Finite Gorenstein representation type implies simple singularity

By L. Christensen, Greg Piepmeyer, J. Striuli, Ryo Takahashi. 2007

56 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8155172026ed1386bad07f860da5d2fd026189ec

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Citations in the corpus

On algebras of finite Cohen-Macaulay type (2011, background): “For instance in the recent paper [37], Christensen, Piepmeyer, Striuli and Takahashi proved that if a commutative Noetherian local ring R is of finite CM-type and admits a non-free finitely generated module of Gorenstein dimension zero, then R is Gorenstein and an isolated singularity, which is a si”
Monomorphism categories, cotilting theory, and Gorenstein-projective modules (2011, background): “ [EJ2], Corollary 11.5.3). Determining all the Cohen-Macaulay A-modules and all the Gorenstein-projective A-modules in explicit way, is a basic requirement in applications (see e.g. [AM], [B], [BGS], [CPST], [EJ2], [GZ], [K]). Corollary 4.1. (i) Let Abe an Artin algebra with inj.dimAA<∞. Then CM(”
On G-Regular Local Rings (2008, background): “( 2008 , Theorem B) says that every non-Gorenstein local ring over which there exist only finitely many isomorphism classes of indecomposable totally reflexive modules is a G-regular ring.”