Pseudo-torsion classes

Abstract

For a finite dimensional algebra $\Lambda$, we consider a torsion class $G$ in $mod$-$\Lambda$, which is not necessarily functorially finite. 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 call these ``pseudo-chambers''. We also consider ``infinitesimal chambers". To each pseudo-chamber $U$ we associate a ``pseudo-torsion class'' $P(U)$ and a ``pseudo-torsionfree class'' $Q(U)$ and show that they determine $U$. Furthermore, we show that if both $P(U)$ and $Q(U)$ are finitely generated, then $U$ is an open set. 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 pseudo-chambers and our results are new even in this classical setting.

arXiv comment42 pages, 7 figures. Version 2: Updated figures to include full stability diagrams. Two new sections on functorially finite pseudo-torsion classes and connections with Algebraic K-theory