Brauer-Thrall for totally reflexive modules

Lars Winther Christensen, David A. Jorgensen, Hamidreza Rahmati, Janet Striuli, Roger Wiegand

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

Let R be a commutative noetherian local ring that is not Gorenstein. It is known that the category of totally reflexive modules over R is representation infinite, provided that it contains a non-free module. The main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation.Local rings (R,m) with m3=0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics. For such rings we obtain conclusive results about the category of totally reflexive modules, modeled on the Brauer-Thrall conjectures. Starting from a non-free cyclic totally reflexive module, we construct a family of indecomposable totally reflexive R-modules that contains, for every n∈N, a module that is minimally generated by n elements. Moreover, if the residue field R/m is algebraically closed, then we construct for every n∈N an infinite family of indecomposable and pairwise non-isomorphic totally reflexive R-modules, each of which is minimally generated by n elements. The modules in both families have periodic minimal free resolutions of period at most 2.

Original languageEnglish
Pages (from-to)340-373
Number of pages34
JournalJournal of Algebra
Volume350
Issue number1
DOIs
StatePublished - Jan 15 2012

Keywords

  • Brauer-thrall conjectures
  • Exact zero divisor
  • Gorenstein representation type
  • Maximal cohen-macaulay module
  • Totally reflexive module

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