Every finite abelian group is the Brauer group of a ring

Abstract

Given an arbitrary finite abelian group G G a ring R R is constructed using cohomological techniques from algebraic geometry whose Brauer group is G G . If G G is a cyclic group, then R R can be taken to be a three-dimensional noetherian integral domain. If G G is not a cyclic group the ring R R is a three-dimensional noetherian ring. At the expense of raising the dimension of R R , R R can be chosen to be a domain. We also calculate B ( R [ x , 1 / x ] ) B(R[x,1/x]) for R R a commutative noetherian regular ring containing a field of characteristic zero.