A survey on the Koszul homology algebra

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.