The relative Brauer group and generalized cyclic crossed products for a ramified covering
type: Item
instance of: scholarly article
title: The relative Brauer group and generalized cyclic crossed products for a ramified covering
zbMATH Open document ID: 1376.14025
author: Timothy J. Ford
published in: Journal of Algebra
publication date: 2016-01-29T00:00:00Z
review text: The crossed product theorem states that every central simple algebra over a field \(K\) split by a Galois extension \(L\) of \(K\) is Brauer equivalent to a crossed product, otherwise stated, the relative Brauer group \(B(L/K)\) is isomorphic to the second cohomology group \(H^2(G,L^*)\). If \(T\) is a Galois extension of a commutative ring \(A\), then we have a homomorphism \(H^2(G,T^*)\to B(T/A)\) that is in general not surjective or injective. This homomomorphism fits into a 7-term exact sequence, due to Chase, Harrison and Rosenberg [\textit{S. U. Chase} et al., Mem. Am. Math. Soc. 52, 15--33 (1965; Zbl 0143.05902)]. At the level of algebras, this can restated in the following form: every \(A\)-Azumaya algebra split by \(T\) is Brauer equivalent to a \textsl{generalized} crossed product, a statement originally due to \textit{T. Kanzaki} [Osaka J. Math. 5, 175--188 (1968; Zbl 0202.04202)]. The generalization comes from the fact that such a generalized crossed product Azumaya algebra is isomorphic as an \(T\)-module to a direct sum of \(|G|\) rank one \textsl{projective} \(T\)-modules, instead of rank one \textsl{free} modules. The starting point of the present paper is the following. Let \(T/A\) be a ramified extension of noetherian integrally closed integral domains for which the extension of quotient fields L/K is Galois with cyclic group \(G=\langle\sigma\rangle\). Now define generalized cyclic crossed modules starting from rank one \textsl{reflexive} \(T\)-modules, or, equivalently, reflexive fractional ideals of \(T\) in \(L\). The fact that \(G\) is cyclic entails that the generalized crossed product is determined by a single reflexive \(T\)-module, allowing the author to make explicit computations. \textit{D. S. Rim} [Proc. Am. Math. Soc. 16, 837--840 (1965; Zbl 0166.30604)] presented a divisorial version of the Chase-Harrison-Rosenberg exact sequence, where the group replacing the relative Brauer group is described in cohomological terms. In some cases, t…
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 13C20
zbMATH DE Number: 6537406
zbMATH Keywords: Brauer group
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: Generalized crossed products
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Divisor class group
reviewed by: Stefaan Caenepeel
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.10.013
OpenAlex ID: W2185937821
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DOI: 10.1016/J.JALGEBRA.2015.10.013
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