On the Brauer group of the product of a torus and a semisimple algebraic group

Abstract

Let T be a torus (not assumed to be split) over a field F, and denote by ${}_n{H}_{et}^2(X,Gm)$ the subgroup of elements of exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism ${}_n{H}_{et}^2(T,Gm){\to }_n{H}_{et}^2(X\times T,Gm)$ is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characteristic p>0.

21 pages