REDUCED UNITARYK-THEORY AND DIVISION RINGS OVER DISCRETELY VALUED HENSEL FIELDS

Abstract

In this paper a Hermitian analog of reduced K-theory is constructed. The author studies the reduced unitary Whitehead groups SUK1(A) of simple finite-dimensional central algebras A over a field K, which arise both in unitary K-theory and in the theory of algebraic groups. In the case of discretely valued Hensel fields K, with this end in mind groups of unitary projective conorms are introduced, with the aid of which the groups SUK1(A) are included in exact sequences whose terms are computable in many important cases. For a number of special fields K of significant interest the triviality of the groups SUK1(A) is deduced from this. In addition, for an important class of simple algebras a formula is proved that reduces the computation of SUK1(A) to the calculation of so-called relative involutory Brauer groups, which are easily computable in many cases. Furthermore, for an arbitrary field K the behavior of SUK1(A) is described when K undergoes a purely transcendental extension, which in the case of division rings of odd index is a stability theorem important for many applications.Bibliography: 31 titles.