On homological smoothness of generalized Weyl algebras over polynomial algebras in two variables

TL;DR

This paper investigates the conditions under which generalized Weyl algebras over polynomial algebras in two variables are homologically smooth, providing explicit criteria and computing their Nakayama automorphisms.

Contribution

It offers a necessary and sufficient condition for homological smoothness and explicitly calculates Nakayama automorphisms based on Jacobian determinants.

Findings

Derived a criterion for homological smoothness of the algebras

Computed Nakayama automorphisms explicitly

Linked algebra properties to Jacobian determinants

Abstract

Homological smoothness and twisted Calabi-Yau property of generalized Weyl algebras over polynomial algebras in two variables is studied. A necessary and sufficient condition to be homologically smooth is given. The Nakayama automorphisms of such algebras are also computed in terms of the Jacobian determinants of defining automorphisms.