Self-orthogonal tau-tilting modules and tilting modules

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