Reduced unitary $K$-theory and division rings over discretely valued Hensel fields

V. I. Yanchevskii

Izvestiya. Mathematics , Tome 13 (1979) no. 1, pp. 175-213

Résumé

In this paper a Hermitian analog of reduced $K$-theory is constructed. The author studies the reduced unitary Whitehead groups $SUK_1(A)$ 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 $SUK_1(A)$ 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 $SUK_1(A)$ is deduced from this. In addition, for an important class of simple algebras a formula is proved that reduces the computation of $SUK_1(A)$ 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 $SUK_1(A)$ 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.