Brauer groups, Hopf algebras and Galois theory
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 \(H\)-\(A\)-Hopf modules. Amitsur, Sweedler and Harrison cohomologies are used to describe the group of \(H\)-Galois objects. Among the examples are when \(H\) is a monogenic non-free Larson order of prime power rank. The third part of this monograph, comprising 150 pages, is an exposition of the Brauer-Long group \(BD(R,H)\) of equivalence classes of \(H\)-Azumaya algebras for \(H\) a commutative, cocommutative \(R\)-Hopf algebra which is faithfully projective of finite rank. \(BD(R,H)\) is a vast generalization of the Brauer-Wall group, first introduced to help understand Clifford algebras [\textit{C. T. C. Wall}, J. Reine Angew. Math. 213, 187-199 (1964; Zbl 0125.01904)]. The split part \(BD^s(R,H)\) of \(BD(R,H)\) is the subgroup consisting of \(H\)-Azumaya algebras which are twisted forms of endomorphism rings of faithfully projective \(H\)-dimodules: \(BD^s(R,H)\) is the product of the Brauer group of \(R\) and the groups of \(H\)-Galois objects and \(H^*\)-Galois objects. The cokernel of \(BD^s\) in \(BD\) is a kind of orthogonal group which is difficult to determine even when \(R=\mathbb{C}\): c.f. section 13.11 (or the reviewer [Trans. Am. Math. Soc. 204, 137-160 (1975; Zbl 0308.13005)], section 5, which includes cases not covered in 13.11). The author presents his results, some with \textit{M. Beattie} [e.g., Trans. Am. Math. Soc. 324, No. 2, 747-775 (1991; Zbl 0724.13003)], giving both a cohomological description of \(BD^s\) and, in some cases, the explicit multiplication of elements of \(BD\). A final chapter introduces the Brauer group of Yetter-Drinfel'd module algebras, which generalizes \(BD(R,G)\) to the case where \(H\) need no longer be faithfully projective, commutative or cocommutative. The bibliography contains 209 items. As Chapter 13 of this work makes evident, the Brauer-Long group is one of the most complicated invariants of a commutative ring yet conceived. This exposition, by the world's leading expert on \(BD\), is likely to be a standard reference for this invariant for years to come. This book will also be a useful resource for researchers interested in the Taylor Brauer group or in Hopf Galois extensions.
scientific article; zbMATH DE number 1115314