Bicompact torsion classes and conjectures on brick infinite algebras

TL;DR

This paper introduces bicompact torsion classes in module categories of finite dimensional algebras, conjectures their equivalence to functorially finite torsion classes, and explores implications for brick infinite algebras.

Contribution

It proposes a new class of torsion classes called bicompact, conjectures their equivalence to functorially finite classes, and proves this for certain algebra types.

Findings

Bicompact torsion classes are conjectured to be the same as functorially finite torsion classes.

The conjecture is proven for hereditary algebras.

Implications for conjectures on brick infinite algebras are discussed.

Abstract

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that $\mathcal{T}$ is the smallest torsion class containing $M$. If a torsion class satisfies this and the dual condition, then we call it a bicompact torsion class. We conjecture that bicompact torsion classes are precisely functorially finite torsion classes, and prove it for hereditary algebras and also for semistable torsion classes. This gives that Demonet Conjecture implies Enomoto Conjecture, both of which are important conjectures on brick infiniteness.