The Brauer ring of a field

Abstract

The topic of this work is the Brauer group of a commutative ring. Recent investigations have yielded generalizations of this invariant in several directions; herein we wish to propose another: that the Brauer group be viewed as a subgroup of the unit group of the “Brauer ring”. In justification, we should expect this ring to store and yield information about separable algebras which the Brauer group does not, and we should hope to be able to recover the Brauer group from purely ring theoretic properties. One purpose of the present paper is to describe the extent to which these goals are achieved. Another is to show that, in a categorical sense, the ring we shall describe is best possible.

To obtain structural results for the Brauer ring, we will make use of the theory of Green-functors, especially of the Burnside ring. Our structure theory will be independent of the construction of the Brauer ring, thus giving promise for applications in other areas. So as to not get too lost in our categorical approach, we have omitted some straight-forward axiom checking proofs; they may be regarded as exercises.

In this paper we will restrict our attention to the field case, leaving the Brauer ring of more general algebraic objects to another time.

The author wishes to express his deep respect and thanks to Professor R. S. Pierce, for his endless insights in to the problems at hand, and for asking the fundamental questions on which this work is based.