Skew Calabi-Yau property of faithfully flat Hopf Galois extensions

TL;DR

This paper demonstrates that in faithfully flat Hopf Galois extensions, the skew Calabi-Yau property of the base algebra and Hopf algebra extends to the larger algebra, with explicit formulas for the Nakayama automorphism.

Contribution

It establishes that the skew Calabi-Yau property is preserved under faithfully flat Hopf Galois extensions and provides a method to compute the Nakayama automorphism for cleft extensions.

Findings

B is skew Calabi-Yau if A and H are.

Nakayama automorphism of B derived from A, H, and homological determinant.

Study of Hopf bimodule structure on Ext groups.

Abstract

This paper shows that if $H$ is a Hopf algebra and $A \subseteq B$ is a faithfully flat $H$-Galois extension, then $B$ is skew Calabi-Yau provided $A$ and $H$ are. Specifically, for cleft extensions $A \subseteq B$, the Nakayama automorphism of $B$ can be derived from those of $A$ and $H$, along with the homological determinant of the $H$-action on $A$. This finding is based on the study of the Hopf bimodule structure on $\mathrm{Ext}^i_{A^e}(A, B^e)$.