Krull relations in Hopf Galois extensions: Lifting and twisting

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t-Lying Over (t-LO), and Going Up (GU), analogous to the classical Krull relations for prime ideals; we also defined three new “dual” Krull relations. We say that H itself is said to have one of the Krull relations if the relation holds for all faithfully flat H -Galois extensions. We showed in [MS] that H has one of the three “dual” Krull relations if and only if the dual Hopf algebra H ∗ of H has the original relation (hence the name). An important example of Hopf Galois extensions is given by Hopf crossed products A = Rσ # H . Moreover, Galois extensions can be useful in studying crossed products, since they satisfy a “transitivity” property which crossed products lack. That is, if K is a normal Hopf subalgebra of H with Hopf quotient H , then, in general, one cannot write A = Rσ # H = (RσK) #τ H , by an example of [S2]. Another basic example of a Hopf Galois extension is given by a Hopf algebra A with a normal Hopf subalgebra R of finite index such that A is faithfully flat over R: for then R ⊂A is faithfully flat H -Galois.