Indecomposable Gorenstein modules of odd rank
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.
Cohen-Macaulay unique factorization domain
scientific article; zbMATH DE number 1275257
scientific article