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Finite Gorenstein representation type implies simple singularity |
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| Authors: | Lars Winther Christensen Janet Striuli Ryo Takahashi |
| Institution: | a Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA b Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA c Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan |
| 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. |
| Keywords: | 14B05 18G25 13C14 |
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