Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. (Q457329)

The authors study the following question, asked by \textit{S. Montgomery} [Hopf algebras and their actions on rings. Providence: AMS (1993; Zbl 0793.16029)]: let \(A\) be a Hopf subalgebra of a Hopf algebra \(H\) over a field; is \(H\) faithfully flat over \(A\)? The question goes back to one of Kaplansky's conjectures, and has a rich history, documented well in the introduction of the paper.NEWLINENEWLINE The main result is the following: if \(H\) is cosemisimple, then \(H\) is faithfully flat over every Hopf subalgebra \(A\). Moreover, the left \(H\)-module coalgebra \(C=H/HA^+\) is cosemisimple. A second result is the following: if the coradical of \(H\) is a Hopf algebra, then \(H\) is faithfully flat over its cosemisimple Hopf subalgebras. In the final Section 3, the left inverse of the inclusion \(A\to H\) is studied, in the case where \(A\) and \(H\) are CQG algebras.