Eventually homological isomorphisms and Gorenstein projective modules

TL;DR

This paper demonstrates how eventually homological isomorphisms induce equivalences between various categories related to Gorenstein projective modules, and applies these results to verify conjectures and describe modules over specific algebras.

Contribution

It introduces the concept that eventually homological isomorphisms induce categorical equivalences and applies this to Gorenstein modules and conjectures.

Findings

Induces triangle equivalences between singularity and Gorenstein categories

Reduces Auslander-Reiten and Gorenstein symmetry conjectures via isomorphisms

Verifies the Auslander-Reiten conjecture for certain algebras

Abstract

We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.