Brauer Groups, Hopf Algebras and Galois Theory

This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and étale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included.

The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph.

Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use.

Résumé
Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra.

Contenu
I The Brauer group of a commutative ring.- 1 Morita theory for algebras without a unit.- 2 Azumaya algebras and Taylor-Azumaya algebras.- 3 The Brauer group.- 4 Central separable algebras.- 5 Amitsur cohomology and étale cohomology.- 6 Cohomological interpretation of the Brauer group.- II Hopf algebras and Galois theory.- 7 Hopf algebras.- 8 Galois objects.- 9 Cohomology over Hopf algebras.- 10 The group of Galois (co)objects.- 11 Some examples.- III The Brauer-Long group of a commutative ring.- 12 H-Azumaya algebras.- 13 The Brauer-Long group of a commutative ring.- 14 The Brauer group of Yetter-Drinfel'd module algebras.- A Abelian categories and homological algebra.- A.1 Abelian categories.- A.2 Derived functors.- B Faithfully flat descent.- C Elementary algebraic K-theory.