Self-orthogonal tau-tilting modules and tilting modules

Let $\mathrm{\Lambda }$ be an artin algebra and $T$ a $\tau$-tilting $\mathrm{\Lambda }$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}{\Lambda}^{i}(T,\Fac T)=0$forall$i\geq 1$,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 ${\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathrm{\Lambda }}^i(T,T)=0$ for all $i\ge 1$. Moreover, we also give an example to show that a support $\tau$-tilting but not $\tau$-tilting module $M$ of finite projective dimension satisfying ${\rm Ext}{\Lambda}^{i}(M, M)=0$forall$i\geq1$ need not be a partial tilting module.