A Bijection theorem for Gorenstein projective \tau-tilting modules

TL;DR

This paper establishes a Gorenstein analog of the support τ-tilting modules bijection theorem, connecting Gorenstein projective support τ-tilting modules with Gorenstein torsion classes, and explores properties of CM-τ-tilting finite algebras.

Contribution

It introduces Gorenstein projective τ-tilting theory and proves a bijection theorem, extending classical support τ-tilting results to the Gorenstein setting.

Findings

Bijection between Gorenstein support τ-tilting modules and Gorenstein torsion classes.

Introduction of CM-τ-tilting finite algebras and their properties.

Bongartz completion may not preserve Gorenstein projectiveness.

Abstract

We introduce the notions of Gorenstein projective $\tau$-rigid modules, Gorenstein projective support $\tau$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support $\tau$-tilting modules. More precisely, for an algebra $\Lambda$, We prove that there is a bijection between the set of Gorenstein projective support $\tau$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$\tau$-tilting finite algebras and show that $\Lambda$ is CM-$\tau$-tilting finite if and only if $\Lambda^{\rm {op}}$ is CM-$\tau$-tilting finite. Moreover, we show that the Bongartz completion of a Gorenstein projective $\tau$-rigid module need not be a Gorenstein projective $\tau$-tilting module.