Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular

TL;DR

This paper characterizes graded twisted Calabi-Yau algebras as generalized Artin-Schelter regular algebras, providing classifications for dimensions 0, 1, and 2, and linking properties like separability, tensor structures, and noetherianity.

Contribution

It establishes a comprehensive characterization of graded twisted Calabi-Yau algebras across dimensions, connecting them to separability, tensor algebras, and GK dimension.

Findings

Dimension 0 algebras are separable k-algebras.

Dimension 1 algebras are tensor algebras over separable algebras.

Dimension 2 algebras are noetherian if and only if they have finite GK dimension.

Abstract

This is a general study of twisted Calabi-Yau algebras that are $\mathbb{N}$-graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi-Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin-Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi-Yau algebras of dimension 0 as separable $k$-algebras, and we similarly characterize graded twisted Calabi-Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi-Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.