TY - JOUR

T1 - Brauer-Thrall for totally reflexive modules

AU - Christensen, Lars Winther

AU - Jorgensen, David A.

AU - Rahmati, Hamidreza

AU - Striuli, Janet

AU - Wiegand, Roger

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2012/1/15

Y1 - 2012/1/15

N2 - 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.

AB - 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.

KW - Brauer-thrall conjectures

KW - Exact zero divisor

KW - Gorenstein representation type

KW - Maximal cohen-macaulay module

KW - Totally reflexive module

UR - http://www.scopus.com/inward/record.url?scp=82255192010&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2011.09.042

DO - 10.1016/j.jalgebra.2011.09.042

M3 - Article

AN - SCOPUS:82255192010

VL - 350

SP - 340

EP - 373

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -