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Elementary symplectic orbits and improved K1-stability

Published online by Cambridge University Press:  10 June 2010

Pratyusha Chattopadhyay  and
Ravi A. Rao
Pratyusha Chattopadhyay
Affiliation:
Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, India, pratyusha@imsc.res.in
Ravi A. Rao
Affiliation:
Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Navy Nagar, Mumbai 400005, India, ravi@math.tifr.res.in
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Abstract

It is shown that the set of orbits of the action of the elementary symplectic group on all unimodular rows over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary general linear group. This result is used to improve injective stability for K1 of the symplectic group over non-singular affine algebras.

Keywords

unimodular rowselementary symplectic group
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 2 , April 2011 , pp. 389 - 403
Copyright
Copyright © ISOPP 2010

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References

1.Bak, A., Hazrat, R., Vavilov, N., Localization-completion strikes again: Relative K 1 is nilpotent by abelian, Journal of Pure and Applied Algebra 213 (2009), 1075–1085.Google Scholar
2.Bak, A., Petrov, V., Tang, G., Stability for quadratic K 1. Special issue in honour of Hyman Bass on his seventieth birthday. Part I. K-Theory 30 (1) (2003), 1–11.CrossRefGoogle Scholar
3.Bass, H., K-theory and stable algebra, Inst. Publ. Math. IHES 22 (1964), 5–60.CrossRefGoogle Scholar
4.Bass, H., Milnor, J., Serre, J.P., Solution of the congruence subgroup problem for SLn, n ≥ 3 and Sp2n, n ≥ 2, Inst. Publ. Math. IHES 33 (1967), 59–137.CrossRefGoogle Scholar
5.Basu, R., Khanna, R., Rao, R.A., On Quillen's local global principle, Contemp. Math. 390Amer. Math. Soc. Providence RI (2005), 17–30.Google Scholar
6.Basu, R., Rao, R.A., Injective Stability for K 1 of Classical Modules, Journal of Algebra 323 (4) (2010), 867–877.Google Scholar
7.Gupta, S.K., Murthy, M.P., Suslin's Work on Linear Groups over Polynomial Rings and Serre Problem, ISI Lecture Notes 8 (1980).Google Scholar
8.van der Kallen, W., A group structure on certain orbit sets of unimodular rows, Journal of Algebra 82 (2) (1983), 363–397.Google Scholar
9.van der Kallen, W., A module structure on certain orbit sets of unimodular rows, J. Pure and Appl. Algebra 57 (1989), 281–316.Google Scholar
10.Kopeĭko, V.I., The stabilisation of symplectic groups over a polynomial ring, Math. USSR. Sbornik 34 (1978), 655–669.Google Scholar
11.Lam, T.Y., Serre's conjecture, Lecture Notes in Mathematics 635, Springer-Verlag, Berlin-New York, 1978. Revised edition: Serre's Problem on Projective Modules. Springer Monographs in Math. ISBN 3-540-23317-2.Google Scholar
12.Lindel, H., On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (2) (1981–1982), 319–323.CrossRefGoogle Scholar
13.Popescu, D., General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85–115.Google Scholar
14.Popescu, D., Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989), 121–128.Google Scholar
15.Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.CrossRefGoogle Scholar
16.Rao, R.A., An elementary transformation of a special unimodular vector to its top coefficient vector, Proc. Amer. Math. Soc. 93 (1) (1985), 21–24.Google Scholar
17.Rao, R.A., van der Kallen, W., Improved stability for K 1 and WMSd of a non-singular affine algebra, K-theory (Strasbourg, 1992). Astérisque no. 226 (1994), 411–420.Google Scholar
18.Suslin, A.A., Vaserstein, L.N., Serre's problem on Projective Modules over Polynomial Rings and Algebraic K-theory, Math. USSR Izvestija 10 (1976), 937–1001.Google Scholar
19.Suslin, A.A., Stably Free Modules. (Russian) Math. USSR Sbornik 102 (144) (1977) no. 4, 537–550; Mat. Inst. Steklov. (LOMI) 114 (1982), 187–195, 222.Google Scholar
20.Suslin, A.A., On the Structure of the Special Linear Group over Polynomial Rings, Math. USSR. Izvestija 11 (1977), 221–238.CrossRefGoogle Scholar
21.Suslin, A.A., Cancellation for affine varieties (Russian) Modules and algebraic groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 187–195, 222.Google Scholar
22.Vaserstein, L.N., On the stabilization of the general linear group over a ring, Mat. Sbornik (N.S.) 79 (121) 405–424 (Russian); English translation in Math. USSR-Sbornik 8 (1969), 383–400.Google Scholar
23.Vaserstein, L.N., Stabilisation of Unitary and Orthogonal Groups over a Ring with Involution, Mat. Sbornik 81 (123) No. 3 (1970), 328–351.Google Scholar
24.Vaserstein, L.N., On the normal subgroups of GLn over a ring, Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) 456–465; Lecture Notes in Math. 854, Springer, Berlin-New York, 1981.Google Scholar
25.Vorst, T., The general linear group of polynomial rings over regular rings, Comm. Algebra 9 (5) (1981), 499–509.Google Scholar