Flatness over PI coideal subalgebras

TL;DR

This paper proves flatness properties of certain Hopf algebras over their coideal subalgebras under specific conditions, revealing broad classes of flat and faithfully flat Hopf algebra extensions.

Contribution

It establishes flatness and faithful flatness of Hopf algebras over polynomial identity coideal and Hopf subalgebras under residual finiteness and Artinian conditions.

Findings

Hopf algebra H is flat over any right coideal subalgebra with a polynomial identity

H is faithfully flat over polynomial identity Hopf subalgebras

Identifies large classes of Hopf algebras with flatness properties

Abstract

Under the assumption that a residually finite dimensional Hopf algebra H has an Artinian ring of fractions it is proved that H is a flat module over any right coideal subalgebra satisfying a polynomial identity and is faithfully flat over any polynomial identity Hopf subalgebra. As a consequence we find a large class of Hopf algebras which are flat over all coideal subalgebras and are faithfully flat over all Hopf subalgebras.