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

TL;DR

This paper investigates the Brauer group of products of tori and semisimple algebraic groups, providing conditions for isomorphisms in cohomology, and applies these results to compute Brauer groups of reductive groups and affine quadrics, also exploring p-torsion in characteristic p.

Contribution

It establishes conditions under which the cohomological Brauer group pull-back is an isomorphism for products involving tori, and applies this to compute Brauer groups of specific algebraic varieties.

Findings

Identifies conditions for isomorphism of Brauer groups in product schemes.

Computes Brauer groups of certain reductive groups and affine quadrics.

Analyzes p-torsion in Brauer groups over fields of characteristic p.

Abstract

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.