Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

TL;DR

This paper proves that cosemisimple Hopf algebras are always faithfully flat over their Hopf subalgebras, extending known results to this class and analyzing related structural properties.

Contribution

It establishes that cosemisimple Hopf algebras are faithfully flat over all Hopf subalgebras, adding a new class to previously known cases.

Findings

Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras.

The third term in the exact sequence is always a cosemisimple coalgebra.

The expectation map is positive for CQG algebras.

Abstract

The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $H$ is a CQG algebra.