On the projectivity of module coalgebras

In this paper, we derive some criteria for the projectivity of a module coalgebra over a finite dimensional Hopf algebra. In particular, we show that any Hopf algebra over a field of characteristic zero is faithfully flat over its group-like subHopf algebra. Finally, we prove that if B is a finite dimensional subHopf algebra of a Hopf algebra A, then B is normal in A if and only if AB(+) = B(+)A. This improves a result by S. Montgomery (1993).