Brauer–thrall for Totally Reflexive Modules

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 nonfree 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.