Failure of flatness over finite-dimensional Hopf subalgebras

TL;DR

This paper demonstrates that finite-dimensional nonsemisimple Hopf algebras can fail to be flat over certain Hopf subalgebras, while identifying classes of infinite-dimensional Hopf algebras with universal flatness properties.

Contribution

It establishes the existence of non-flat extensions for finite-dimensional nonsemisimple Hopf algebras and characterizes infinite-dimensional Hopf algebras with universal flatness over their subalgebras.

Findings

Finite-dimensional nonsemisimple Hopf algebras are not necessarily flat over subalgebras.

Existence of Hopf algebra extensions where flatness fails.

Certain infinite-dimensional Hopf algebras are faithfully flat over all their subalgebras.

Abstract

It is proved in this paper that for any finite-dimensional nonsemisimple Hopf algebra $A$ there exists a Hopf algebra $H$ containing $A$ as a Hopf subalgebra such that $H$ is not flat over $A$. On the other hand, there is a class of infinite-dimensional Hopf algebras with the property that all Hopf algebras without exception are faithfully flat modules over Hopf subalgebras from this class.