Hopf-Galois extensions and an exact sequence for $H$-Picard groups

TL;DR

This paper explores the structure of $H$-Galois extensions, introduces the $H$-Picard group, and establishes an exact sequence relating it to the Picard group of the coinvariant subalgebra, advancing understanding of symmetries in Hopf algebra extensions.

Contribution

It introduces the $H$-Picard group and proves an exact sequence connecting it with the Picard group of the coinvariant subalgebra in $H$-Galois extensions.

Findings

Defined the $H$-Picard group for $H$-Galois extensions.

Established an exact sequence linking $H$-Picard and Picard groups.

Analyzed $H$-Morita autoequivalences of $A$.

Abstract

Let $H$ be a Hopf algebra, and $A$ an $H$-Galois extension. We investigate $H$-Morita autoequivalences of $A$, introduce the concept of $H$-Picard group, and we establish an exact sequence linking the $H$-Picard group of $A$ and the Picard group of $A^{{\rm co}H}$.