Group Rings and Class Groups pp 179-193 | Cite as

# 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 *p*th root of unity, we use Kummer theory to give an explicit description of \(
PH(\mathfrak{B}) \)
and of the kernel of *ψ*.

## Keywords

Isomorphism Class Galois Group Galois Theory Algebra Homomorphism Galois Extension## Preview

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