Alternating forms and the Brauer group of a geometric field

Abstract

We compute the theory of H 2 ( G , Q / Z ) H^{2}(G,\mathbb {Q}/\mathbb {Z}) for any proabelian group G G , using a natural isomorphism with the group Alt ⁡ ( G , Q / Z ) \operatorname {Alt}(G,\mathbb {Q}/\mathbb {Z}) of continuous alternating forms. We use this to establish a sort of generic behavioral ideal, or role model, for the Brauer group Br ( F ) \text {Br}(F) of a geometric field F F of characteristic zero. We show this ideal is attained in several interesting cases.