Homogenized skew PBW extensions

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 $$\sigma $$ σ -filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a $$\sigma $$ σ -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.

Noncommutative scheme theory and the Serre-Artin-Zhang-Verevkin theorem for semi-graded rings

“…A detailed list of examples of semi-graded rings and its relationships with other algebras can be found in Fajardo et al [14]. Ringtheoretical, algebraic and geometrical properties of semi-graded rings have been investigated in the literature by several authors (e.g., [2,11,21,35,36,39,45,46,47] and references therein).…”

On Σ-skew reflexive-nilpotents-property for rings

“…Since its introduction, ring-theoretical and homological properties of skew PBW extensions have been studied by some people (e.g. [5], [21], [22], [23], [44], [47], [70], [25], [56], [61], [66], [68], [69]). As a matter of fact, a book that includes several of the works carried out for these extensions has been published (Fajardo et al [16]).…”