$\tau$-tilting finiteness of two-point algebras II

TL;DR

This paper develops a strategy based on g-vectors to identify new minimal tau-tilting infinite two-point algebras and determines the tau-tilting finiteness of several classes, including radical cube zero cases.

Contribution

It introduces a novel g-vector based approach to classify tau-tilting finiteness in two-point algebras and explores derived equivalence classes of the Kronecker algebra.

Findings

Identified new minimal tau-tilting infinite two-point algebras.

Determined tau-tilting finiteness for various monomial algebras.

Found the derived equivalence class of the Kronecker algebra contains only itself and its opposite.

Abstract

In this paper, we explain a strategy on $g$-vectors to discover some new minimal $\tau$-tilting infinite two-point algebras. Consequently, the $\tau$-tilting finiteness of various two-point monomial algebras, including all radical cube zero cases, could be determined. Moreover, we find that the derived equivalence class of the Kronecker algebra contains only itself and its opposite algebra.