Pseudo-torsion classes

TL;DR

This paper develops a geometric wall-and-chamber framework for torsion classes in module categories, introducing pseudo-torsion classes and analyzing stability paths, extending classical results with new chamber structures.

Contribution

It constructs a novel wall-and-chamber structure for torsion classes, including pseudo-torsion classes, and explores their stratifications via stability paths, enriching classical module theory.

Findings

Defined pseudo-torsion and pseudo-torsionfree classes for each chamber.

Established a correspondence between green paths and Harder-Narasimhan stratifications.

Extended classical torsion theory with new chamber structures and stratifications.

Abstract

For a finite dimensional algebra $\Lambda$, we consider a torsion class $G$ in $mod$-$\Lambda$, which is not necessarily finitely generated. We construct a wall-and-chamber structure for $G$ where the chambers are the connected components of the complement of the union of walls. We also consider ``infinitesimal chambers". To each chamber we associate a ``pseudo-torsion class'' and a ``pseudo-torsionfree class'' and show that they are all distinct. We consider ``green paths'' in the stability space and associate to them Harder-Narasimhan stratifications of $G$. This paper is part of a series of papers whose goal is to study the ``ghosts'' which are remnants of the indecomposable $\Lambda$-modules which do not lie in $G$. In the special case when our torsion class is all of $mod$-$\Lambda$, we are in the classical well-known setting. All of our results apply to this classical setting. The ``pseudo-torsion classes'' are torsion classes. We point out that we do not take the closure of the set of walls. So, we get more chambers and our results are new even in this classical setting.