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 linear and quadratic Schur subgroups over the $ S$-integers of a number field


Author: Carl R. Riehm
Journal: Proc. Amer. Math. Soc. 107 (1989), 83-87
MSC: Primary 11R65; Secondary 11R54, 13A20, 20C10
DOI: https://doi.org/10.1090/S0002-9939-1989-0979218-0
MathSciNet review: 979218
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be an algebraic number field and let $ \mathfrak{O}$ be a ring of $ S$-integers in $ K$ (where $ S$ is a set of primes of $ K$ containing all the archimedean primes); that is to say, $ \mathfrak{O}$ is a Dedekind domain whose field of quotients is $ K$. In analogy with a theorem of T. Yamada in the case of a field of characteristic 0, it is shown that if $ S\left( \mathfrak{O} \right)$ is the Schur subgroup of the Brauer group $ B\left( \mathfrak{O} \right)$ and if $ \mathfrak{o} = \mathfrak{O} \cap k$, where $ k$ is any field containing the maximal abelian extension of $ \mathbb{Q}$ in $ K$, then $ S\left( \mathfrak{O} \right) = \mathfrak{O} \otimes S\left( \mathfrak{o} \right)$, i.e. the Brauer classes in $ S\left( \mathfrak{O} \right)$ are those obtained from $ S\left( \mathfrak{o} \right)$ by extension of the scalars to $ \mathfrak{O}$. A similar theorem is proved as well in the case of the Schur subgroup $ S\left( {\mathfrak{O},\omega } \right)$ of the quadratic Brauer group $ B\left( {\mathfrak{O},\omega } \right)$, where $ \omega $ is an involution of $ \mathfrak{O}$.


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

  • [H-T-W] I. Hambleton, L. Taylor, and E. B. Williams, An introduction to the maps between surgery obstruction groups, Algebraic Topology, Aarhus 1982, pp. 49-127; Lecture Notes in Math., vol. 1051, Springer-Verlag, New York, 1984. MR 764576 (86b:57017)
  • [I] I. M. Isaacs, Character theroy of finite groups, Academic Press, New York, 1976. MR 0460423 (57:417)
  • [J] G. J. Janusz, Tensor products of orders, J. London Math. Soc. (2) 20 (1979), 186-192. MR 551444 (81g:16009)
  • [M] R. Mollin, The Schur group of a field of characteristic zero, Pacific J. Math. 76 (1978), 471-478. MR 506148 (80c:12018)
  • [MO] I. Reiner, Maximal orders, London Math. Soc. Monographs, no. 5, Academic Press, London, 1975.
  • [O-S] M. Orzech and C. Small, The Brauer group of commutative rings, Lecutre Notes in Pure and Appl. Math., no. 11, Dekker, New York, 1975. MR 0457422 (56:15627)
  • [R1] C. Riehm, The Schur subgroup of the Brauer group of cyclotomic rings of integers, Proc. Amer. Math. Soc. 103(1988), 27-30. MR 938638 (89c:13005)
  • [R2] C. Riehm, The quadratic Schur subgroup over local and global fields Math. Annalen, 283 ((1989), 479-489. MR 985243 (90c:12003)
  • [S] D. Saltman, Azumaya algebras with involution, J. Algebra 52 (1978), 526-539. MR 495234 (80a:16013)
  • [Y] T. Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Math., vol. 397, Springer-Verlag, New York, 1974. MR 0347957 (50:456)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11R65, 11R54, 13A20, 20C10

Retrieve articles in all journals with MSC: 11R65, 11R54, 13A20, 20C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0979218-0
Keywords: Schur subgroup, integral Brauer group, integral representations, representations of finite groups, quadratic Brauer group, Azumaya algebras, algebraic number fields
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society