Brauer–Thrall for totally reflexive modules

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 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.

On the existence of non-free totally reflexive modules

“…Most of the constructions of totally reflexive modules in the literature start with a pair of exact zero divisors, which can then be used to construct more complicated modules. We are only aware of one example (Proposition 9.1 in [6]) of a ring which admits non-free totally reflexive modules, but does not have exact zero divisors. This example occurs over a characteristic two field, and can be considered a pathological case (the ring defined by the same equations over a field of characteristic different from two will have exact zero divisors).…”

“…More complex totally reflexive modules can be constructed using a pair of exact zero divisor, see [9] and [6].…”

Filtrations of Totally Reflexive Modules

“…Something that is known from [7] is that if there exists a nonfree totally reflexive module over a non-Gorenstein local ring with the residue field of characteristic zero, then there exist infinitely many nonisomorphic indecomposable totally reflexive modules over the ring as well. In fact, in [6] and [10] infinite families of nonisomorphic indecomposable totally reflexive modules are constructed, both of which arise from exact zerodivisors.…”

Brauer–Thrall for Totally Reflexive Modules over Local Rings of Higher Dimension

“…It is quite natural to ask whether there are analogues of the Brauer-Thrall conjectures for totally reflexive modules or not. Motivated by such a question, Christensen, Jorgensen, Rahmati, Striuli and Wiegand [12] proved the following result: (1) There exists a family {M n } n∈N of nonisomorphic indecomposable totally reflexive Rmodules of length ne. Moreover the minimal free resolution of each M n is periodic of period at most 2.…”