Azumaya algebras with involutions
Let R be a commutative ring and S/R an étale quadratic extension. Br(-) is the Brauer group and \({}_ 2Br(\)-) its 2-torsion. This paper studies the problem of when the sequence \[ _ 2Br(R)\quad \to^{Res}\quad_ 2Br(S)\quad \to^{Cores}\quad_ 2Br(R) \] is exact. If R, S are fields and char \(R\neq 2\), the sequence is known to be exact. From the authors' introduction: ``Let A be an Azumaya S-algebra whose class [A] belongs to kernel (Cores: \({}_ 2Br(S)\to_ 2Br(R))\). For [A] to belong to \(Res(_ 2Br(R))\), it is necessary and sufficient that there exists an Azumaya S-algebra B Brauer equivalent to A admitting involutions of first and second kinds which commute. Results of Saltman assert that there exist algebras \(B_ i\), \(i=1,2\), Brauer equivalent to A which support involutions of the first and second kind, respectively. We construct a homomorphism \[ h: \ker (Cores:_ 2Br(S)\to_ 2Br(R))\to Pic(R)/Cores Pic(S) \] whose kernel consists precisely of classes of Azumaya S-algebras which support commuting involutions of both kinds. Thus, h is a measure of the obstruction to the sequence being exact... Finally, we give an example of an étale quadratic extension S/R for which (the sequence) is not exact.''
scientific article; zbMATH DE number 4137973