Hopf-Galois extensions and isomorphisms of small categories

TL;DR

This paper establishes isomorphisms between two linear categories associated with modules over Hopf-Galois extensions, providing a new perspective on structure theorems for cleft extensions and lifting theorems.

Contribution

It introduces a novel categorical framework linking modules over Hopf-Galois extensions to isomorphic linear categories, simplifying proofs of key theorems.

Findings

Proves isomorphism between two linear categories associated with modules over Hopf-Galois extensions

Provides new proofs for the Structure Theorem for cleft extensions

Derives the Militaru-Stefan lifting Theorem using these categorical isomorphisms

Abstract

We associate two linear categories with two objects to a module over the subalgebra of coinvariants of a Hopf-Galois extension, and prove that they are isomorphic. The structure Theorem for cleft extensions, and the Militaru \cStefan lifting Theorem can be obtained using these isomorphisms.