Finite Gorenstein representation type implies simple singularity

Lars Winther Christensen, Greg Piepmeyer, Janet Striuli, Ryo Takahashi

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

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.

Original languageEnglish
Pages (from-to)1012-1026
Number of pages15
JournalAdvances in Mathematics
Volume218
Issue number4
DOIs
StatePublished - Jul 10 2008

Keywords

  • Approximations
  • Cohen-Macaulay representation type
  • Covers
  • Gorenstein dimension
  • Precovers
  • Simple singularity
  • Totally reflexive modules

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