Homogenized skew PBW extensions

TL;DR

This paper introduces a new filtration called $\sigma$-filtered for skew PBW extensions, explores their homological properties, and shows how homogenization preserves key algebraic features, advancing the understanding of noncommutative algebra structures.

Contribution

It proposes the $\sigma$-filtered skew PBW extension, analyzes its homological properties, and demonstrates how homogenization maintains important algebraic properties.

Findings

Homogenization of a $\sigma$-filtered skew PBW extension over a ring $R$ results in a graded skew PBW extension.

If the homogenization of $R$ is Auslander-regular, then the homogenization of $A$ is a domain, Noetherian, and Artin-Schelter regular.

The algebra $A$ is shown to be Noetherian, Zariski, and skew Calabi-Yau under certain conditions.

Abstract

In this paper, we provide a new and more general filtration to the family of noncommutative rings known as skew PBW extensions. We introduce the notion of $\sigma$-filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a $\sigma$-filtered skew PBW extension $A$ over a ring $R$ is a graded skew PBW extension over the homogenization of $R$. Using this fact, we prove that if the homogenization of $R$ is Auslander-regular, then the homogenization of $A$ is a domain Noetherian, Artin-Schelter regular, and $A$ is Noetherian, Zariski and (ungraded) skew Calabi-Yau.