Pseudo-torsion classes

For a finite dimensional algebra $\mathrm{\Lambda }$, we consider a torsion class $G$ in $mod$-$\mathrm{\Lambda }$, 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 $Λ$-modules which do not lie in $G$. In the special case when our torsion class is all of $mod$-$Λ$, 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.