Finite Gorenstein representation type implies simple singularity

TL;DR

This paper shows that a finite set of indecomposable totally reflexive modules over a noetherian local ring implies the ring is Gorenstein with a simple singularity, highlighting a deep link between module theory and singularity classification.

Contribution

It establishes a new criterion connecting finite Gorenstein representation type with simple singularities in local rings.

Findings

Finite Gorenstein representation type implies the ring is Gorenstein with a simple hypersurface singularity.

If the residue field has a totally reflexive cover, then the ring is Gorenstein or all totally reflexive modules are free.

The set of indecomposable totally reflexive modules is either singleton or corresponds to a simple singularity.

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.