Brauer-Thrall for totally reflexive modules

TL;DR

This paper explores the complexity of totally reflexive modules over certain local rings, demonstrating the existence of infinitely many indecomposable modules with controlled generation properties, extending the Brauer-Thrall conjectures.

Contribution

It provides a detailed classification of totally reflexive modules over rings with m^3=0, constructing infinite families with specific generation and isomorphism properties.

Findings

Constructs a family of indecomposable totally reflexive modules with arbitrary minimal generators.

Shows existence of infinite non-isomorphic modules over algebraically closed residue fields.

Modules have periodic minimal free resolutions of period at most 2.

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 m^3=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 in 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 in N an infinite family of indecomposable and pairwise non-isomorphic totally reflexive R-modules, that are all minimally generated by n elements. The modules in both families have periodic minimal free resolutions of period at most 2.