Hopf Orders and a Generalization of a Theorem of L. R. McCulloh

AbstractLet K be an algebraic number field with ring of integers O and let G be an elementary abelian group of order lk. Let A be a Hopf order in KG and let B be its dual. An order X in a Galois G-extension of K is a semilocal principal homogeneous space over B if Xl is a principal homogeneous space over Bl and X is integrally closed away from l. We define a map ψ from the group of such X to the locally free classgroup Cl(A). Assuming that A admits C ≅ Flk×⊆ Aut(G), we describe the image of ψ in terms of a Stickelberger ideal in ZC. This generalizes a result of L. R. McCulloh on the classes in Cl(OG) realized by tame rings of integers