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.