The Brauer Group of a Commutative Ring

Abstract

The Brauer group Br(R) of a commutative ring was introduced by Auslander and Goldman in their 1960 paper The Brauer Group of a Commutative Ring, building on earlier work of Azumaya. This group coincides with the “classical” Brauer group (cf. Chapter 4) in the case when R is a field. One of the points of extending the theory to rings is that one can relate Brauer groups of fields to Brauer groups of related rings in exact sequences; one then hopes that this will help compute the classical Brauer group. The Brauer group of a commutative ring is also part of a Galois theory of commutative rings. For more on these matters, the reader may consult Galois Theory and Cohomology of Commutative Rings by Chase, Harrison and Rosenberg, The Brauer Group of Commutative Rings by Orzech and Small, Separable Algebras Over Commutative Rings by DeMeyer and Ingraham, or the paper of Auslander and Goldman quoted above.