Computing the Brauer-Long group of \({\mathbb{Z}}/2\) dimodule algebras (Q1113977)

The authors compute the Brauer-Long group BD(R,\({\mathbb{Z}}/2)\) of classes of \({\mathbb{Z}}/2{\mathbb{Z}}\)-dimodule algebras over a connected ring R in which 2 is invertible. They show that \[ BD(R,{\mathbb{Z}}/2)\simeq {\mathbb{Z}}/2\times H^ 1(R,{\mathbb{Z}}/2)\times H^ 1(R,{\mathbb{Z}}/2)\times Br(R), \] where \(H^ 1\) is the etale cohomology and Br the Brauer group. An important part of the paper consists in giving explicit product rules. The paper concludes with some applications and examples.