From CM-finite to CM-free

TL;DR

This paper generalizes stability results for Gorenstein categories, explores the properties of CM-finite algebras, and establishes categorical equivalences involving Gorenstein-projective objects and singularity categories.

Contribution

It proves a generalized stability for Gorenstein categories and shows that the relative Auslander algebra of a CM-finite algebra is CM-free, linking various derived and defect categories.

Findings

The relative Auslander algebra of a CM-finite algebra is CM-free.

The Gorenstein defect category is equivalent to the singularity category of the relative Auslander algebra.

The paper extends stability results for Gorenstein categories.

Abstract

The aim of this paper is twofold. On one hand, we prove a slight generalization of the stability for Gorenstein categories in [SWSW] and [Huang]; and show that the relative Auslander algebra of a CM-finite algebra is CM-free. On the other hand, we describe the bounded derived category, and the Gorenstein defect category introduced in [BJO], via Gorenstein-projective objects; and we show that the Gorenstein defect category of a CM-finite algebra is triangle-equivalent to the singularity category of its relative Auslander algebra.