Normal \(\mathbb{Z}_ r\)-graded rings and normal cyclic covers
type: Item
instance of: scholarly article
title: Normal \(\mathbb{Z}_ r\)-graded rings and normal cyclic covers
zbMATH Open document ID: 0780.13001
DOI: 10.1007/BF02567764
author: Masataka Tomari
author: Kei-Ichi Watanabe
published in: Manuscripta Mathematica
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/155757
review text: Let \(\bigoplus^{r-1}_{i=0}S_ i\) be a noetherian \(\mathbb{Z}_ r\)-graded domain (i.e. \(S_ iS_ j\subseteq S_ k\) with \(k=i+j \text{mod} r)\). Put \(R=S_ 0\); then each \(S_ i\) turns out to be an \(R\)-module of rank 1 and if \(S\) is normal, then each \(S_ i\) is isomorphic to a divisorial ideal of \(R\). The pair \((S,u)\), for any choice of \(u\neq 0\) in \(S_ 1\), determines a fractional divisor \(D\) of \(R\) with \(rD=\text{div}_ R(u^ r)\); conversely, starting with \(R\) and taking \(f\in R\), \(D\in\text{Div}(R)\otimes_ \mathbb{Z}\mathbb{Q}\) with \(rD=\text{div}_ R(P)\), we may reconstruct a \(\mathbb{Z}_ r\)-graded extension \(S\) of \(R\). Such extensions are the ``cyclic covers'' of \(R\). The authors study relations between properties of \(S\) and \(R\); in particular they look for conditions which make \(S\) normal, Gorenstein or regular. These results are applied to the theory of normal graded rings: if such a ring has an isolated singularity, then for all closed points \(x\in\text{Proj}(R)\), the local ring \(A_ x\) must have a regular cyclic cover; the structure of these covers is determined by the paper.
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 219906
zbMATH Keywords: normal cyclic covers
zbMATH Keywords: noetherian graded domain
zbMATH Keywords: divisorial ideal
zbMATH Keywords: fractional divisor
zbMATH Keywords: normal graded rings
zbMATH Keywords: isolated singularity
MaRDI profile type: Publication
cites work: Q5670698
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description: scientific article; zbMATH DE number 219906