$\tau$-Tilting modules over one-point extensions by a simple module at a source point

TL;DR

This paper classifies $ au$-tilting modules over one-point extensions of finite dimensional algebras at source points and provides formulas to compute their counts, with applications to Dynkin type algebras.

Contribution

It introduces a classification of $ au$-tilting modules over one-point extensions and derives formulas for counting them, extending understanding of module categories.

Findings

Derived formulas for $| ilt B|$ and $| ext{stilt } B|$ in terms of $A$ and $A/ ext{e}_i$

Calculated the number of $ au$-tilting modules for linearly Dynkin type algebras with radical square zero

Provided a classification framework for $ au$-tilting modules over one-point extensions

Abstract

Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $\tau$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|.$$ As a consequence, we can calculate the numbers of $\tau$-tilting modules and support $\tau$-tilting modules over linearly Dynkin type algebras whose square radical are zero.