A Variety Associated to an Algebra with Involution

Abstract

A variety IV( A, *) is canonically associated to a central simple algebra A with orthogonal involution of the first kind *. The properties of this variety are the main topic of this paper. The paper is divided into two parts. We first define this variety to be the subvariety of the Grassmannian of the n-dimensional subspaces of the n 2-dimensional vector space A consisting of all the n-dimensional right ideals I of A, such that I* I = 0. The geometric properties of IV( A, *) are investigated. In particular the generic property of this variety is identified. Further, the kernel of the scalar extension map Br( F) → Br( K), where K is the function field of IV( A, *), is also computed. Lastly, a criterion on the embeddability of the field K into the function field of the Brauer-Severi variety of A is given. In the second part, the Quillen K-theory of IV( A, *) is computed. The K-theory result then leads to an index reduction formula for the function field K.