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 Mat. Sb. (N.S.), 1979, Volume 110(152), Number 4(12), Pages 579–596 (Mi msb2511)

Reduced unitary $K$-theory. Aplications to algebraic groups

V. I. Yanchevskii

Abstract: In this paper a number of facts of reduced $K$-theory are proved, and some applications to algebraic groups are also considered.
In § 2 a stability theorem is proved for reduced unitary Whitehead groups under certain imbeddings in the skew field of fractions of noncommutative polynomial rings.
In §§ 3,4 the author considers reduced unitary Whitehead groups of skew fields over doubly henselian discretely valued fields; exact sequences are given for their computation. In particular, the nontriviality of the reduced unitary functor is proved for cyclic algebras.
The fifth and sixth sections are devoted to applications to algebraic groups. In particular, general problems of weak approximation and rationality for special unitary groups are solved in the negative.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1981, 38:4, 533–548

Bibliographic databases:

UDC: 513.6
MSC: Primary 18F25; Secondary 20G15, 14L10

Citation: V. I. Yanchevskii, “Reduced unitary $K$-theory. Aplications to algebraic groups”, Mat. Sb. (N.S.), 110(152):4(12) (1979), 579–596; Math. USSR-Sb., 38:4 (1981), 533–548

Citation in format AMSBIB
\Bibitem{Yan79} \by V.~I.~Yanchevskii \paper Reduced unitary $K$-theory. Aplications to algebraic groups \jour Mat. Sb. (N.S.) \yr 1979 \vol 110(152) \issue 4(12) \pages 579--596 \mathnet{http://mi.mathnet.ru/msb2511} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=562210} \zmath{https://zbmath.org/?q=an:0462.20037|0426.20032} \transl \jour Math. USSR-Sb. \yr 1981 \vol 38 \issue 4 \pages 533--548 \crossref{https://doi.org/10.1070/SM1981v038n04ABEH001460} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981LQ11400006} 

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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. P. Platonov, “The arithmetic theory of algebraic groups”, Russian Math. Surveys, 37:3 (1982), 1–62
2. Yu. L. Ershov, “Henselian valuations of division rings and the group $SK_1$”, Math. USSR-Sb., 45:1 (1983), 63–71
3. Zhuk I., “Rationality of Some Homogeneous Spaces of the So(Q) Group”, Dokl. Akad. Nauk Belarusi, 26:9 (1982), 773–775
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. Kaskevich V., “The Structure of Unitary and Orthogonal Groups Over Henselian Discrete Valuation Azymaya Algebras”, Dokl. Akad. Nauk Belarusi, 31:12 (1987), 1073–1076
6. A. P. Monastyrnyi, V. I. Yanchevskii, “Whitehead groups of spinor groups”, Math. USSR-Izv., 36:1 (1991), 61–100
7. Monastyrnyi A. Janchevskii V., “Whitehead-Groups of the Classical Type Algebraic-Groups and the R-Equivalence”, 314, no. 1, 1990, 110–114
8. Ulf Rehmann, Sergey V. Tikhonov, Vyacheslav I. Yanchevskiǐ, “Prescribed behavior of central simple algebras after scalar extension”, Journal of Algebra, 2011
9. 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
10. Wadsworth A.R., “Unitary Sk1 of Semiramified Graded and Valued Division Algebras”, Manuscr. Math., 139:3-4 (2012), 343–389
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