The interval neighborhoods in the real Grothendieck groups

Abstract

For a finite dimensional algebra $A$, the TF equivalence on the real Grothendieck group $K_0(\mathrm{proj}A{)}_{\mathbb{R}}$ can be regarded as a completion of the $g$-fan. For example, the silting cones $C^{\circ }(U)$ of 2-term presilting complexes $U$ give the most fundamental family of TF equivalence classes. The next step is studying the TF equivalence classes around each silting cone $C^{\circ }(U)$. Thus, in this paper, we investigate the closed interval neighborhood $D(U)$ of $C^{\circ }(U)$. As our main result, we give a $2^{|U|}:1$ correspondence between the TF equivalence classes in $D(U)$ and those in $K_0(\mathrm{proj}B{)}_{\mathbb{R}}$, where $B$ is the algebra appearing in the $\tau$-tilting reduction at $U$. For this purpose, we give an explicit description of defining inequalities and the faces of $D(U)$ as a polyhedral cone, by using 2-term simple-minded collections and $M$-TF equivalences.

55 pages