Ozone groups of Artin–Schelter regular algebras satisfying a polynomial identity

Abstract We study the ozone group of noetherian Artin–Schelter regular algebras satisfying a polynomial identity. The ozone group was shown in previous work by the authors to be an important invariant in the study of PI skew polynomial rings and their centers. In this paper, we show that skew polynomial rings are in fact characterized as those algebras with maximal order abelian ozone groups. We also classify three-dimensional quadratic PI Artin–Schelter regular algebras with trivial ozone group, which must necessarily be Calabi–Yau. This class includes most three-dimensional PI Sklyanin algebras. Further examples and applications are given, including applications to the Zariski Cancellation Problem.