Self-orthogonal tau-tilting modules and tilting modules

Abstract

Let $Λ$ be an artin algebra and $T$ a $τ$-tilting $Λ$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}_Λ^{i}(T,\Fac T)=0$ for all $i\geq 1$, where $\Fac T$ is the full subcategory consisting of modules generated by $T$. Consequently, a $τ$-tilting module $T$ of finite projective dimension is a tilting module if and only if ${\rm Ext}_Λ^{i}(T, T)=0$ for all $i\geq 1$. Moreover, we also give an example to show that a support $τ$-tilting but not $τ$-tilting module $M$ of finite projective dimension satisfying ${\rm Ext}_Λ^{i}(M, M)=0$ for all $i\geq1$ need not be a partial tilting module.