Some homological properties of skew PBW extensions arising in non-commutative algebraic geometry

TL;DR

This paper investigates homological properties like Auslander-Gorenstein regularity, Cohen-Macaulayness, and noetherianity in skew PBW extensions, which encompass many important non-commutative algebraic structures.

Contribution

It provides new insights into the homological properties of skew PBW extensions within non-commutative algebraic geometry, including parametrizations of point modules for key examples.

Findings

Skew PBW extensions include various non-commutative polynomial-like rings.

Homological properties such as Auslander-Gorenstein regularity are studied for these extensions.

Point modules are parametrized for several key examples.

Abstract

In this short paper we study for the skew PBW (Poincar\'e-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the parametrization of its point modules.