Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hom-orthogonal modules and brick-Brauer-Thrall conjectures

Kaveh Mousavand, Charles Paquette

TL;DR

This paper develops a Hom-orthogonal framework to study bricks and brick-Brauer-Thrall (bBT) conjectures for finite-dimensional algebras over algebraically closed fields. By relating Hom-orthogonal modules to semibricks of bricks and leveraging geometry of representation varieties and generalized standard AR-quiver components, it derives sharp size bounds, constructs infinite families of bricks of fixed dimension, and verifies several bBT conjectures for broad algebra classes, including those with generalized standard components. Key contributions include a geometric characterization of brick-finite algebras, equivalences among bBT conjectures in the generalized standard setting, and reductions to quotient algebras that extend CKW results to algebras beyond preprojective components, with implications for tame algebras and stability notions. Overall, the work connects Hom-orthogonality, bricks, and representation-theoretic geometry to provide new tools and criteria for understanding brick distribution and the validity of bBT conjectures in a wide range of 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.

Hom-orthogonal modules and brick-Brauer-Thrall conjectures

TL;DR

This paper develops a Hom-orthogonal framework to study bricks and brick-Brauer-Thrall (bBT) conjectures for finite-dimensional algebras over algebraically closed fields. By relating Hom-orthogonal modules to semibricks of bricks and leveraging geometry of representation varieties and generalized standard AR-quiver components, it derives sharp size bounds, constructs infinite families of bricks of fixed dimension, and verifies several bBT conjectures for broad algebra classes, including those with generalized standard components. Key contributions include a geometric characterization of brick-finite algebras, equivalences among bBT conjectures in the generalized standard setting, and reductions to quotient algebras that extend CKW results to algebras beyond preprojective components, with implications for tame algebras and stability notions. Overall, the work connects Hom-orthogonality, bricks, and representation-theoretic geometry to provide new tools and criteria for understanding brick distribution and the validity of bBT conjectures in a wide range of 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.
Paper Structure (8 sections, 18 theorems, 6 equations)

This paper contains 8 sections, 18 theorems, 6 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries and Background
  3. Representation varieties and their components
  4. brick-Brauer-Thrall conjectures
  5. Generalized standard components
  6. Hom-orthogonal modules and brick-finiteness
  7. Orthogonal orbits and irreducible components
  8. Bricks and generalized standard components

Key Result

Lemma 2.1

With the above notations, there always exists a monomorphism Moreover, we get the isomorphism $T_M(\underline{d})/T_M(O_M) \simeq {\rm Ext}^1_A(M,M)$, provided that ${\rm Ext}^2_A(M,M)=0$.

Theorems & Definitions (38)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Conjecture 2.1: Second brick-Brauer-Thrall Conjecture -- 2nd bBT
  • Conjecture 2.2: Semibrick Conjecture
  • Conjecture 2.3: Demonet's Conjecture
  • Conjecture 2.4: Stable brick Conjecture
  • Conjecture 2.5: Stable Second brick-Brauer-Thrall Conjecture -- Stable 2nd bBT
  • Theorem 2.4
  • ...and 28 more