Brauer groups, Hopf algebras and Galois theory
type: Item
instance of: scholarly article
title: Brauer groups, Hopf algebras and Galois theory
zbMATH Open document ID: 0898.16001
author: Stefaan Caenepeel
published in: K-Monographs in Mathematics
publication date: 1998-02-09T00:00:00Z
review text: This ambitious monograph presents the theory of the Brauer-Long group of equivalence classes of \(H\)-dimodule algebras, where \(H\) is a faithfully projective, commutative and cocommutative \(R\)-Hopf algebra, \(R\) a commutative ring. The Brauer-Long group, introduced by \textit{F. W. Long} [J. Algebra 30, 559-601 (1974; Zbl 0282.16007)], has embedded in it the usual Brauer group of Azumaya algebras of \(R\), as well as the group of \(H\)-Galois objects, so the first two parts of this work give an exposition of the Brauer group and the group of \(H\)-Galois objects. Much of the material has appeared previously only in the periodical literature, and some of the theory and examples are new. The approach is primarily categorical and homological. The presentation of the Brauer group takes up the first 170 pages. The treatment is novel. Taylor Azumaya algebras are introduced and presented from a Morita-theoretic viewpoint: classical Azumaya algebras are Taylor Azumaya algebras with unit. Etale and related cohomology theories are introduced over commutative rings (not schemes), leading to Taylor's theorem [\textit{J. L. Taylor}, Pac. J. Math. 103, 163-203 (1982; Zbl 0528.13007)]\ identifying the Taylor Brauer group with a second étale cohomology group, and Gabber's theorem identifying the usual Brauer group with the torsion subgroup of \(H^2\) [\textit{O. Gabber}, Lect. Notes Math. 844, 129-209 (1981; Zbl 0472.14013)]. The relation between the Brauer group and \(H^2\) follows the approach of \textit{O. E. Villamayor} and \textit{D. Zelinsky} [J. Pure Appl. Algebra 10, 19-55 (1977; Zbl 0383.13002)]. The treatment is self-contained except for Artin's Refinement Theorem. The second part, comprising 130 pages, studies Hopf Galois extensions of commutative rings. As with Azumaya algebras, a module-theoretic point of view is used: an \(H\)-comodule \(R\)-algebra \(A\) is an \(H\)-Galois object if the category of \(R\)-modules is equivalent to the category of right \(H\)-\(A\)…
reviewed by: Lindsay N. Childs
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 13B05
zbMATH DE Number: 1115314
zbMATH Keywords: Brauer groups
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: Brauer-Long groups
zbMATH Keywords: Hopf Galois extensions
zbMATH Keywords: Amitsur cohomology
zbMATH Keywords: Sweedler cohomology
zbMATH Keywords: Galois objects
zbMATH Keywords: Yetter-Drinfeld module algebras
zbMATH Keywords: dimodule algebras
zbMATH Keywords: commutative Hopf algebras
zbMATH Keywords: Larson orders
zbMATH Keywords: cocommutative Hopf algebras
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description: scientific article; zbMATH DE number 1115314