Indecomposable Gorenstein modules of odd rank
type: Item
instance of: scholarly article
title: Indecomposable Gorenstein modules of odd rank
zbMATH Open document ID: 0936.13007
author: C. Rotthaus
author: Dana Weston
author: Roger Wiegand
published in: Journal of Algebra
publication date: 2000-05-14T00:00:00Z
review text: If a Cohen-Macaulay local ring \(R\) has a Gorenstein module, then \(R\) has a unique indecomposable Gorenstein module \(G\) and every Gorenstein \(R\)-module is isomorphic to a direct sum of copies of \(G\). \textit{R. Fossum, H.-B. Foxby, P. Griffith} and \textit{I. Reiten} [Publ. Math., Inst. Hautes Étud. Sci. 45, 193-215 (1975; Zbl 0321.13013)] claimed that an indecomposable Gorenstein module of rank greater than one must have even rank. This note gives a counter-example to this claim by producing a Cohen-Macaulay unique factorization domain having an indecomposable Gorenstein module of arbitrary rank. For this the authors consider the complete normal domain \(A = k[[x^r,x^{r-1}y,\ldots,y^r]]\), \(r \geq 2\), where \(k\) is an algebraically closed field of characteristic zero. By a result of \textit{R. C. Heitmann} [Trans. Am. Math. Soc. 337, No. 1, 379-387 (1993; Zbl 0792.13011)] there is a unique factorization local domain \(R\) such that \(A\) is the completion of \(R\). It is shown that \(R\) has an indecomposable Gorenstein module of rank \(r\) if \(r\) is odd and of rank \(r/2\) if \(r\) is even.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 1275257
zbMATH Keywords: indecomposable Gorenstein module
zbMATH Keywords: Cohen-Macaulay unique factorization domain
reviewed by: Trung Ngo Viet
MaRDI profile type: Publication
OpenAlex ID: W2002835680
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DOI: 10.1006/JABR.1998.7698
description: scientific article; zbMATH DE number 1275257