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Mat. Sb. (N.S.), 1978, Volume 106(148), Number 1(5), Pages 94–107 (Mi msb2556)  

This article is cited in 38 scientific papers (total in 38 papers)

The stabilization of symplectic groups over a polynomial ring

V. I. Kopeiko


Abstract: We prove that if $B$ is a polynomial ring over a field, then for $r\geqslant2$, any element of $Sp_{2r}B$ can be written as a product of elementary symplectic matrices over $B$.
We also prove a stabilization theorem for the symplectic $K_1$-functor in the case of polynomial rings and Laurent rings.
Bibliography: 6 titles.

Full text: PDF file (1310 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1978, 34:5, 655–669

Bibliographic databases:

UDC: 512.864
MSC: 20G35, 18F25
Received: 21.03.1977

Citation: V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring”, Mat. Sb. (N.S.), 106(148):1(5) (1978), 94–107; Math. USSR-Sb., 34:5 (1978), 655–669

Citation in format AMSBIB
\Bibitem{Kop78}
\by V.~I.~Kopeiko
\paper The stabilization of symplectic groups over a~polynomial ring
\jour Mat. Sb. (N.S.)
\yr 1978
\vol 106(148)
\issue 1(5)
\pages 94--107
\mathnet{http://mi.mathnet.ru/msb2556}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=497932}
\zmath{https://zbmath.org/?q=an:0398.20051|0414.20037}
\transl
\jour Math. USSR-Sb.
\yr 1978
\vol 34
\issue 5
\pages 655--669
\crossref{https://doi.org/10.1070/SM1978v034n05ABEH001337}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Golubchik I., Mikhalev A., “The Elementary Subgroup of a Unitary-Group Over Pi-Rings”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1985, no. 1, 30–36  mathscinet  zmath  isi
    2. Vavilov N., “The Structure of Splittable Classical-Groups Over a Commutative Ring”, 299, no. 6, 1988, 1300–1303  mathscinet  zmath  isi
    3. Grunewald F., Mennicke J., Vaserstein L., “On Symplectic Groups Over Polynomial-Rings”, Math. Z., 206:1 (1991), 35–56  crossref  mathscinet  zmath  isi
    4. V. I. Kopeiko, “On the structure of the symplectic group of polynomial rings over regular rings of dimension $\le 1$”, Russian Math. Surveys, 47:4 (1992), 210–211  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. Anthony Bak, Nikolai Vavilov, “Normality for elementary subgroup functors”, Math Proc Camb Phil Soc, 118:1 (1995), 35  crossref  mathscinet  zmath
    6. V. I. Kopeiko, “O strukture simplekticheskoi gruppy kolets mnogochlenov nad regulyarnymi koltsami”, Fundament. i prikl. matem., 1:2 (1995), 545–548  mathnet  mathscinet  zmath
    7. Leonid N. Vaserstein, Hong You, “Normal subgroups of classical groups over rings”, Journal of Pure and Applied Algebra, 105:1 (1995), 93  crossref
    8. Nikolai Vavilov, Eugene Plotkin, “Chevalley groups over commutative rings: I. Elementary calculations”, Acta Appl Math, 45:1 (1996), 73  crossref  mathscinet  zmath  isi
    9. N. A. Vavilov, V. A. Petrov, “On supergroups of $\mathrm{Ep}(2l,R)$”, St. Petersburg Math. J., 15:4 (2004), 515–543  mathnet  crossref  mathscinet  zmath
    10. Roozbeh Hazrat, Nikolai Vavilov, “K1 of Chevalley groups are nilpotent”, Journal of Pure and Applied Algebra, 179:1-2 (2003), 99  crossref
    11. N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-proof of structure theorem for Chevaller groups of type $\mathrm E_6$ and $\mathrm E_7$”, St. Petersburg Math. J., 16:4 (2005), 649–672  mathnet  crossref  mathscinet  zmath
    12. Hong You, “Overgroups of symplectic group in linear group over commutative rings”, Journal of Algebra, 282:1 (2004), 23  crossref
    13. Basu R., Rao R., Khanna R., “On Quillen's Local Global Principle”, Commutative Algebra and Algebraic Geometry, Contemporary Mathematics Series, 390, eds. Ghorpade S., Srinivasan H., Verma J., Amer Mathematical Soc, 2005, 17–30  crossref  zmath  isi
    14. E. I. Bunina, “Elementary equivalence of Chevalley groups over fields”, J. Math. Sci., 152:2 (2008), 155–190  mathnet  crossref  mathscinet  zmath  elib  elib
    15. N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book”, J. Math. Sci. (N. Y.), 140:5 (2007), 626–645  mathnet  crossref  mathscinet  zmath  elib  elib
    16. N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948  mathnet  crossref  mathscinet  elib  elib
    17. N. A. Vavilov, V. A. Petrov, “Overgroups of $\mathrm{EO}(n,R)$”, St. Petersburg Math. J., 19:2 (2008), 167–195  mathnet  crossref  mathscinet  zmath  isi  elib
    18. V. Petrov, A. Stavrova, “Elementary subgroups of isotropic reductive groups”, St. Petersburg Math. J., 20:4 (2009), 625–644  mathnet  crossref  mathscinet  zmath  isi  elib
    19. J. Math. Sci. (N. Y.), 168:3 (2010), 334–348  mathnet  crossref
    20. A. Bak, R. Hazrat, N. Vavilov, “Localization–completion strikes again: Relative is nilpotent by abelian”, Journal of Pure and Applied Algebra, 213:6 (2009), 1075  crossref
    21. V. Petrov, R. Hazrat, N. Vavilov, “Relative subgroups in Chevalley groups”, J K-Theory, 2010, 1  crossref
    22. Pratyusha Chattopadhyay, Ravi A. Rao, “Elementary symplectic orbits and improved K1-stability”, J K-Theory, 2010, 1  crossref
    23. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    24. Rabeya Basu, Ravi A. Rao, “Injective stability for of classical modules”, Journal of Algebra, 323:4 (2010), 867  crossref
    25. Roozbeh Hazrat, Nikolai Vavilov, Zuhong Zhang, “Relative unitary commutator calculus, and applications”, Journal of Algebra, 2011  crossref
    26. J. Math. Sci. (N. Y.), 179:6 (2011), 662–678  mathnet  crossref
    27. N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550  mathnet  crossref  mathscinet
    28. Roozbeh Hazrat, Nikolai Vavilov, Zuhong Zhang, “Relative commutator calculus in Chevalley groups”, Journal of Algebra, 385 (2013), 262  crossref
    29. J. Math. Sci. (N. Y.), 200:6 (2014), 742–768  mathnet  crossref
    30. A. V. Stepanov, “Non-Abelian $K$-theory for Chevalley groups over rings”, J. Math. Sci. (N. Y.), 209:4 (2015), 645–656  mathnet  crossref  mathscinet
    31. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  mathnet  crossref  mathscinet
    32. Andrei Lavrenov, “Another presentation for symplectic Steinberg groups”, Journal of Pure and Applied Algebra, 2015  crossref
    33. Pratyusha Chattopadhyay, R.A.. Rao, “Equality of linear and symplectic orbits”, Journal of Pure and Applied Algebra, 2015  crossref
    34. E. Yu. Voronetsky, “Normality of elementary subgroup in $\operatorname{Sp}(2,A)$”, J. Math. Sci. (N. Y.), 222:4 (2017), 386–393  mathnet  crossref  mathscinet
    35. J. Math. Sci. (N. Y.), 222:4 (2017), 466–515  mathnet  crossref  mathscinet
    36. J. Math. Sci. (N. Y.), 232:5 (2018), 591–609  mathnet  crossref  mathscinet
    37. V. I. Kopeiko, “Nilpotentnaya po Bassu unitarnaya $K_1$-gruppa unitarnogo koltsa”, Voprosy teorii predstavlenii algebr i grupp. 32, Zap. nauchn. sem. POMI, 460, POMI, SPb., 2017, 134–157  mathnet
    38. N. A. Vavilov, “Towards the reverse decomposition of unipotents”, Voprosy teorii predstavlenii algebr i grupp. 33, Zap. nauchn. sem. POMI, 470, POMI, SPb., 2018, 21–37  mathnet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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