Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Schur group conjecture for the ring of integers of a number field


Author: Peter Nelis
Journal: Proc. Amer. Math. Soc. 114 (1992), 307-318
MSC: Primary 11R21; Secondary 11R18, 16S34, 20C05
DOI: https://doi.org/10.1090/S0002-9939-1992-1070529-X
MathSciNet review: 1070529
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ R$ is the ring of $ \mathbb{S}$ integers of a subcyclotomic number field $ K$, then the Schur group conjecture asserts that the Schur group of $ R$ equals the intersection of the Brauer group of $ R$ and the Schur group of $ K$. We prove this assertion in case $ \mathbb{S}$ is the set of all Archimedian primes, i.e. when $ R$ is the ring of integers of $ K$.


References [Enhancements On Off] (What's this?)

  • [1] S. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955), 361-386. MR 0074393 (17:577c)
  • [2] S. Caenepeel and F. M. J. Van Oystaeyen, A note on generalized Clifford algebras and projective representations, Comm. Algebra, 17 (1989), 93-102. MR 970865 (90b:15021)
  • [3] C. W. Curtis and I. Reiner, Methods of representation theory with applications to finite groups and orders 1, Pure Appl. Math., Wiley, New York, 1981. MR 632548 (82i:20001)
  • [4] F. De Meyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Math., vol. 181, Springer-Verlag, Berlin, 1978.
  • [5] F. De Meyer and R. Mollin, The Schur group, orders and their applications, Lecture Notes in Math., vol. 1142, Springer-Verlag, Berlin, 1985, pp. 205-209. MR 812500 (87a:13003)
  • [6] -, The Schur group of a commutative ring, J. Pure Appl. Algebra 35 (1985), 117-122. MR 775464 (86c:13001)
  • [7] L. Dornhoff, Group representation theory, vol. 1, Pure Appl. Math., vol. 7, Dekker, 1971. MR 0347959 (50:458a)
  • [8] W. Feit, The computation of Schur indices, Israel J. Math. 46 (1983), 275-300. MR 730344 (85j:20004)
  • [9] J. M. Fontaine, Sur les decompositions des algèbres de groupes, Ann. Sci. École Norm. Sup. (4), 4 (1971), 121-180. MR 0313370 (47:1925)
  • [10] G. J. Janusz, Simple components of $ \mathbb{Q}[\operatorname{Sl}(2,q)]$, Comm. Algebra 1 (1974), 1-22. MR 0344323 (49:9062)
  • [11] P. Nelis and F. M. J. Van Oystaeyen, The projective Schur subgroup of the Brauer group and root groups of finite groups, J. Algebra (to appear).
  • [12] P. Nelis, Integral matrices spanned by finite groups, J. Pure Appl. Algebra, submitted.
  • [13] M. Orzech and Ch. Small, The Brauer group of commutative rings, Lecture Notes in Pure and Appl. Math., vol. 11, Dekker, New York, 1975. MR 0457422 (56:15627)
  • [14] I. Reiner, Maximal orders L. M. S. Monographs, vol. 5, 1975. MR 0393100 (52:13910)
  • [15] C. R. Riehm, Linear and quadratic Schur subgroups, lecture notes, 1988, preprint.
  • [16] -, The linear and quadratic Schur subgroups over the $ S$-integers of a number field, Proc. Amer. Math. Soc. 107 (1989), 83-87. MR 979218 (90a:11146)
  • [17] -, The Schur subgroup of the Brauer group of cyclotomic rings of integers, Proc. Amer. Math. Soc., 103 (1988), 27-30. MR 938638 (89c:13005)
  • [18] J. P. Serre, Représentations linéaires des groupes finis, Collection Méthodes, Hermann Paris, 1971.
  • [19] S. Tanaka, Construction and classification of irreducible representations of special linear group of the second order over a finite field, Osaka J. Math. 4 (1967), 65-84. MR 0219635 (36:2714)
  • [20] J. G. Thompson, Finite groups and even lattices, J. Algebra. 38 (1976), 523-524. MR 0399257 (53:3108)
  • [21] M. F. Vigneras, Arithmétique des algèbres de quaternions, Lecture Notes in Math., vol. 800, Springer-Verlag, Berlin, 1980. MR 580949 (82i:12016)
  • [22] T. Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Math., vol. 397, Springer-Verlag, Berlin, 1974. MR 0347957 (50:456)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11R21, 11R18, 16S34, 20C05

Retrieve articles in all journals with MSC: 11R21, 11R18, 16S34, 20C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1070529-X
Keywords: Group rings, Azumaya algebras, the integral Brauer group, the Schur group, linear and modular representations
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society