Artin-Schelter Gorenstein property of Hopf Galois extensions

Abstract: This paper investigates the homological properties of the faithfully flat Hopf Galois extension $A \subseteq B$. It establishes that when $B$ is a noetherian affine PI algebra and $A$ is AS Gorenstein, $B$ inherits the AS Gorenstein property. Furthermore, we demonstrate that injective dimension serves as a monoidal invariant for AS Gorenstein Hopf algebras. Specifically, if two such Hopf algebras have equivalent monoidal categories of comodules, then their injective dimensions are equal.