Push-forward of Hopf--Galois extensions: the non central case

TL;DR

This paper explores how Hopf--Galois extensions behave under push-forward operations, using twisted tensor products to understand algebraic structures similar to pullbacks of principal bundles.

Contribution

It introduces a method to construct push-forward of Hopf--Galois extensions via twisted tensor products, extending the algebraic framework for these extensions.

Findings

Push-forward of an H-Galois extension remains an H-Galois extension.

Application of twisted tensor product algebras to covariant module extensions.

Comparison of Ehresmann--Schauenburg algebroids before and after push-forward.

Abstract

We study the push-forward of Hopf--Galois extensions as the algebraic counterpart of the pullback of principal bundles. We apply the theory of twisted tensor product algebras to endow covariant extensions of modules along a map $\mathsf{F}$ with an algebra structure, under compatibility conditions between $\mathsf{F}$ and the twisting map. The push-forward of an $H$-Galois extension $B \subset A$ along a map $\mathsf{F} : B \to C$ is an $H$-Galois extension of $C$. The corresponding Ehresmann--Schauenburg algebroids are compared.