Homogeneous skew-fields of non-commutative rational functions and their reduced Whitehead groups

V. I. Yanchevskiǐ

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 21, Tome 388 (2011), pp. 270-308

Résumé

A construction of skew-fields of non-commutative rational functions is studied. We discuss and prove criterions for such skew-fields to be homogeneous and finite-dimensional over their centers and describe relations between some objects defined in terms of the skew-fields of constants, which help to compute reduced Whitehead groups of corresponding skew-fields of non-commutative rational functions. In particular we present a proof of one previous result of V. P. Platonov and the author about reduced Whitehead groups of skew-fields of non-commutative rational functions announced in 1979 and obtain in non-Henselian case of such skew-fields analogues of all results of Yu. L. Ershov for Henselian situation.

Bibliographie

[1] N. Dzhekobson, Teoriya kolets, IL, M., 1947

[2] Yu. L. Ershov, “Genzelevy normirovaniya tel i gruppa $SK_1$”, Mat. sb., 117(159):1 (1982), 60–68 | MR | Zbl

[3] Dzh. Kassels, A. Frelikh (red.), Algebraicheskaya teoriya chisel, Mir, M., 1969 | MR

[4] P. Kon, Svobodnye koltsa i ikh svyazi, Mir, M., 1975 | MR

[5] S. Leng, Algebra, Mir, M., 1968

[6] R. Pirs, Assotsiativnye algebry, Mir, M., 1986 | MR

[7] V. P. Platonov, “O probleme Tannaka–Artina”, Dokl. AN SSSR, 221:5 (1975), 1038–1041 | MR | Zbl

[8] V. P. Platonov, “Problema Tannaka–Artina i gruppy proektivnykh konorm”, Dokl. AN SSSR, 222:6 (1975), 1299–1302 | MR | Zbl

[9] V. P. Platonov, “Problema Tannaka–Artina i privedennaya $K$-teoriya”, Izv. AN SSSR, ser. matem., 40:2 (1976), 227–261 | MR | Zbl

[10] V. P. Platonov, “Privedennaya $K$-teoriya i approksimatsiya v algebraicheskikh gruppakh”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 142, 1976, 198–207 | MR | Zbl

[11] V. P. Platonov, V. I. Yanchevskii, “$\mathrm{SK}_1$ dlya tel nekommutativnykh ratsionalnykh funktsii”, Dokl. AN SSSR, 249:5 (1979), 1064–1068 | MR | Zbl

[12] A. V. Prokopchuk, S. V. Tikhonov, V. I. Yanchevskii, “Ob obschikh elementakh v gruppakh $\mathrm{SK}_1$ dlya tsentralnykh prostykh algebr”, Vestsi NAN Belarusi. Seryya fiz-mat. navuk, 2008, no. 3, 35–42 | MR

[13] A. V. Prokopchuk, S. V. Tikhonov, V. I. Yanchevskii, “O spetsialnykh svoistvakh tel nekommutativnykh ratsionalnykh funktsii”, Dokl. NAN Belarusi (to appear)

[14] Zh.-P. Serr, Kogomologii Galua, Mir, M., 1968 | MR | Zbl

[15] A. A. Suslin, “Algebraicheskaya K-teoriya i gomomorfizm normennogo vycheta”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Nov. dostizh., 25, 1984, 115–207 | MR | Zbl

[16] V. I. Yanchevskii, “Kommutanty prostykh algebr s syurektivnoi privedennoi normoi”, Dokl. AN SSSR, 221:5 (1975), 1056–1058 | MR | Zbl

[17] V. I. Yanchevskii, “Tela nad genzelevymi diskretno normirovannymi polyami i problema Tannaka–Artina”, Dokl. AN SSSR, 226:2 (1976), 281–283 | MR | Zbl

[18] V. I. Yanchevskii, Privedennaya unitarnaya $K$-teoriya, Dis. na soisk. uch. st. d. f.-m. n., In-t matematiki AN BSSR, Minsk, 1980

[19] S. A. Amitsur, D. Saltman, “Generic Abelian Crossed Products and $p$-Algebras”, J. Algebra, 51 (1978), 76–87 | DOI | MR | Zbl

[20] P. Draxl, M. Kneser (eds.), $\mathrm{SK}_1$ von Schiefkörpern, Lecture Notes in Math., 778, Springer-Verlag, 1980 | DOI | MR | Zbl

[21] P. Gille, “Le Probleme de Kneser–Tits”, Séminaire Bourbaki, Exp. No. 983, v. 2007/2008, Astérisque, 326, 2009, 39–81 | MR

[22] R. Hazrat, “Wedderburn's Factorization Theorem, Application to Reduced $K$-Theory”, Proc. Amer. Math. Soc., 130 (2002), 311–314 | DOI | MR | Zbl

[23] R. Hazrat, A. R. Wadsworth, “$\mathrm{SK}_1$ of graded division algebras”, Israel J. Math. (to appear)

[24] A. S. Merkurjev, “Generic element in $\mathrm{SK}_1$ for simple algebras”, $K$-Theory, 7 (1993), 1–3 | DOI | MR | Zbl

[25] T. Nakayama, Y. Matsushima, “Über die multiplicative Gruppe einer $p$-adischen Divisionalgebra”, Proc Imp. Acad. Japan, 19 (1943), 622–628 | DOI | MR | Zbl

[26] V. P. Platonov, “Algebraic groups and reduced $K$-theory”, Proc. Intern. Congr. Math., v. 1, Helsinki, 1980, 311–323 | MR

[27] U. Rehmann, S. V. Tikhonov, V. I. Yanchevskii, “Symbols and cyclicity of algebras after a scalar extension”, Fundam. Prikl. Mat., 14:6 (2008), 193–209 | MR

[28] A. A. Suslin, “$\mathrm{SK}_1$ of Division Algebras and Galois Cohomology”, Algebraic $K$-Theory, Adv. Soviet Math., 4, Amer. Math. Soc., Providence, 1991, 75–99 | MR

[29] A. A. Suslin, “$\mathrm{SK}_1$ of Division Algebras and Galois Cohomology revisted”, Proceeding of the St. Petersburg Math. Soc., v. XII, Amer. Math. Soc. Trasl. Ser. 2, 219, 2006, 125–147 | MR | Zbl

[30] S. Wang, “On the commutator group of a simple algebra”, Amer. J. Math., 72 (1950), 323–334 | DOI | MR | Zbl

[31] V. I. Yanchevskiĭ, “Whitehead groups and groups of $R$-equivalence classes of linear algebraic groups of non-commutative classical type over some virtual fields”, Algebraic Groups and Arithmetic (TIFR), Mumbai, 2004, 491–505 | MR | Zbl