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Class groups for integral representations of metacyclic groups

Published online by Cambridge University Press:  26 February 2010

S. Galovich ,
I. Reiner  and
S. Ullom
S. Galovich
Affiliation:
Carleton College, Northfield, Minnesota 55057. University of Illinois, Urbana, Illinois 61801.
I. Reiner
Affiliation:
Carleton College, Northfield, Minnesota 55057. University of Illinois, Urbana, Illinois 61801.
S. Ullom
Affiliation:
Carleton College, Northfield, Minnesota 55057. University of Illinois, Urbana, Illinois 61801.
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Extract

Let R be a Dedekind domain whose quotient field K is an algebraic number field, and let Λ be an R-order in a semisimple K-algebra A with 1. A Λ-lattice is a finitely generated R-torsionfree left Λ-module. We shall call a Λ-lattice M locally free of rank n if for each maximal ideal p of R, Mp is Λp,-free on n generators. (The subscript p denotes localization.) The (locally free) class group of Λ is the additive group C(Λ) generated by symbols

where

and where xM = 0 if and only if M is stably free (that is, M + Λ(k) ≅ Λ + Λ(k) for some k).

Type
Research Article
Information
Mathematika , Volume 19 , Issue 1 , June 1972 , pp. 105 - 111
Copyright
Copyright © University College London 1972

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References

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