Faithful flatness of Hopf algebras over coideal subalgebras with a bimodule conditional expectation

TL;DR

This paper proves that Hopf algebras are faithfully flat over certain coideal subalgebras when these subalgebras are direct summands as bimodules, providing a direct and self-contained proof.

Contribution

It offers a new, direct proof of faithful flatness of Hopf algebras over coideal subalgebras under specific bimodule conditions.

Findings

Hopf algebra $H$ is faithfully flat over $A$ when $A$ is a direct summand as an $A$-bimodule

Provides a self-contained proof of faithful flatness in this context

Clarifies conditions under which flatness holds for coideal subalgebras

Abstract

We give a direct and self-contained proof that if $H$ is a Hopf algebra and $A\subset H$ is a right coideal subalgebra such $A$ is a direct summand in $H$ as an $A$-bimodule, then $H$ is faithfully flat as a left and right $A$-module.