On unique and almost unique factorization of complete ideals. II
type: Item
instance of: scholarly article
title: On unique and almost unique factorization of complete ideals. II
zbMATH Open document ID: 0715.13012
DOI: 10.1007/BF01388844
author: Steven Dale Cutkosky
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143720
review text: Let (R,m) be a complete normal local domain of dimension \( 2.\) The set of complete m-primary ideals m(R) of R has a natural semigroup structure. This paper generalizes results obtained by the same author in part I of this paper [Am. J. Math. 111, No.3, 417-433 (1989; Zbl 0697.13007)] in the case where R/m is algebraically closed. - Consider the following properties of R: (a1) m(R) has unique factorization; (a2) R is a UFD; (b1) R satisfies condition (N); (b2) Cl(R) is a torsion group; (b3) m(R) is semifactorial. In the present paper (theorem 5) it is proved that \((a1)\Rightarrow(a2)\), and an example is given to show that the implication \((a2)\Rightarrow(a1)\) is false. (However Lipman has shown that \((a2)\Rightarrow(a1)\) is true if \(R/m\) is an algebraically closed field.) The implications \((b1)\Rightarrow(b3)\) were proved by \textit{H. Göhner} [J. Algebra 24, 403-429 (1975; Zbl 0308.13023)]. Theorem 4 of the present paper shows that \((b1)\Leftrightarrow(b2)\). Condition (N) was introduced by \textit{H. T. Muhly} and \textit{M. Sakuma} in J. Lond. Math. Soc. 38, 341-350; 494 (1963; Zbl 0142.288): R satisfies condition (N) if any proper birational morphism \(X\to Spec(R)\) factors as a sequence of blow-ups of height 2 ideals with irreducible exceptional divisors. The paper concludes with an example of a complete normal local domain of dimension 2 such that the normalization of the blow-up of any m-primary ideal has reducible exceptional divisor. This answers a question of J. Sally.
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 4177233
zbMATH Keywords: complete primary ideals
zbMATH Keywords: complete normal local domain
zbMATH Keywords: UFD
zbMATH Keywords: birational morphism
zbMATH Keywords: blow-ups
reviewed by: Leslie G. Roberts
MaRDI profile type: Publication
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cites work: Q3973362
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cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
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description: scientific article; zbMATH DE number 4177233