A note on the Hasse quiver of $\tau$-tilting modules

TL;DR

This paper refines the understanding of the Hasse quiver structure for support τ-tilting modules by restricting Adachi's construction specifically to τ-tilting modules, enhancing the combinatorial framework.

Contribution

It demonstrates that Adachi's method for support τ-tilting modules can be effectively restricted to τ-tilting modules, providing a more precise combinatorial description.

Findings

The Hasse quiver for support τ-tilting modules can be restricted to τ-tilting modules.

The restriction simplifies the combinatorial structure of the quiver.

The method applies to algebras with an indecomposable projective-injective module.

Abstract

Let $\Lambda$ be an algebra with a indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support $\tau$-tilting $\Lambda$-modules. In this paper, we will show that it can be restricted to $\tau$-tilting modules.