Brauer-Thrall for totally reflexive modules over local rings of higher dimension

TL;DR

This paper extends Brauer-Thrall theorems to totally reflexive modules over certain local rings with specific zero-divisor pairs, showing the existence of infinitely many indecomposable modules of various multiplicities.

Contribution

It proves the Brauer-Thrall theorems for totally reflexive modules over local rings with particular zero-divisor conditions, a novel extension in the theory.

Findings

Existence of infinitely many indecomposable totally reflexive modules of various multiplicities.

For infinite residue fields, infinitely many isomorphism classes of such modules exist.

The results apply under specific conditions on zero-divisors and the dimension of the quotient ring.

Abstract

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive $R$-modules of multiplicity $n$.