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The Schur subgroup of the Brauer group of cyclotomic rings of integers


Author: C. Riehm
Journal: Proc. Amer. Math. Soc. 103 (1988), 27-30
MSC: Primary 13A20; Secondary 11S25, 12E15, 20C10
DOI: https://doi.org/10.1090/S0002-9939-1988-0938638-X
MathSciNet review: 938638
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Abstract: Let $ K$ be a finite abelian extension of the rational numbers $ \mathbb{Q}$. Let $ {\mathbf{S}}$ be a finite set of primes of $ K$ including the infinite ones, and let $ \mathfrak{o}$ be the ring of $ {\mathbf{S}}$-integers in $ K$. Then the Schur subgroup $ S(\mathfrak{o})$ of the Brauer group $ B(\mathfrak{o})$ is defined, in analogy with $ S(K)$, via representations of finite groups on finitely generated projective $ \mathfrak{o}$-modules. It is easy to see that $ S(\mathfrak{o}) \subseteq S(K) \cap B(\mathfrak{o})$. We shall show that there is equality in the case of $ K$ a purely cyclotomic extension $ \mathbb{Q}({\varepsilon _m})$ of $ \mathbb{Q}$ (where $ {\varepsilon _m}$ is an $ m$th root of 1).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938638-X
Keywords: Schur subgroup, Brauer group, Azumaya algebras, integral representations, representations of finite groups
Article copyright: © Copyright 1988 American Mathematical Society

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