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Homotopy invariance of non-stable K1-functors

  • A. Stavrova (a1)


Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.



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A.Abe, E., Whitehead groups of Chevalley groups over polynomial rings, Comm. Algebra 11 (1983), 12711307.
BHV.Bak, A., Hazrat, R., Vavilov, N., Localizaion—completion strikes again: Relative K 1 is nilpotentby abelian, J. of Pure and Appl. Algebra 213 (2009), 10751085.
BBR.Bak, A., Basu, R., Rao, R. A., Local-global principle for transvection groups, Proceedings of the AMS 138 (2010), 11911204.
B.Bass, H., K-theory and stable algebra, Publ. Math. IHÉS 22 (1964), 560.
Ba.Basu, R., Topics in classical algebraic K-theory, PhD Thesis, 2006.
BT1.Borel, A., Tits, J., Groupes rÉductifs, Publ. Math. I.H.É.S. 27 (1965), 55151.
BT2.Borel, A., Tits, J., Compléments à l'article “Groupes réductifs”, Publ. Math. I.H.É.S. 41 (1972) 253276.
C.Cohn, P.M., On the structure of GL 2 of a ring, Publ. Math. I.H.É.S. 30 (1966), 365413.
CTS.Colliot-Thélène, J.-L., Sansuc, J.-J., Principal homogeneous spaces under flasque tori: applications, Journal of Algebra 106 (1987), 148205.
CTO.Colliot-Thélène, J.-L., Ojanguren, M., Espaces Principaux Homogènes Localement Triviaux, Publ. Math. I.H.É.S. 75(2) (1992), 97122.
SGA3.Demazure, M., Grothendieck, A., Schémas en groupes, Lecture Notes in Mathematics 151-153, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
Ge.Gersten, S. M., Higher K-theory of Rings, Lecture Notes in Mathematics 341, 342, Springer-Verlag, Berlin-Heidelberg-New York, 1973.
G.Gille, Ph., Le problème de Kneser-Tits, Sém. Bourbaki 983 (2007), 983–01–983-39.
GMV1.Grunewald, F., Mennicke, J., Vaserstein, L., On symplectic groups over polynomial rings, Math. Z. 206 (1991), 3556.
GMV2.Grunewald, F., Mennicke, J., Vaserstein, L., On the groups SL2(ℤ[x]) and SL2(k[x, y]), Israel J. Math. 86 (1994), 157193.
J.Jardine, J.F., On the homotopy groups of algebraic groups, J. Algebra 81 (1983), 180201.
K78.Kopeiko, V.I., Stabilization of symplectic groups over a ring of polynomials (Russian), Mat. Sb. (N.S.) 106 (148)(1) (1978), 94107.
K95a.Kopeiko, V.I., On the structure of the symplectic group of polynomial rings over regular rings (Russian), Fundam. Prikl. Mat. 1(2) (1995), 545548.
K95b.Kopeiko, V.I., On the structure of the special linear group over Laurent polynomial rings (Russian), Fundam. Prikl. Mat. 1 (4) (1995), 11111114.
K96.Kopeiko, V.I., Letter to the editors: “On the structure of the symplectic group of polynomial rings over regular rings” and “On the structure of the special linear group over Laurent polynomial rings” (Russian), Fundam. Prikl. Mat. 2(3) (1996), 953.
K99.Kopeiko, V.I., Symplectic groups over rings of Laurent polynomials, and patching diagrams (Russian), Fundam. Prikl. Mat. 5(3) (1999), 943945.
KrMC.Krstić, S., McCool, J., Free quotients of SL2(R[X]), Proc. Amer. Math. Soc. 125 (1997), 15851588.
L.Lindel, H., On the Bass—Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981), 319323.
LSt.Luzgarev, A., Stavrova, A., Elementary subgroup of an isotropic reductive group is perfect, St. Petersburg Math. J. 23 (2012), 881890.
M.Margaux, B., The structure of the group G(k[t]): Variations on a theme of Soulé, Algebra and Number Theory 3 (2009), 393409.
Ma.Matsumura, H., Commutative algebra, second ed., Math. Lect. Note Series 56, Benjamin/Cummings Publishing Co., Inc., Reading, Massachusetts, 1980.
Mo.Morel, F., -Algebraic topology over a field, Lecture Notes in Mathematics 2052, 2012.
PaStV.Panin, I., Stavrova, A., Vavilov, N., On Grothendieck—Serre's conjecture concerning principal G-bundles over reductive group schemes: I, preprint,
PSt1.Petrov, V., Stavrova, A., Elementary subgroups of isotropic reductive groups, St. Petersburg Math. J. 20 (2009), 625644.
PSt2.Petrov, V., Stavrova, A., Tits indices over semilocal rings, Transf. Groups 16 (2011), 193217.
Po.Popescu, D., Letter to the Editor: General Néron desingularization and approximation, Nagoya Math. J. 118 (1990), 4553.
Q.Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167171.
Sou.Soulé, C., Chevalley groups over polynomial rings, Homological group theory (Proc. Sympos., Durham, 1977), 359367, London Math. Soc. Lecture Note Ser. 36 (1979), Cambridge Univ. Press.
St.Stavrova, A., Stroenije isotropnyh reduktivnyh grupp, PhD thesis, St. Petersburg State University, 2009.
Ste.Stepanov, A., private communication.
Su.Suslin, A.A., On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11 (1977), 221238.
SuK.Suslin, A.A., Kopeiko, V.I., Quadratic modules and the orthogonal group over polynomial rings, J. of Soviet Math. 20 (1982), 26652691.
Sw.Swan, R. G., Néron-Popescu desingularization, in Algebra and Geometry (Taipei, 1995), Lect. Alg. Geom. 2 (1998), 135198. Int. Press, Cambridge, MA.
T1.Tits, J., Algebraic and abstract simple groups, Ann. of Math. 80 (1964), 313329.
T2.Tits, J., Classification of algebraic semisimple groups, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence RI, 1966, 3362.
vdK.van der Kallen, W., A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra 57 (1989), 281316.
VW.Völkel, K., Wendt, M., On -fundamental groups of isotropic reductive groups, 2012,
V.Vorst, T., The general linear group of polynomial rings over regular rings, Comm. Algebra 9 (1981), 499509.
W.Wendt, M., -homotopy of Chevalley groups, J. K-Theory 5 (2010), 245287.



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