Semistable torsion classes and canonical decompositions

We study two classes of torsion classes which generalize functorially finite
torsion classes, that is, semistable torsion classes and morphism torsion
classes. Semistable torsion classes are parametrized by the elements in the
real Grothendieck group up to TF equivalence. We give a description of TF
equivalence classes in terms of the canonical decompositions of the spaces of
projective presentations. We also give a description of semistable torsion
classes in terms of morphism torsion classes. Moreover, we prove a
stabilization property of the rigid part of elements in the Grothendieck group.
As an application of our results, we give explicit descriptions of TF
equivalence classes of path algebras of extended Dynkin quivers, preprojective
algebras of type $\widetilde{\mathbb{A}}$, and a gentle algebra of rank $3$.