A survey on the Koszul homology algebra

TL;DR

This survey reviews classical and recent results on the structure of Koszul homology algebras of rings, highlighting their role in characterizing ring properties and discussing open questions in the field.

Contribution

It provides a comprehensive overview of the classical and recent developments in the study of Koszul homology algebras, emphasizing their applications and open problems.

Findings

Characterization of rings via Koszul homology algebra structure

Recent progress on Koszul algebras' homology structures

Open questions on the algebraic properties of Koszul homologies

Abstract

The Koszul homology algebra of a commutative local (or graded) ring $R$ tends to reflect important information about the ring $R$ and its properties. In fact, certain classes of rings are characterized by the algebra structure on their Koszul homologies. In this paper we survey some classical results on the Koszul homology algebras of such rings and highlight some applications. We report on recent progress on the Koszul homology algebras of Koszul algebras and examine some open questions on the topic.