Normal basis and transitivity of crossed products for Hopf algebras (Q1204406)

Let \(H\) be a Hopf algebra, \(A\) an \(H\)-comodule algebra, and let \(B\) denote the subalgebra of \(H\)-coinvariants of \(A\). Furthermore, let \(p: H \to H'\) be a surjective Hopf algebra map and let \(B'\) denote the subalgebra of \(H'\)-coinvariants of \(A\), viewing \(A\) as \(H'\)-comodule algebra in the obvious fashion. The basic questions that are investigated in this interesting article are the following: (1) If the extension \(B\subseteq A\) is \(H\)-Galois, is \(B' \subseteq A\) \(H'\)-Galois? (2) If \(B\subseteq A\) is an \(H\)-crossed product (i.e., \(H\)-Galois with a normal basis), is \(B'\subseteq A\) an \(H'\)-crossed product? As to (1), it is shown that the answer is usually positive. For example, working over a base field, it suffices to assume that \(H\) is faithfully coflat as left \(H'\)-comodule. Problem (2) is more delicate, but some interesting sufficient conditions are established. Over a field, it suffices to assume that \(H\) is injective as right \(H'\)-comodule and that the coradical of \(H'\) is liftable along \(p\).