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

Abstract

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) ≥ 2 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.