Artin-Schelter Gorenstein property of Hopf Galois extensions

TL;DR

This paper studies the homological properties of Hopf Galois extensions, showing that under certain conditions, the AS Gorenstein property is preserved and that injective dimension is a monoidal invariant for AS Gorenstein Hopf algebras.

Contribution

It proves the preservation of the AS Gorenstein property in noetherian affine PI Hopf Galois extensions and establishes injective dimension as a monoidal invariant.

Findings

B inherits the AS Gorenstein property from A.

Injective dimension is invariant under monoidal equivalence.

Results apply to noetherian affine PI algebras.

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.