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Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 5, Pages 993–1054 (Mi izv2227)  

This article is cited in 34 scientific papers (total in 34 papers)

Serre's problem on projective modules over polynomial rings, and algebraic $K$-theory

L. N. Vaserstein, A. A. Suslin


Abstract: We obtain new results on the freeness of projective modules over polynomial rings. In particular we prove the freeness of all projective modules over polynomial rings in five variables with coefficients from an arbitrary field.
Bibliography: 46 titles.

Full text: PDF file (6253 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:5, 937–1001

Bibliographic databases:

UDC: 519.4
MSC: Primary 13C10, 13F20, 14F05; Secondary 13B25, 18F25
Received: 04.04.1975

Citation: L. N. Vaserstein, A. A. Suslin, “Serre's problem on projective modules over polynomial rings, and algebraic $K$-theory”, Izv. Akad. Nauk SSSR Ser. Mat., 40:5 (1976), 993–1054; Math. USSR-Izv., 10:5 (1976), 937–1001

Citation in format AMSBIB
\Bibitem{VasSus76}
\by L.~N.~Vaserstein, A.~A.~Suslin
\paper Serre's problem on projective modules over polynomial rings, and algebraic $K$-theory
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1976
\vol 40
\issue 5
\pages 993--1054
\mathnet{http://mi.mathnet.ru/izv2227}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=447245}
\zmath{https://zbmath.org/?q=an:0338.13015}
\transl
\jour Math. USSR-Izv.
\yr 1976
\vol 10
\issue 5
\pages 937--1001
\crossref{https://doi.org/10.1070/IM1976v010n05ABEH001822}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Suslin, “On stably free modules”, Math. USSR-Sb., 31:4 (1977), 479–491  mathnet  crossref  mathscinet  zmath  isi
    2. V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring”, Math. USSR-Sb., 34:5 (1978), 655–669  mathnet  crossref  mathscinet  zmath
    3. A. A. Suslin, “Reciprocity laws and the stable rank of polynomial rings”, Math. USSR-Izv., 15:3 (1980), 589–623  mathnet  crossref  mathscinet  zmath  isi
    4. Suslin A., “Mennicke Symbols and their Applications in the K-Theory of Fields”, 966, 1982, 334–356  mathscinet  zmath  isi
    5. Wilberd van der Kallen, “A group structure on certain orbit sets of unimodular rows”, Journal of Algebra, 82:2 (1983), 363  crossref
    6. Vaserstein L., “Bass 1st Stable Range Condition”, J. Pure Appl. Algebr., 34:2-3 (1984), 319–330  crossref  mathscinet  zmath  isi
    7. Gustavo Corach, Fernando Daniel Suárez, “Extension problems and stable rank in commutative Banach algebras”, Topology and its Applications, 21:1 (1985), 1  crossref
    8. M. Ojanguren, R. Parimala, R. Sridharan, “Symplectic bundles over affine surfaces”, Comment Math Helv, 61:1 (1986), 491  crossref  mathscinet  zmath  isi
    9. Leonid N Vaserstein, “Operations on orbits of unimodular vectors”, Journal of Algebra, 100:2 (1986), 456  crossref
    10. S. Mandal, A. Roy, “Generating ideals in polynomial rings”, Math Z, 195:3 (1987), 315  crossref  mathscinet  zmath  isi
    11. L.N. Vaserstein, B.A. Magurnt, “Prestabilization for K1 of Banach algebras”, Linear Algebra and its Applications, 95 (1987), 69  crossref
    12. Wilberd Van Der Kallen, “A module structure on certain orbit sets of unimodular rows”, Journal of Pure and Applied Algebra, 57:3 (1989), 281  crossref
    13. Andreas Wiemers, “Some properties of projective modules over discrete Hodge algebras”, Journal of Algebra, 150:2 (1992), 402  crossref
    14. F.D.. Veldkamp, “n-Barbilian Domains”, Results. Math, 23:1-2 (1993), 177  crossref
    15. Ravi A Rao, “An Abelian Group Structure on Orbits of “Unimodular Squares” in Dimension 3”, Journal of Algebra, 210:1 (1998), 216  crossref
    16. Michael Maltenfort, “Addition and Subtraction of Ideals”, Journal of Algebra, 214:2 (1999), 519  crossref
    17. 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
    18. 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
    19. 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
    20. Manoj Kumar Keshari, “Cancellation problem for projective modules over affine algebras”, J K-Theory, 2008, 1  crossref  adsnasa  isi
    21. S.A. Katre, R.A. Rao, D.N. Sheth, “Solving linear systems via Pfaffians”, Linear Algebra and its Applications, 430:4 (2009), 968  crossref
    22. Anuradha S. Garge, “The Steinberg formula for orbit groups”, Expositiones Mathematicae, 27:4 (2009), 341  crossref
    23. Pratyusha Chattopadhyay, Ravi A. Rao, “Elementary symplectic orbits and improved K1-stability”, J K-Theory, 2010, 1  crossref
    24. Alpesh M. Dhorajia, Manoj K. Keshari, “A note on cancellation of projective modules”, Journal of Pure and Applied Algebra, 2011  crossref
    25. 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
    26. Andrei Suslin, “Quillen's solution of Serre's Problem”, J. K-Theory, 2013, 1  crossref
    27. Anjan Gupta, Anuradha Garge, R.A.. Rao, “A nice group structure on the orbit space of unimodular rows-II”, Journal of Algebra, 407 (2014), 201  crossref
    28. Anjan Gupta, “Optimal injective stability for the symplectic K1Sp group”, Journal of Pure and Applied Algebra, 2014  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. Pratyusha Chattopadhyay, R.A.. Rao, “Equality of linear and symplectic orbits”, Journal of Pure and Applied Algebra, 2015  crossref
    32. J. Math. Sci. (N. Y.), 222:4 (2017), 466–515  mathnet  crossref  mathscinet
    33. J. Math. Sci. (N. Y.), 232:5 (2018), 591–609  mathnet  crossref  mathscinet
    34. T. N. Hoi, N. H. T. Nhat, “Subgroups of the general linear group containing the elementary subgroup over a commutative ring extension of rank 2”, Voprosy teorii predstavlenii algebr i grupp. 31, Zap. nauchn. sem. POMI, 455, POMI, SPb., 2017, 209–225  mathnet  mathscinet
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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