Non-stable K1-functors of Multiloop Groups

Stavrova, Anastasia

Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 150-178

Résumé

Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R\,=\,k\left[ x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1} \right]$ . Assume that $G$ contains a maximal $R$ -torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_{m}^{2}$ . We show that the natural map of non-stable ${{K}_{1}}$ -functors, also called Whitehead groups, $K_{1}^{G}\left( R \right)\,\to \,K_{1}^{G}\left( k\left( \left( {{x}_{1}} \right) \right)\cdots \left( \left( {{x}_{n}} \right) \right) \right)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

Détail

DOI : 10.4153/CJM-2015-035-2

Mots-clés : 20G35, 19B99, 17B67, loop reductive group, non-stable K1-functor, Whitehead group, Laurent polynomial, Lie torus

Bibliographie

[A] [A] Abe, E., Whitehead groups of Chevalley groups over polynomial rings. Comm. Algebra 11(1983), 1271–1307. Google Scholar | DOI

[Ab] [Ab] Abramenko, P., On finite and elementary generation of SL(R). arxiv:0808.1095 Google Scholar

[AABGP] [AABGP] Allison, B., Azam, S., Berman, S., Gao, Y., and Pianzola, A., Extended affine Lie algebras and their root systems. Mem. Amer. Math. Soc. 126(1997), no. 603. http://dx.doi.Org/10.1090/memo/0603 Google Scholar

[ABFP] [ABFP] Allison, B., Berman, S., Faulkner, J., and Pianzola, A., Multiloop realization of extended affine Lie algebras and Lie tori. Trans. Amer. Math. Soc. 361(2009), 4807–4842. Google Scholar | DOI

[BaMo] [BaMo] Bachmuth, S. and Mochizuki, H. Y., E ≠ SL for most Laurent polynomial rings. Amer. J.Math. 104(1982), no. 6, 1181–1189. Google Scholar | DOI

[B] [B] Bass, H., K-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. 22(1964), 5–60. Google Scholar

[BT1] [BT1] Borel, A. and Tits, J., Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27(1965), 55–151. Google Scholar

[BT2] [BT2] Borel, A. and Tits, J., Homomorphismes “abstraits” de groupes algébriques simples. Ann. of Math. 97(1973), 499–571. http://dx.doi.Org/10.2307/1970833 Google Scholar

[Bou] [Bou] Bourbaki, N., Groupes et algèbres de Lie. Chapitres 4–6. Hermann, Paris, 1968. Google Scholar

[ChGPl] [ChGPl] Chernousov, V., Gille, P., and Pianzola, A., Torsors over the punctured affine Une. Amer. J. Math. 134(2012), no. 6, 1541–1583. Google Scholar | DOI

[ChGP2] [ChGP2] Chernousov, V., Gille, P., and Pianzola, A., Conjugacy classes for loop reductive group schemes and Lie algebras. Bull. Math. Sci. 4(2014), 281–324. Google Scholar | DOI

[ChGP3] [ChGP3] Chernousov, V., Gille, P., and Pianzola, A., Whitehead groups of loop group schemes of nullity one. J. Ramanujan Math. Soc. 29(2014), no. 1, 1–26. Google Scholar

[ChM] [ChM] Chernousov, V. and Merkurjev, A. S., R-equivalence and special unitary groups. J. Algebra 209(1998), 175–198. http://dx.doi.Org/10.1006/jabr.1998.7534 Google Scholar

[Che] [Che] Chevalley, C., Certains schémas de groupes semi-simples. Sera. Bourbaki 6(1995), 219–234. Google Scholar

[SGA3] [SGA3] Demazure, M. and Grothendieck, A. , Schémas en groupes. Lecture Notes in Math., 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970. Google Scholar

[FGIKNV] [FGIKNV] Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S., Nitsure, N., and Vistoli, A., Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123, American Mathematical Society, Providence, RI, 2005. Google Scholar

[Gl] [Gl] Gille, P., Spécialisation de la R-équivalence pour les groupes réductifs. Trans. Amer. Math. Soc. 356(2004), 4465–4474. Google Scholar | DOI

[G2] [G2] Gille, P., Le problème de Kneser-Tits. Séminaire Bourbaki Vo. 2007/2008, Astérisque 326(2009), Exp. No. 983, vii, 39–81. Google Scholar

[GP1] [GP1] Gille, P. and Pianzola, A., Galois cohomology and forms of algebras over Laurent polynomial rings. Math. Ann. 338(2007), 497–543. http://dx.doi.Org/10.1007/s00208-007-0086-2 Google Scholar

[GP2] [GP2] Gille, P. and Pianzola, A., Isotriviality and étale cohomology of Laurent polynomial rings. J. Pure Appl. Algebra 212(2008), 780–800. http://dx.doi.Org/10.1016/j.jpaa.2007.07.005 Google Scholar

[GP3] [GP3] Gille, P. and Pianzola, A., Torsors, reductive group schemes and extended affine Lie algebras. Mem. Amer. Math. Soc. 226(2013), no. 1063. Google Scholar

[HV] [HV] Hazrat, R. and Vavilov, N., K of Chevalley groups are nilpotent. J. Pure Appl. Algebra 179(2003), 99–116. Google Scholar | DOI

[J] [J] Jardine, J. F., On the homotopy groups of algebraic groups. J. Algebra 81(1983), 180–201. http://dx.doi.Org/10.1016/0021-8693(83)90215-6 Google Scholar

[Ma] [Ma] Manin, Yu. I., Cubic forms: algebra, geometry, arithmetic. 2nd ed., North Holland Mathematical Library, 4, North Holland, Amsterdam, 1986. Google Scholar

[M] [M] Margaux, B., The structure of the group G(k[t]): Variations on a theme of Soulé. Algebra Number Theory 3(2009), 393–409. Google Scholar | DOI

[N] [N] Neher, E., Lie tori. C. R. Math. Acad. Sci. Soc. R. Can. 26(2004), no. 3, 84–89. Google Scholar

[OPa] [OPa] Ojanguren, M. and Panin, I., Rationally trivial hermitian spaces are locally trivial. Math. Z. 237(2001), 181–198. Google Scholar | DOI

[PaStV] [PaStV] Panin, I., Stavrova, A., and Vavilov, N., On Grothendieck-Serre's conjecture concerning principal G-bundles over reductive group schemes: I. Compos. Math. 151(2015), no. 3,535–567. http://dx.doi.Org/10.1112/S0010437X14007635 Google Scholar

[PStl] [PStl] Petrov, V. and Stavrova, A., Elementary subgroups of isotropic reductive groups. St. Petersburg Math. J. 20(2009), no. 4, 625–644. http://dx.doi.Org/10.1090/S1061-0022-09-01064-4 Google Scholar

[PSt2] [PSt2] Petrov, V. and Stavrova, A., Tits indices over semilocal rings. Transform. Groups 16(2011), 193.–217 http://dx.doi.Org/10.1007/s00031-010-9112-7 Google Scholar

[Q] [Q] Quillen, D., Projective modules over polynomial rings. Invent. Math. 36(1976), 167–171. Google Scholar | DOI

[Se] [Se] Serre, J.-P., Galois cohomology. English transi, by P. Ion, Springer-Verlag, Berlin Heidelberg, 1997. Google Scholar

[StO9] [StO9] Stavrova, A., Stroenije isotropnyh reduktivnyh grupp. Ph.D. dissertation, St. Petersburg State University, 2009. Google Scholar

[Stl3] [Stl3] Stavrova, A., Homotopy invariance of non-stable K-functors. J. K-Theory 13(2014), 199–248. http://dx.doi.Org/10.1017/isOl3006012jkt232 Google Scholar

[S78] [S78] Stein, M. R., Stability theorems for K, K and related functors modeled on Chevalley groups. Japan J. Math. (N.S.) 4(1978), no. 1, 77–108. Google Scholar

[Su] [Su] Suslin, A. A., On the structure of the special linear group over polynomial rings. Math. USSR Izv. 11(1977), 221–238. Google Scholar

[Tl] [Tl] Tits, J., Algebraic and abstract simple groups. Ann. of Math. 80(1964), 313–329. Google Scholar | DOI

[V] [V] Voskresenskiĭ, V. E., Algebraic groups and their birational invariants. Translations of Mathematical Monographs, 179, American Mathematical Society, Providence, RI, 1998. Google Scholar

[W] [W] Wendt, M., -homotopy of Chevalley groups. J. K-Theory 5(2010), 245–287. http://dx.doi.Org/10.1017/isO10001014jktO96 Google Scholar

[Y] [Y] Yoshii, Y., Lie tori-A simple characterization of extended affine Lie algebras. Publ. Res. Inst. Math. Sci. 42(2006), 739–762. Google Scholar | DOI