The question of whether or not a Hopf algebra is faithfully flat over a Hopf subalgebra has received positive answers in several particular cases: when (or more generally, just ) is commutative, cocommutative, or pointed, or when contains the coradical of . 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” is always a cosemisimple coalgebra, and that the expectation is positive when is a CQG algebra.
"Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras." Algebra Number Theory 8 (5) 1179 - 1199, 2014. https://doi.org/10.2140/ant.2014.8.1179