Koszul algebras and regularity

TL;DR

This survey explores the properties, techniques, and variants of Koszul algebras and their regularity, highlighting open problems and recent developments in the field.

Contribution

It provides a comprehensive overview of methods to determine Koszulness and discusses new variants and open questions in the study of Koszul algebras.

Findings

Various techniques to establish Koszulness

Discussion of strongly, absolutely, and universally Koszul variants

Presentation of open problems in the field

Abstract

This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. We describe several techniques to establish the Koszulness of algebras. We discuss variants of the Koszul property such as strongly Koszul, absolutely Koszul and universally Koszul. We present several open problems related with these notions and their local variants.