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.