Bicompact torsion classes and conjectures on brick infinite algebras

Abstract

In the category $\mathrm{mod}A$ of finitely generated modules over a finite dimensional algebra $A$, a torsion class $\mathcal{T}\subset \mathrm{mod}A$ is said to be compact if there exists a module $M\in \mathrm{mod}A$ 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's Lattice Point Conjecture implies Enomoto's Infinite Semibrick Conjecture, both of which are important conjectures on brick infiniteness.

arXiv comment13 pages