Brauer groups of algebraic function fields

Abstract

Let E be a countable field. Then B(E),, the p-primary component of the Brauer group of E, if a countable abelian torsion group. As such, the Ulm invariants of B(E), classify that group modulo its maximal divisible subgroup. For an exposition of the Ulm theory, see [I 11. In [9] (see also [7]) the authors and Jack Sonn determine the Ulm invariants of B(E), when E is a rational function field over a global field. The present paper is motivated by this result, and is concerned with two further lines of investigation suggested by this work. In Section 2 we attempt to extend the results of [9] to the case when E is an algebraic function field over a global field. This case seems considerably more difficult than the rational function field case. If E is a rational function field, B(E), is determined by means of the AuslanderBrumer-Faddeev Theorem [ 7, p. 5 1 ] in terms of Brauer groups and character groups of fields of lower transcendence degree. No such result is available in the algebraic function field case. Nevertheless, we are able to completely determine the Ulm invariants of B(E), at finite ordinals when E is an algebraic function field over a global field; this result is obtained using the Merkurjev-Suslin Theorem [14], Saltman’s theory of generic Galois extensions [ 161, and the results of [8]. For E as above we have very little information about the Ulm invariants at infinite ordinals. We do, however,