This article is cited in 9 scientific papers (total in 9 papers)
Reduced unitary $K$-theory and division rings over discretely valued Hensel fields
V. I. Yanchevskii
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.
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Mathematics of the USSR-Izvestiya, 1979, 13:1, 175–213
MSC: Primary 16A54, 16A39; Secondary 16A28
V. I. Yanchevskii, “Reduced unitary $K$-theory and division rings over discretely valued Hensel fields”, Izv. Akad. Nauk SSSR Ser. Mat., 42:4 (1978), 879–918; Math. USSR-Izv., 13:1 (1979), 175–213
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\paper Reduced unitary $K$-theory and division rings over discretely valued Hensel fields
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\jour Math. USSR-Izv.
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This publication is cited in the following articles:
V. I. Yanchevskii, “Reduced unitary $K$-theory. Aplications to algebraic groups”, Math. USSR-Sb., 38:4 (1981), 533–548
A. E. Zalesskii, “Linear groups”, Russian Math. Surveys, 36:5 (1981), 63–128
Yu. L. Ershov, “Henselian valuations of division rings and the group $SK_1$”, Math. USSR-Sb., 45:1 (1983), 63–71
V. P. Platonov, V. I. Yanchevskii, “Dieudonné's conjecture on the structure of unitary groups over a division ring, and Hermitian $K$-theory”, Math. USSR-Izv., 25:3 (1985), 573–599
A. P. Monastyrnyi, V. I. Yanchevskii, “Whitehead groups of spinor groups”, Math. USSR-Izv., 36:1 (1991), 61–100
J.-F Renard, J.-R Tignol, A.R. Wadsworth, “Graded Hermitian forms and Springer's theorem”, Indagationes Mathematicae, 18:1 (2007), 97
V. I. Yanchevskii, “Reduced Whitehead groups and conjugacy problem for special unitary groups of anisotropic hermitian forms”, J. Math. Sci. (N. Y.), 192:2 (2013), 250–262
A. R. Wadsworth, “Unitary SK1 of semiramified graded and valued division algebras”, manuscripta math, 139:3-4 (2012), 343
S. V. Tikhonov, V. I. Yanchevskii, “Homomorphisms and involutions of unramified henselian division algebras”, J. Math. Sci. (N. Y.), 209:4 (2015), 657–664
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