Comparing $\tau$-tilting modules and $1$-tilting modules

TL;DR

This paper explores the relationship between $ au$-tilting modules and 1-tilting modules, providing characterizations and studying a conjecture related to self-orthogonal modules using delooping levels.

Contribution

It offers new characterizations of $ au$-tilting and 1-tilting modules via quotient algebras and tensor/tor conditions, and investigates the Self-orthogonal $ au$-tilting Conjecture.

Findings

Characterized $ au$-tilting modules as 1-tilting over certain quotient algebras.

Characterized 1-tilting modules as $ au$-tilting modules with ${ m Tor}^1$-vanishing.

Confirmed the conjecture when the endomorphism algebra has finite global delooping level.

Abstract

We characterize $\tau$-tilting modules as $1$-tilting modules over quotient algebras satisfying a tensor-vanishing condition, and characterize $1$-tilting modules as $\tau$-tilting modules satisfying a ${\rm Tor}^1$-vanishing condition. We use delooping levels to study \emph{Self-orthogonal $\tau$-tilting Conjecture}: any self-orthogonal $\tau$-tilting module is $1$-tilting. We confirm the conjecture when the endomorphism algebra of the module has finite global delooping level.