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We study two classes of torsion classes which generalize functorially finitetorsion classes, that is, semistable torsion classes and morphism torsionclasses. Semistable torsion classes are parametrized by the elements in thereal Grothendieck group up to TF equivalence. We give a close connectionbetween TF equivalence classes and the cones given by canonical decompositionsof the spaces of projective presentations due to Derksen-Fei. More strongly,for $E$-tame algebras and hereditary algebras, we prove that TF equivalenceclasses containing lattice points are exactly the cones given by canonicaldecompositions. One of the key steps in our proof is a general description ofsemistable torsion classes in terms of morphism torsion classes. We also answera question by Derksen-Fei negatively by giving examples of algebras which donot satisfy the ray condition. As an application of our results, we give anexplicit description of TF equivalence classes of preprojective algebras oftype $\widetilde{\mathbb{A}}$.

Semistable torsion classes and canonical decompositions in Grothendieck groups