  Let T be a torus (not assumed to be split) over a field F, and denote by
$_n{H^{2}_{et}(X,Gm)}$ 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
$_n{H^2_{et}(T,Gm)}\to_n{H^2_{et}(X\times T,Gm)}$ 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.
