Homological regularities and concavities

Kirkman, Ellen; Won, Robert; Zhang, James J.

doi:10.2140/ant.2026.20.1559

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Homological regularities and concavities

Ellen Kirkman, Robert Won and James J. Zhang

Vol. 20 (2026), No. 8, 1559–1595

DOI: 10.2140/ant.2026.20.1559

Abstract

This paper concerns homological notions of regularity for noncommutative algebras. Properties of an algebra $A$ are reflected in the regularities of certain (complexes of) $A$ -modules. We study the classical Tor regularity and Castelnuovo–Mumford regularity, which were generalized from the commutative setting to the noncommutative setting by Jørgensen, Dong and Wu. We also introduce two new numerical homological invariants: concavity and Artin–Schelter regularity. Artin–Schelter regular algebras occupy a central position in noncommutative algebra and noncommutative algebraic geometry, and we use these invariants to establish criteria which can be used to determine whether a noetherian connected graded algebra is Artin–Schelter regular.

Keywords

Artin–Schelter regular algebra, Castelnuovo–Mumford regularity, Tor regularity, Koszul algebra, concavity

Mathematical Subject Classification

Primary: 16E10, 16E65, 20J99

Milestones

Received: 16 February 2024

Revised: 21 March 2025

Accepted: 5 August 2025

Published: 25 September 2026

Authors

Ellen Kirkman
Department of Mathematics Wake Forest University Winston-Salem, NC United States

Robert Won
Department of Mathematics The George Washington University Washington, DC United States

James J. Zhang
Department of Mathematics University of Washington Seattle, WA United States

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Vol. 20, No. 8, 2026