A variety associated to an algebra with involution
Let \(A\) be a central simple algebra of degree \(n\) over a field \(F\) with an orthogonal involution \(*\). In analogy with the classical construction of the Brauer-Severi variety \(BS(A)\) of \(A\), the author defines a Brauer- Severi variety \(IV(A,*)\), which he calls the involution variety, for the pair \((A,*)\) and the main topic of this paper is to discuss properties of \(IV(A,*)\). The function field \(K\) of \(IV(A,*)\) is a generic isotropic splitting field for \(IV(A,*)\), i.e. it splits \(A\) and \(*\) is over \(K\) the adjoint involution of an isotropic quadratic form. The kernel of the map of Brauer groups \(\text{Br}(F)\to\text{Br}(K)\) is the subgroup generated by the class of \(A\), except when \((A,*)\) is the tensor product of two quaternion algebras \(Q_ 1\) and \(Q_ 2\) with canonical involutions, in which case the kernel is generated by the classes of \(Q_ 1\) and \(Q_ 2\). This case is quite special (``\(D_ 2= A_ 1+ A_ 1\)'') and has to be treated separately. Further the author gives some criteria to determine when \(K\), which has transcendence degree \(n-2\), can be embedded into \(K_ B\), the function field \(K_ B\) of \(BS(A)\), which has transcendence degree \(n-1\). Finally the author computes the Quillen \(K\)- theory of \(IV(A,*)\) and applies the result to obtain an index reduction formula for the function field \(K\) of \(IV(A,*)\).
scientific article; zbMATH DE number 645092