The interval neighborhoods in the real Grothendieck groups

TL;DR

This paper explores the structure of TF equivalence classes in the real Grothendieck group of a finite dimensional algebra, focusing on neighborhoods around silting cones and their relation to $ au$-tilting reduction.

Contribution

It provides a detailed description of the interval neighborhoods of silting cones and establishes a 2^{|U|}:1 correspondence with TF equivalence classes in a reduced algebra.

Findings

Explicit polyhedral description of the neighborhood $D(U)$ as a cone.

A 2^{|U|}:1 correspondence between classes in $D(U)$ and a reduced algebra.

Characterization of faces and inequalities of $D(U)$ using simple-minded collections.

Abstract

For a finite dimensional algebra $A$, the TF equivalence on the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ 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 $K_0(\operatorname{\mathsf{proj}} B)_\mathbb{R}$, 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.