Indecomposable Gorenstein Modules of Odd Rank

Abstract

Let (R,m) be a local Cohen–Macaulay ring withm-adic completionR̂. AGorenstein R-module is a non-zero finitely generatedR-module whosem-adic completion is isomorphic to a direct sum of copies of the canonical module ωR̂. Therankof the Gorenstein moduleGis the positive integerrsuch thatĜ≅r·ωR̂(the direct sum ofrcopies of ωR̂). In this note we show that for any given positive integerrthere is a Cohen–Macaulay ringRwith an indecomposable Gorenstein moduleGof rankr.