Applications of Gorenstein projective τ-rigid modules

We introduce the notion of [Formula: see text]-[Formula: see text]-tilting free algebras as the generalization of [Formula: see text]-free algebras and show the construction of [Formula: see text]-[Formula: see text]-tilting free algebras. We give a bijection between Gorenstein projective [Formula: see text]-rigid modules and certain modules over an endomorphism algebra. We give a partial answer to Tachikawa’s first conjecture by using Gorenstein projective [Formula: see text]-rigid modules.