A Bijection theorem for Gorenstein projective τ-tilting modules

Abstract

In this paper, we introduce the notions of Gorenstein projective τ -rigid modules, Gorenstein projective support τ -tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support τ -tilting modules. More precisely, for an algebra Λ, we show that there is a bijection between the set of Gorenstein projective τ -rigid pairs in mod Λ and the set of Gorenstein injective τ −1 -rigid pairs in mod Λ op . We prove that there is a bijection between the set of Gorenstein projective support τ -tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-τ -tilting finite algebras and show that Λ is CM-τ -tilting finite if and only if Λ op is CM-τ -tilting finite. MSC2020: 16G10, 18G25.

A construction of Gorenstein projective $\tau $-tilting modules

“…We also need the following definitions of Gorenstein projective support τ -tilting modules and Gorenstein projective τ -tilting modules [XZ21].…”

“…It has been shown in [XZ21] that quotient algebras of CM-finite algebras need not be CM-finite. However, we have the following result.…”

“…Recently, Xie and the second author [XZ21] combined Gorenstein projective modules with τ -tilting modules and built a bijection from Gorenstein projective support τ -tilting modules to Gorenstein injective support τ −1 -tilting modules which is analogous to Adachi-Iyama-Reiten's bijection from support τ -tilting modules to support τ −1 -tilting modules. But there is little evidence for the existence of non-trivial Gorenstein projective [support] τ -tilting modules except support τ -tilting modules over self-injective algebras.…”

“…Recall from [XZ21] that an algebra is called Cohen-Macaulay-τ -tilting finite (CM-τ -tilting finite for short) if it admits finitely many isomorphism classes of indecomposable Gorenstein projective τ -rigid modules. The CMτ -tilting finite algebras are generalizations of both algebras of finite Cohen-Macaulay type (CM-finite algebras for short) [B11, C08, LZ10a] and τ -tilting finite algebras [DIJ19].…”