A bijection theorem for Gorenstein projective τ-tilting modules

Abstract

We introduce the notions of Gorenstein projective [Formula: see text]-rigid modules, Gorenstein projective support [Formula: see text]-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi–Iyama–Reiten’s bijection theorem on support [Formula: see text]-tilting modules. More precisely, for an algebra [Formula: see text], we prove that there is a bijection between the set of Gorenstein projective support [Formula: see text]-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-[Formula: see text]-tilting finite algebras and show that [Formula: see text] is CM-[Formula: see text]-tilting finite if and only if [Formula: see text] is CM-[Formula: see text]-tilting finite. Moreover, we show that the Bongartz completion of a Gorenstein projective [Formula: see text]-rigid module need not be a Gorenstein projective [Formula: see text]-tilting module.