Semistable torsion classes and canonical decompositions in Grothendieck groups (Q6639749)

Derived categories are basic in homological algebra and appear in many branches of mathematics, and tilting theory is a powerful tool to study equivalences of the derived categories. There are two important notions in tilting theory: tilting/silting complexes and \(t\)-structures. Two rings \(A\) and \(B\) are derived equivalent if and only if there exists a tilting complex of \(A\) whose endomorphism ring is isomorphic to \(B\). The class of silting complexes is a generalization of the class of tilting complexes from the point of view of mutation, which is a categorical operation to construct a new silting complex from a given one by replacing a direct summand. A \(t\)-structure is a pair of two full subcategories satisfying certain axioms, and intermediate \(t\)-structures correspond bijectively with torsion classes in the module category.\N\NS. Asai and O. Iyama study two classes of torsion classes that generalize functorially finite torsion classes, i.e., semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. They give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations. For \(E\)-tame algebras and hereditary algebras, the authors prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. They also answer a question by Derksen-Fei negatively by giving examples of algebras that do not satisfy the ray condition. As an application, Asai-Iyama give an explicit description of TF equivalence classes of preprojective algebras of type \(\widetilde{\mathbb{A}}\).