Gorenstein curves and symmetry of the semigroup of values
type: Item
instance of: scholarly article
title: Gorenstein curves and symmetry of the semigroup of values
zbMATH Open document ID: 0692.13017
DOI: 10.1007/BF01258440
author: F. Delgado
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155316
review text: Let \({\mathfrak O}\) be the local ring of a reduced algebroid curve over an algebraically closed field K, and let \(p_ 1,...,p_ d\) be its minimal primes. Then \({\mathfrak O}_ i={\mathfrak O}/p_ i\) is an integral domain and its normalization \(\bar{\mathfrak O}_ i\) is a discrete valuation ring; let \(v_ i\) be the corresponding valuation. The natural injection \({\mathfrak O}\to {\mathfrak O}_ 1\times...\times {\mathfrak O}_ d\) and the \(v_ i\) define an additive subsemigroup S of \({\mathbb{N}}^ d\). The conductor \(\{z\in \bar {\mathfrak O}: z\bar {\mathfrak O}\subset {\mathfrak O}\}\), where \(\bar{\mathfrak O}\) is the integral closure of \({\mathfrak O}\) in its total quotient ring, is a non-zero principal ideal \(\zeta\) \(\bar{\mathfrak O}\) in \(\bar{\mathfrak O}=\bar{\mathfrak O}_ 1\times...\times \bar{\mathfrak O}_ d\), and if \(\zeta =(z_ 1,...,z_ d)\) and \(\delta =(v_ 1(z_ 1),...,v_ d(z_ d))\) then \(\delta+{\mathbb{N}}^ d\subset S\). Set \(\tau=\delta - (1,...,1)\). For \(\alpha\in {\mathbb{Z}}^ d\) set \(\Delta_ i(\alpha)=\{\beta \in S:\quad \beta_ i=\alpha_ i\) and \(\beta_ j>\alpha_ j\) for \(j\neq i\}\) and \(\Delta(\alpha)= \Delta_ 1(\alpha)\cup...\cup \Delta_ d(\alpha)\). The main result of this paper says that \({\mathfrak O}\) is Gorenstein if and only if the semigroup S satisfies the following ``symmetry'' condition: for \(\alpha \in {\mathbb{Z}}^ d\), one has \(\alpha\in S\) if and only if \(\Delta(\tau -\alpha)=\emptyset\). This generalizes the well known result for \(d=1\), due to \textit{E. Kunz} [Proc. Am. Math. Soc. 25, 748-751 (1970; Zbl 0197.314)]. The proof is considerably more complicated than the case of \(d=1\).
reviewed by: H.Matsumura
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 4132313
zbMATH Keywords: Gorenstein curves
zbMATH Keywords: semigroup of values
zbMATH Keywords: reduced curve singularity
zbMATH Keywords: algebroid curve
zbMATH Keywords: conductor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4132313