Brauer Groups of Homogenecus Spaces, I

Abstract

This work originates with my desire to find an algebraic homogeneous space, with respect to an affine algebraic group over (C the field of complex numbers which is not a rational algebraic variety. The question in fact is open, but I believe that a non- rational homogeneous space will be found. A quick perusal of the facts involved shows that the chief candidate for such a homogeneous space would be one of the form G/K for G and K both semi-simple over (E and K not contained in any parabolic subgroup of G. In fact this last condition is not essential, but by considering Levi factors one sees that the essential part of the problem lies here.