Applications of Gorenstein projective $τ$-rigid modules

We first introduce the notion of $CM$-$τ$-tilting free algebras as the generalization of $CM$-free algebras and show the homological properties of $CM$-$τ$-tilting free algebras. Then we give a bijection between Gorenstein projective $τ$-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 $τ$-rigid modules.

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