Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T18:59:37.588Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Fröhlich, A., “On the classgroup of integral group rings of finite Abelian groups”, Mathematika, 16 (1969), 143–152. MR 41, 5512.CrossRefGoogle Scholar
2.Jacobinski, H., “Two remarks about hereditary orders”, Proc. Amer. Math. Soc., 28 (1971), 1–8.CrossRefGoogle Scholar
3.Kervaire, M. A. and Murthy, M. P., “On the projective class group of cyclic groups of prime power order ” (to appear).Google Scholar
4.Lee, M. P., “Integral representations of dihedral groups of order 2p”, Trans. Amer. Math, Soc. 110 (1964), 213–231. MR 28, 139.Google Scholar
5.Martinet, J., “Modules sur l'algebre du groupe quaternionien”, Ann. Sci. Ecole Norm. Sup., 4 (1971), 399–408.CrossRefGoogle Scholar
6.Reiner, I., “Integral representations of cyclic groups of prime order”, Proc. Amer. Math. Soc. 8 (1957), 142–146. MR 18, 717.CrossRefGoogle Scholar
7.Reiner, I. and Ullom, S., “Class groups of integral group rings, Trans. Amer. Math. Soc. (to appear)Google Scholar
8.Reiner, I. and Ullom, S., “A Mayer-Vietoris sequence for class groups ” J. Algebra (to appear).Google Scholar
9.Rim, D. S., “Modules over finite groups”, Ann. of Math. (2), 69 (1959), 700–712. MR 21, 3474.CrossRefGoogle Scholar
10.Rosen, M., “Representations of twisted group rings ”, Thesis, Princeton University (Princeton, N.J., 1963).Google Scholar
11.Ullom, S., “A note on the classgroups of integral group rings of some cyclic groups”, Mathematika, 17 (1970), 79–81. MR 42, 4650.CrossRefGoogle Scholar