Artin-Schelter Regular Algebras

TL;DR

This survey reviews the development and significance of Artin-Schelter regular algebras, highlighting their role as noncommutative analogs of polynomial rings and their impact on noncommutative geometry over the past 30 years.

Contribution

It provides a comprehensive overview of foundational results and recent research themes in Artin-Schelter regular algebras, emphasizing their importance in noncommutative projective geometry.

Findings

Artin-Schelter regular algebras model noncommutative projective spaces.

They have been central to developments in noncommutative algebraic geometry.

The survey summarizes 30 years of research and key open problems.

Abstract

Artin-Schelter regular algebras can be thought of as noncommutative versions of commutative polynomial rings, modeled after the special homological properties polynomial rings have as graded rings. First defined by Artin and Schelter in 1987, their introduction formed the beginning of the subject of noncommutative projective geometry. Artin-Schelter regular algebras have continued to play a large role in that subject, since geometrically they represent noncommutative (weighted) projective spaces. This is a survey of Artin-Schelter regular algebras, based on a talk at the 2022 meeting "Recent Advances and New Directions in the Interplay of Noncommutative Algebra and Geometry" at the University of Washington, in honor of the 65th birthday of S. Paul Smith. We review the earliest foundational results in the subject, and then describe some of the major themes of the last 30 years of research.