On the global dimension of the endomorphism algebra of a $\tau$-tilting module

TL;DR

This paper explores the relationship between the global dimension of an algebra and that of the endomorphism algebra of a $ au$-tilting module, revealing finiteness conditions and bounds in specific algebra classes.

Contribution

It establishes conditions under which the global dimension of endomorphism algebras of $ au$-tilting modules is finite, including explicit bounds for certain algebra types.

Findings

Global dimension of endomorphism algebra can be infinite in general.

For monomial and special biserial algebras of global dimension two, the endomorphism algebra's global dimension is always finite.

Explicit bounds are provided for special biserial algebras.

Abstract

We find a relationship between the global dimension of an algebra $A$ and the global dimension of the endomorphism algebra of a $\tau$-tilting module, when $A$ is of finite global dimension. We show that, in general, the global dimension of the endomorphism algebra is not always finite. For monomial algebras and special biserial algebras of global dimension two, we prove that the global dimension of the endomorphism algebra of any $\tau$-tilting module is always finite. Moreover, for special biserial algebras, we give an explicit bound.