Geodesic
Schur and Projective Schur Groups of Number Rings
Canadian journal of mathematics, Tome 43 (1991) no. 3, pp. 540-558

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Nelis, Peter. Schur and Projective Schur Groups of Number Rings. Canadian journal of mathematics, Tome 43 (1991) no. 3, pp. 540-558. doi: 10.4153/CJM-1991-033-5
@article{10_4153_CJM_1991_033_5,
     author = {Nelis, Peter},
     title = {Schur and {Projective} {Schur} {Groups} of {Number} {Rings}},
     journal = {Canadian journal of mathematics},
     pages = {540--558},
     year = {1991},
     volume = {43},
     number = {3},
     doi = {10.4153/CJM-1991-033-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-033-5/}
}
TY  - JOUR
AU  - Nelis, Peter
TI  - Schur and Projective Schur Groups of Number Rings
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 540
EP  - 558
VL  - 43
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-033-5/
DO  - 10.4153/CJM-1991-033-5
ID  - 10_4153_CJM_1991_033_5
ER  - 
%0 Journal Article
%A Nelis, Peter
%T Schur and Projective Schur Groups of Number Rings
%J Canadian journal of mathematics
%D 1991
%P 540-558
%V 43
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-033-5/
%R 10.4153/CJM-1991-033-5
%F 10_4153_CJM_1991_033_5

[1] 1. Childs, L., Azumayas over number rings as crossed products, preprint 1986. Google Scholar

[2] 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] 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] 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] 5. De Meyer, F. and Ingraham, E., Separable algebras over commutative rings. Lecture Notes in Mathematics, 181, Springer-Verlag, Berlin, 1971. Google Scholar

[6] 6. Karpilovsky, G., Projective representations of finite groups. Marcel Dekkerlnc, 1985. Google Scholar

[7] 7. Kargapolov, M. and Merzliakov, I., Éléments de la théorie des groupes. Traduction Française, Éditions Mir, Moscou, 1985. Google Scholar

[8] 8. Lorenz, F. and Opolka, H., Einfache Algebren und projective Darstellungen endlicher Gruppen, Mathematische Zeitschrift 162(1978), 175–186. Google Scholar

[9] 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] 10. Nelis, P., Integral matrices spanned by finite groups, U.I.A. preprint, 12, 1990. Google Scholar

[11] 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] 12. Orzech, M. and Small, C., The Brauer Group of Commutative Rings. Lecture Notes in Pure and Applied Mathematics, 11, 1975. Google Scholar

[13] 13. Reiner, I., Maximal orders. S, L.M.. Monographs5, 1975. Google Scholar

[14] 14. Riehm, C.R., Linear and quadratic Schur subgroups, Lecture Notes, 1988, preprint. Google Scholar

[15] 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] 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] 17. Saltman, D.J., Azumaya algebras with involution, Journal of Algebra, 52(1978), 526–539. Google Scholar

[18] 18. Scharlau, W., Involutions on simple algebras and orders, Canadian Mathematical Society Conference Proceedings, 4(1984), 141–157. Google Scholar

[19] 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] 20. Serre, J.P., Cours d'Arithmétique, le Mathématicien (P.U.F.), 2(1970). Google Scholar

[21] 21. Serre, J.P., Corps Locaux. Hermann, Paris, 1962. Google Scholar

[22] 22. Swan, R.G., Projective modules over group rings and maximal orders, Annals ofMathematics, 76-1(1962), 55–61. Google Scholar

[23] 23. Thompson, J.G., Finite groups and even lattices, Journal of Algebra, 38(1970), 523–524. Google Scholar

[24] 24. Vigneras, M.F., Arithmétique des Algèbres de Quaternions. Lecture Notes in Mathematics, 800, Springer- Verlag, Berlin, 1980. Google Scholar

[25] 25. Yamada, T., The Schur subgroup of the Brauer group. Lecture Notes in Mathematics, 397, Springer-Verlag, Berlin, 1974. Google Scholar

[26] 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

Cité par Sources :