TY - JOUR

T1 - Finite Gorenstein representation type implies simple singularity

AU - Christensen, Lars Winther

AU - Piepmeyer, Greg

AU - Striuli, Janet

AU - Takahashi, Ryo

N1 - Funding Information:
Keywords: Approximations; Cohen–Macaulay representation type; Covers; Gorenstein dimension; Precovers; Simple singularity; Totally reflexive modules ✩ This work was done while L.W.C. visited University of Nebraska–Lincoln (UNL), partly supported by a grant from the Carlsberg Foundation. Part of it was done during R.T.’s visit to UNL supported by NSF grant DMS 0201904. J.S. was supported by NSF grant DMS 0201904. * Corresponding author. E-mail addresses: lars.w.christensen@ttu.edu (L.W. Christensen), greg@math.missouri.edu (G. Piepmeyer), jstriuli2@math.unl.edu (J. Striuli), takahasi@math.shinshu-u.ac.jp (R. Takahashi). 1 Current address: Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.

PY - 2008/7/10

Y1 - 2008/7/10

N2 - 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.

AB - 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.

KW - Approximations

KW - Cohen-Macaulay representation type

KW - Covers

KW - Gorenstein dimension

KW - Precovers

KW - Simple singularity

KW - Totally reflexive modules

UR - http://www.scopus.com/inward/record.url?scp=43049165781&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2008.03.005

DO - 10.1016/j.aim.2008.03.005

M3 - Article

AN - SCOPUS:43049165781

VL - 218

SP - 1012

EP - 1026

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -