Support $\tau$-tilting modules over one-point extensions

TL;DR

This paper investigates how support τ-tilting modules over an algebra A can be extended to its one-point extension B, establishing a relationship between their counts and providing methods for such extensions.

Contribution

It introduces two methods for extending support τ-tilting modules from A to B and proves an inequality relating their numbers.

Findings

Every support τ-tilting A-module can be extended to B in two ways.

The number of support τ-tilting B-modules is at least twice that of A.

Provides explicit extension methods for support τ-tilting modules.

Abstract

Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $\tau$-tilting $A$-module can be extended to be a support $\tau$-tilting $B$-module by two different ways. As a consequence, it is shown that there is an inequality $$|\stilt B|\geqslant 2|\stilt A|.$$