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
SN - 0001-8708
VL - 218
SP - 1012
EP - 1026
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 4
ER -