The wall-chamber structures of the real Grothendieck groups

Abstract

Let $A$ be a finite-dimensional algebra over a field $K$. By using stability conditions for modules introduced by King, we can define the wall-chamber structure on the real Grothendieck group $K_0(\mathsf{proj}A{)}_{\mathbb{R}}:=K_0(\mathsf{proj}A){\otimes }_{\mathbb{Z}}\mathbb{R}$, as in the works of Brüstle, Smith and Treffinger, and of Bridgeland. In this article, we explain our result that the chambers in this wall-chamber structure are precisely the open cones associated to the basic 2-term silting objects in the perfect derived category ${\mathsf{K}}^{\mathrm{b}}(\mathsf{proj}A)$. As one of the key steps, we introduce an equivalence relation, called TF equivalence, by using the numerical torsion pairs of Baumann–Kamnitzer–Tingley.