl 0836.13013)] on two-dimensional regular local rings. The author shows a one-to-one correspondence between the set of immediate quadratic transforms of the ring and the set of first neighborhood complete ideals. He also gives a criterion for the regularity of a two-dimensional Muhly local domain in terms of the set of Rees valuations of the first neighborhood complete ideals. When \(R\) has embedding dimension three, the author characterizes those domains that are rational singularities in terms of their first neighborhood complete ideals. To be specific, he shows that \((R, \mathfrak m)\) is a rational singularity if and only if for each first neighborhood complete ideal \(I\) of \(R\) there exists a corresponding minimal reduction \((x, y)\) of \(\mathfrak m\) such that \(I \mathfrak m = I(x, y)\). Finally, a necessary and sufficient condition for an immediate prime divisor of the ring to possess an asymptotically irreducibly ideal is derived, and this ideal is described in the case that the ring is a rational singularity.
reviewed by: Sandra Spiroff
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5534982
zbMATH Keywords: first neighborhood ideals
zbMATH Keywords: Muhly domains
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.11.002
OpenAlex ID: W2045966588
cites work: On Adjacent Ideals in Two-Dimensional Rational Singularities
cites work: Semifactoriality and Muhly's condition (N) in two dimensional local rings
cites work: Integral closedness of complete-intersection ideals
cites work: Rational singularities, with applications to algebraic surfaces and unique factorization
cites work: Some Multiplicative Properties of Complete Ideals
cites work: On the Existence of Asymptotically Irreducible Ideals
cites work: Powers of simple complete ideals in two-dimensional regular local rings
cites work: Q5829838
cites work: Genera and arithmetic genera of commutative rings
cites work: Q3847037
cites work: On a Theorem of B. Teissier on Multiplicities of Ideals in Local Rings
cites work: Rees Algebras and Mixed Multiplicities
cites work: Q5732721
DOI: 10.1016/J.JPAA.2008.11.002
description: scientific article; zbMATH DE number 5534982