Artin-Schelter Gorenstein property of Hopf Galois extensions

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.

💡 Research Summary

This paper investigates two intertwined homological questions concerning Hopf algebras and their Galois extensions. The first problem asks whether the Artin‑Schelter (AS) Gorenstein property is preserved under a faithfully flat Hopf‑Galois extension (A\subseteq B). Assuming that (B) is a Noetherian affine PI algebra and that the coinvariant subalgebra (A=B^{\operatorname{co}H}) is AS‑Gorenstein, the authors prove that (B) is also AS‑Gorenstein, with the same Gorenstein dimension. Moreover, they obtain an exact formula for the injective dimension: \