Applications of Gorenstein projective $τ$-rigid modules

TL;DR

The paper develops and interrelates concepts around Gorenstein projective modules and $\tau$-tilting theory by introducing CM-$\tau$-tilting free algebras and the $E$-rigid/$E$-Gorenstein framework. It proves symmetry results (e.g., CM-$\tau$-tilting freeness is preserved under taking opposites and under $T_2$-extensions), and establishes a key bijection between indecomposable $E$-Gorenstein projective $E$-rigid modules and indecomposable Gorenstein projective $\tau$-rigid modules via a Kong–Zhang type equivalence. This leads to a transfer principle between CM-$E$-free and CM-$\tau$-tilting free algebras and yields new structural insight into Tachikawa's conjectures, providing a set of equivalent conditions that characterize self-injective algebras in terms of Gorenstein projective and $\tau$-rigid properties. Collectively, the results contribute to a deeper understanding of when Gorenstein projective $\tau$-rigid modules are forced to be projective and how these conditions interact with extensions and endomorphism algebras, with implications for Nakayama–Tachikawa type conjectures.

Abstract

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.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Conjecture 1.4
• Theorem 1.5
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5