Principal Homogeneous Spaces

  • Klaus W. Roggenkamp
  • Martin J. Taylor
Part of the DMV Seminar book series (OWS, volume 18)


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.


Isomorphism Class Galois Group Galois Theory Algebra Homomorphism Galois Extension 
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Copyright information

© Springer Basel AG 1992

Authors and Affiliations

  • Klaus W. Roggenkamp
    • 1
  • Martin J. Taylor
    • 2
  1. 1.Mathematisches Institut BUniversität StuttgartStuttgart 80Germany
  2. 2.Dept. of Mathematics U.M.I.S.T.ManchesterEngland

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