Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-04T18:04:59.470Z Has data issue: false hasContentIssue false
  • English
  • Français

Schur and Projective Schur Groups of Number Rings

Published online by Cambridge University Press:  20 November 2018

Peter Nelis
Peter Nelis*
Affiliation:
Department of Mathematics University of Antwerp, U.I.A. 2610 Antwerp, Belgium
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Schur or projective Schur group of a field consists of the classes of central simple algebras which occur in the decomposition of a group algebra or a twisted group algebra. For number fields, the projective Schur group has been determined in [8], whereas the Schur group is extensively studied in [25]. Recently, some authors have generalized these concepts to commutative rings. One then studies the classes of Azumaya algebras which are epimorphic images of a group ring or a twisted group ring. Though several properties of the Schur or projective Schur group defined in this way have been obtained, they remain rather obscure objects.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 3 , 01 June 1991 , pp. 540 - 558
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Childs, L., Azumayas over number rings as crossed products, preprint 1986.Google Scholar
2. Curtis, C.W. and Reiner, I., Methods of representation theory with applications to finite groups and orders. Pure & Applied Mathematics. Wiley Interscience Series of Texts, Monographs and Tracts, 1981.Google Scholar
3. De Meyer, F. and Mollin, R., The Schur group, orders and their applications. Lecture Notes Mathematics 1142, Springer-Verlag, Berlin, 205–209. 1985.Google Scholar
4. De Meyer, F. and Mollin, R., The Schur group of a commutative ring, Journal of Pure and Applied Algebra 35(1985), 117–122.Google Scholar
5. De Meyer, F. and Ingraham, E., Separable algebras over commutative rings. Lecture Notes in Mathematics, 181, Springer-Verlag, Berlin, 1971.Google Scholar
6. Karpilovsky, G., Projective representations of finite groups. Marcel Dekkerlnc, 1985.Google Scholar
7. Kargapolov, M. and Merzliakov, I., Éléments de la théorie des groupes. Traduction Française, Éditions Mir, Moscou, 1985.Google Scholar
8. Lorenz, F. and Opolka, H., Einfache Algebren und projective Darstellungen endlicher Gruppen, Mathematische Zeitschrift 162(1978), 175–186.Google Scholar
9. Nelis, P. and J, F.M.. Van Oystaeyen, The projective Schur subgroup of the Brauer group and root groups of finite groups, Journal of Algebra, to appear.Google Scholar
10. Nelis, P., Integral matrices spanned by finite groups, U.I.A. preprint, 12, 1990.Google Scholar
11. Nelis, P., The Schur group conjecture for the ring of integers of a number field, Proc. American Math. Soc, to appear.Google Scholar
12. Orzech, M. and Small, C., The Brauer Group of Commutative Rings. Lecture Notes in Pure and Applied Mathematics, 11, 1975.Google Scholar
13. Reiner, I., Maximal orders. S, L.M.. Monographs5, 1975.Google Scholar
14. Riehm, C.R., Linear and quadratic Schur subgroups, Lecture Notes, 1988, preprint.Google Scholar
15. Riehm, C.R., The linear and quadratic Schur subgroups over the S-integers of a number field, American Mathematical Society, 107- 1(1989), 83–87.Google Scholar
16. Riehm, C.R., The Schur subgroup of the Brauer group of cylotomic rings of integers, American Mathematical Society, 103-1(1988), 27–30.Google Scholar
17. Saltman, D.J., Azumaya algebras with involution, Journal of Algebra, 52(1978), 526–539.Google Scholar
18. Scharlau, W., Involutions on simple algebras and orders, Canadian Mathematical Society Conference Proceedings, 4(1984), 141–157.Google Scholar
19. Schur, I., Untersuchungen iiber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. Gesammelte Abhandlungenl, 198–250. Springer Verlag, Berlin, 1973.Google Scholar
20. Serre, J.P., Cours d'Arithmétique, le Mathématicien (P.U.F.), 2(1970).Google Scholar
21. Serre, J.P., Corps Locaux. Hermann, Paris, 1962.Google Scholar
22. Swan, R.G., Projective modules over group rings and maximal orders, Annals ofMathematics, 76-1(1962), 55–61.Google Scholar
23. Thompson, J.G., Finite groups and even lattices, Journal of Algebra, 38(1970), 523–524.Google Scholar
24. Vigneras, M.F., Arithmétique des Algèbres de Quaternions. Lecture Notes in Mathematics, 800, Springer- Verlag, Berlin, 1980.Google Scholar
25. Yamada, T., The Schur subgroup of the Brauer group. Lecture Notes in Mathematics, 397, Springer-Verlag, Berlin, 1974.Google Scholar
26. Zmud, E.M., Symplectic geometries of finite abelian groups, Mat. Sbomik 86–1.(128)(1971), English translation: Mathematics of the USSR Sbornik, American Mathematical Society, 15-1(1971), 7–29.Google Scholar