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.

在本文中，我们为称为斜 PBW 扩展的非交换环族提供了一种新的更通用的过滤。我们引入了\(\sigma \) -filtered skew PBW 扩展的概念，并研究了这些代数的一些同调性质。我们证明了\(\sigma \) -filtered skew PBW extension A在环R上的均匀化是在R均匀化上的渐变倾斜 PBW 扩展。利用这个事实，我们证明如果R的均质化是 Auslander-regular，那么A的均质化是域 Noetherian、Artin–Schelter regular，并且A是 Noetherian、Zariski 和（未分级的）斜 Calabi–Yau。