The relative Brauer group and generalized cyclic crossed products for a ramified covering

Let T/AT/A be an integral extension of noetherian integrally closed integral domains whose quotient field extension is a finite cyclic Galois extension. Let S/RS/R be a localization of this extension which is unramified. Using a generalized cyclic crossed product construction it is shown that certain reflexive fractional ideals of T with trivial norm give rise to Azumaya R-algebras that are split by S  . Sufficient conditions on T/AT/A are derived under which this construction can be reversed and the relative Brauer group of S/RS/R is shown to fit into the exact sequence of Galois cohomology associated to the ramified covering T/AT/A. Many examples of affine algebraic varieties are exhibited for which all of the computations are carried out.