  We first introduce the notion of $CM$-$\tau$-tilting free algebras as the
generalization of $CM$-free algebras and show the homological properties of
$CM$-$\tau$-tilting free algebras. Then we give a bijection between Gorenstein
projective $\tau$-rigid modules and certain modules by using an equivalence
established by Kong and Zhang. Finally, we give a partial answer to Tachikawa's
first conjecture by using Gorenstein projective $\tau$-rigid modules.
