Finite Gorenstein representation type implies simple singularity

2007/04/25 by Christensen, Lars Winther, Piepmeyer, Greg, Striuli, Janet +1 · 4 citations
#13C14 #14B05 #18G25 #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.0704.3421

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.

Cited by

• Gorenstein-Projective Modules over the Ring of Dual Integers
• On local rings of finite syzygy representation type
• Arf rings, simple singularities and reflexive modules
• Claus Michael Ringel's main contributions to Gorenstein-projective modules

Related

• Totally reflexive modules over rings that are close to Gorenstein
• Existence of Totally Reflexive Modules via Gorenstein Homomorphisms
• On the number of indecomposable totally reflexive modules
• On modules whose dual is of finite Gorenstein dimension
• On local rings of finite syzygy representation type