Homological regularities and concavities

TL;DR

This paper explores homological regularities in noncommutative algebras, introducing new invariants like concavity and Artin-Schelter regularity to characterize algebraic properties and identify Artin-Schelter regular algebras.

Contribution

It introduces two novel homological invariants, concavity and Artin-Schelter regularity, and establishes criteria for recognizing Artin-Schelter regular algebras among noetherian graded algebras.

Findings

Defined and analyzed properties of concavity and Artin-Schelter regularity.

Provided criteria to identify Artin-Schelter regular algebras.

Extended classical homological regularities to the noncommutative setting.

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 J{\o}rgensen and Dong-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.