Principal Homogeneous Spaces

Abstract

As before, let K be the field of fractions of a Dedekind domain \( mathfrak{D} \) of characteristic 0, and let G be a finite abelian group. We will continue to work with Hopf orders \( mathfrak{A} \) in A = KG and B = Map(G,K). In this chapter we will be concerned with the objects on which a Hopf order in A acts. Rather than studying all \( mathfrak{A} \) -modules, we will make use of the comultiplication in \( mathfrak{A} \) by considering only those \( mathfrak{A} \) -modules which have the structure of an \( mathfrak{D} \) -algebra, and are in fact “twisted” versions of \( mathfrak{B} = \mathfrak{A}D \) . These objects are the principal homogeneous spaces for the Hopf order \( mathfrak{B} \) , and the set of isomorphism classes of principal homogeneous spaces can be given the structure of an abelian group \( PH(\mathfrak{B}) \) . As in the previous chapter, we will first work at the level of K-algebras, and then see how the theory lifts to integral level. We shall then construct a group homomorphism ψ from \( PH(\mathfrak{B}) \) to the locally free classgroup \( C1(\mathfrak{A}) \) . Finally, in the case that G is cyclic of order p and K contains a primitive pth root of unity, we use Kummer theory to give an explicit description of \( PH(\mathfrak{B}) \) and of the kernel of ψ.KeywordsIsomorphism ClassGalois GroupGalois TheoryAlgebra HomomorphismGalois ExtensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.