This article is cited in
13 papers
Reduced unitary $K$-theory and division rings over discretely valued Hensel fields
V. I. Yanchevskii
Abstract:
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.
UDC:
513.6
MSC: Primary
16A54,
16A39; Secondary
16A28 Received: 21.07.1977
Fulltext:
PDF file (4275 kB)
