Geodesic
Reduced unitary $K$-theory. Aplications to algebraic groups
Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 533-548 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_1981_38_4_a5,
     author = {V. I. Yanchevskii},
     title = {Reduced unitary $K$-theory. {Aplications} to algebraic groups},
     journal = {Sbornik. Mathematics},
     pages = {533--548},
     year = {1981},
     volume = {38},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_4_a5/}
}
TY  - JOUR
AU  - V. I. Yanchevskii
TI  - Reduced unitary $K$-theory. Aplications to algebraic groups
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 533
EP  - 548
VL  - 38
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1981_38_4_a5/
LA  - en
ID  - SM_1981_38_4_a5
ER  - 
%0 Journal Article
%A V. I. Yanchevskii
%T Reduced unitary $K$-theory. Aplications to algebraic groups
%J Sbornik. Mathematics
%D 1981
%P 533-548
%V 38
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1981_38_4_a5/
%G en
%F SM_1981_38_4_a5
V. I. Yanchevskii. Reduced unitary $K$-theory. Aplications to algebraic groups. Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 533-548. http://geodesic.mathdoc.fr/item/SM_1981_38_4_a5/

[1] V. P. Platonov, “Problema Tankaka–Artina i privedennaya $K$-teoriya”, Izv. AN SSSR, seriya matem., 40 (1976), 227–260 | MR

[2] V. P. Platonov, “Privedennaya $K$-teoriya i approksimatsiya v algebraicheskikh gruppakh”, Trudy Matem. in-ta im. V. A. Steklova, CXLII (1976), 198–207 | MR

[3] V. P. Platonov, “Algebraic groups and reduced $K$-theory”, Proc. Internat. Congress Math., Helsinki, 1978

[4] V. P. Platonov, “Biratsionalnye svoistva privedennoi gruppy Uaitkheda”, DAN BSSR, 21:3 (1977), 197–198 | MR

[5] V. I. Yanchevskii, “Privedennaya unitarnaya $K$-teoriya i tela nad genzelevymi diskretno normirovannymi polyami”, Izv. AN SSSR, seriya matem., 42 (1978), 879–918 | MR

[6] V. P. Platonov, V. I. Yanchevskii, “O stabilnosti v privedennoi $K$-teorii”, DAN SSSR, 242:4 (1978), 769–772 | MR | Zbl

[7] V. P. Platonov, “Beskonechnost privedennoi gruppy Uaitkheda v probleme Tannaka–Artina”, Matem. sb., 100(142) (1976), 191–200 | MR | Zbl

[8] V. I. Yanchevskii, “Obratnaya zadacha privedennoi unitarnoi $K$-teorii”, Matem. zametki, 26:3 (1979), 475–482 | MR | Zbl

[9] M. Kneser, In Coll. Theorie des groupes algebriques, Bruxelles, 1962

[10] G. E. Wall, “The structure of a unitary factor group”, Publ. Math. IHES, 1:3 (1959) | MR

[11] V. E. Voskresenskii, “O privedennoi gruppe Uaitkheda prostoi algebry”, Uspekhi matem. nauk, XXVI:6(162) (1978), 247–248 | MR

[12] N. Dzhekobson, Teoriya kolets, IL, Moskva, 1949

[13] C. Riehm, “The congruence subgroup problem over local fields”, Amer. J. Math., 92:3 (1970), 771–778 | DOI | MR | Zbl

[14] S. Wang, “On Grunwald's theorem”, Ann. Math., 51:2 (1950), 471–484 | DOI | MR | Zbl