Gorenstein Homological Algebra of Artin Algebras

TL;DR

This paper reviews the foundational aspects of Gorenstein homological algebra for artin algebras, aiming to facilitate understanding of the Gorenstein symmetric conjecture and prepare for studying virtually Gorenstein algebras.

Contribution

It provides a comprehensive, self-contained overview of Gorenstein homological algebra of artin algebras, emphasizing Gorenstein-projective modules and resolutions, and discusses related open problems.

Findings

Summarizes key concepts in Gorenstein homological algebra.

Discusses Gorenstein-projective modules and resolutions.

Includes open problems in the field.

Abstract

Gorenstein homological algebra is a kind of relative homological algebra which has been developed to a high level since more than four decades. In this report we review the basic theory of Gorenstein homological algebra of artin algebras. It is hoped that such a theory will help to understand the famous Gorenstein symmetric conjecture of artin algebras. With only few exceptions all the results in this report are contained in the existing literature. We have tried to keep the exposition as self-contained as possible. This report can be viewed as a preparation for learning the newly developed theory of virtually Gorenstein algebras. In Chapter 2 we recall the basic notions in Gorenstein homological algebra with particular emphasis on finitely generated Gorenstein-projective modules, Gorenstein algebras and CM-finite algebras. In Chapter 3 based on a theorem by Beligiannis we study the Gorenstein-projective resolutions and various Gorenstein dimensions; we also discuss briefly Gorenstein derived categories in the sense of Gao and Zhang. We include three appendixes: Appendix A treats cotorsion pairs; Appendix B sketches a proof of the theorem by Beligiannis; Appendix C provides a list of open problems in Gorenstein homological algebra of artin algebras.