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 Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 4, Pages 879–918 (Mi izv1851)

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.

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English version:
Mathematics of the USSR-Izvestiya, 1979, 13:1, 175–213

Bibliographic databases:

UDC: 513.6
MSC: Primary 16A54, 16A39; Secondary 16A28

Citation: 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

Citation in format AMSBIB
\Bibitem{Yan78} \by V.~I.~Yanchevskii \paper Reduced unitary $K$-theory and division rings over discretely valued Hensel fields \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1978 \vol 42 \issue 4 \pages 879--918 \mathnet{http://mi.mathnet.ru/izv1851} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=508832} \zmath{https://zbmath.org/?q=an:0422.20032|0389.20035} \transl \jour Math. USSR-Izv. \yr 1979 \vol 13 \issue 1 \pages 175--213 \crossref{https://doi.org/10.1070/IM1979v013n01ABEH002018} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1979JB17800011} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Yanchevskii, “Reduced unitary $K$-theory. Aplications to algebraic groups”, Math. USSR-Sb., 38:4 (1981), 533–548
2. A. E. Zalesskii, “Linear groups”, Russian Math. Surveys, 36:5 (1981), 63–128
3. Yu. L. Ershov, “Henselian valuations of division rings and the group $SK_1$”, Math. USSR-Sb., 45:1 (1983), 63–71
4. 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
5. A. P. Monastyrnyi, V. I. Yanchevskii, “Whitehead groups of spinor groups”, Math. USSR-Izv., 36:1 (1991), 61–100
6. J.-F Renard, J.-R Tignol, A.R. Wadsworth, “Graded Hermitian forms and Springer's theorem”, Indagationes Mathematicae, 18:1 (2007), 97
7. 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
8. A. R. Wadsworth, “Unitary SK1 of semiramified graded and valued division algebras”, manuscripta math, 139:3-4 (2012), 343
9. 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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