A new characterization of hereditary algebras

TL;DR

This paper characterizes hereditary algebras by showing they have no loops in their quiver and all τ-tilting modules are tilting, providing a new criterion for identifying such algebras.

Contribution

It introduces a novel characterization of hereditary algebras based on quiver loops and τ-tilting modules, linking structural and module-theoretic properties.

Findings

Hereditary algebras have no loops in their quiver.

All τ-tilting modules in hereditary algebras are tilting.

The characterization is both necessary and sufficient.

Abstract

In this short paper we prove that a finite dimensional algebra is hereditary if and only if there is no loop in its ordinary quiver and every $\tau$-tilting module is tilting.