Finite Gorenstein representation type implies simple singularity
Commutative Algebra
2008-02-22 v2 Representation Theory
Abstract
Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.
Cite
@article{arxiv.0704.3421,
title = {Finite Gorenstein representation type implies simple singularity},
author = {Lars Winther Christensen and Greg Piepmeyer and Janet Striuli and Ryo Takahashi},
journal= {arXiv preprint arXiv:0704.3421},
year = {2008}
}