Hom-orthogonal modules and brick-Brauer-Thrall conjectures

TL;DR

This paper investigates Hom-orthogonal modules over finite-dimensional algebras, providing new conditions for the existence of infinite families of bricks, and proves some brick-Brauer-Thrall conjectures for specific algebra classes using algebraic and geometric methods.

Contribution

It introduces necessary and sufficient conditions for infinite families of bricks and verifies the brick-Brauer-Thrall conjectures for algebras with certain Auslander-Reiten quiver components.

Findings

Characterization of infinite families of bricks via Hom-orthogonality

Verification of bBT conjectures for algebras with generalized standard components

New algebraic and geometric criteria for brick-finite algebras

Abstract

For finite dimensional algebras over algebraically closed fields, we study the sets of pairwise Hom-orthogonal modules and obtain new results on some open conjectures on the behaviour of bricks and several related problems, which we generally refer to as brick-Brauer-Thrall (bBT) conjectures. Using some algebraic and geometric tools, and in terms of the notion of Hom-orthogonality, we find necessary and sufficient conditions for the existence of infinite families of bricks of the same dimension. This sheds new light on the bBT conjectures and we prove some of them for new families of algebras. Our results imply some interesting algebraic and geometric characterizations of brick-finite algebras as conceptual generalizations of local algebras. We also verify the bBT conjectures for any algebra whose Auslander-Reiten quiver has a generalized standard component, which particularly extends some results of Chindris-Kinser-Weyman on the algebras with preprojective components.