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On the Completability of Unimodular Rows of Length Three

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Leavitt Path Algebras and Classical K-Theory

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Abstract

In this paper, we use a result of Suslin to give a necessary and sufficient condition under which unimodular rows of length three over rings are completable. We use this criterion to prove that certain unimodular rows over rings of dimension 2 are completable, and reprove results of Seshadri, Bass and Serre. We also give a different proof of a result of Bhatwadekar–Keshari.

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References

  1. H. Bass, \(K\)-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. 22, 5–60 (1964)

    Article  MathSciNet  Google Scholar 

  2. R. Basu, Topics in classical algebraic \(K\)-theory. PhD thesis, School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 2006

    Google Scholar 

  3. R. Basu, R. Sridharan, On Forster’s conjecture and related results. Punjab Univ. Res. J. (Sci.) 57, 13–66 (2007)

    Google Scholar 

  4. S.M. Bhatwadekar, R. Sridharan, On Euler classes and stably free projective modules, Algebra, Arithmetic and Geometry, Part I, II (Mumbai, 2000). Tata Institute of Fundamental Research Studies in Mathematics, vol. 16 (Tata Institute of Fundamental Research, Bombay, 2002), pp. 139–158

    Google Scholar 

  5. S.M. Bhatwadekar, R. Sridharan, Projective generation of curves in polynomial extensions of an affine domain and a question of Nori. Invent. Math. 133(1), 161–192 (1998)

    Article  MathSciNet  Google Scholar 

  6. S.M. Bhatwadekar, R. Sridharan, The Euler class group of a Noetherian ring. Compos. Math. 122(2), 183–222 (2000)

    Article  MathSciNet  Google Scholar 

  7. S.M. Bhatwadekar, M.K. Keshari, A question of Nori: projective generation of ideals. \(K\)-Theory, 28(4), 329–351 (2003)

    Article  MathSciNet  Google Scholar 

  8. S.M. Bhatwadekar, H. Lindel, R.A. Rao, The Bass-Murthy question: Serre dimension of Laurent polynomial extensions. Invent. Math. 81(1), 189–203 (1985)

    Article  MathSciNet  Google Scholar 

  9. S.M. Bhatwadekar, M.K. Das, S. Mandal, Projective modules over smooth real affine varieties. Invent. Math. 166(1), 151–184 (2006)

    Article  MathSciNet  Google Scholar 

  10. M.K. Das, R. Sridharan, Good invariants for bad ideals. J. Algebra 323(12), 3216–3229 (2010)

    Article  MathSciNet  Google Scholar 

  11. N.S. Gopalakrishnan, Commutative Algebra (1984)

    Google Scholar 

  12. S.K. Gupta, M.P. Murthy, Suslin’s Work on Linear Groups over Polynomial Rings and Serre Problem. ISI Lecture Notes, vol. 8 (Macmillan Co. of India Ltd, New Delhi, 1980)

    Google Scholar 

  13. M.K. Keshari, Euler class group of a Noetherian ring. PhD thesis, School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 2001

    Google Scholar 

  14. M. Krusemeyer, Skewly completable rows and a theorem of Swan and Towber. Commun. Algebra 4(7), 657–663 (1976)

    Article  MathSciNet  Google Scholar 

  15. N.M. Kumar, On a theorem of Seshadri, Connected at Infinity. Texts and Readings in Mathematics, vol. 25 (Hindustan Book Agency, New Delhi, 2003), pp. 91–104

    Google Scholar 

  16. N.M. Kumar, Complete intersections. J. Math. Kyoto Univ. 17(3), 533–538 (1977)

    Article  MathSciNet  Google Scholar 

  17. N.M. Kumar, A note on the cancellation of reflexive modules. J. Ramanujan Math. Soc. 17(2), 93–100 (2002)

    MathSciNet  MATH  Google Scholar 

  18. T.Y. Lam, Serre’s Problem on Projective Modules, Springer Monographs in Mathematics (Springer, Berlin, 2006)

    Book  Google Scholar 

  19. S. MacLane, Homology. Die Grundlehren der mathematischen Wissenschaften, vol. 114, 1st edn. (Springer, Berlin-New York, 1967)

    Google Scholar 

  20. S. Mandal, Homotopy of sections of projective modules. J. Algebraic Geom. 1(4), 639–646 (1992). With an appendix by Madhav V. Nori

    Google Scholar 

  21. S. Mandal, Projective modules and complete intersections. Lecture Notes in Mathematics, vol. 1672 (Springer, Berlin, 1997)

    Book  Google Scholar 

  22. M.P. Murthy, R.G. Swan, Vector bundles over affine surfaces. Invent. Math. 36, 125–165 (1976)

    Article  MathSciNet  Google Scholar 

  23. D. Quillen, Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976)

    Article  MathSciNet  Google Scholar 

  24. J.P. Serre, Sur les modules projectifs. SÃl’minaire Dubreil. AlgÃíbre et thÃl’orie des nombres 14, 1–16 (1960–1961)

    Google Scholar 

  25. C.S. Seshadri, Triviality of vector bundles over the affine space \(K^{2}\). Proc. Natl. Acad. Sci. USA 44, 456–458 (1958)

    Article  MathSciNet  Google Scholar 

  26. C.S. Seshadri, Algebraic vector bundles over the product of an affine curve and the affine line. Proc. Am. Math. Soc. 10, 670–673 (1959)

    Article  MathSciNet  Google Scholar 

  27. A.A. Suslin, Stably free modules. Mat. Sb. (N.S.) 102(144)(4), 537–550, 632 (1977)

    Google Scholar 

  28. R.G. Swan, Algebraic vector bundles on the \(2\)-sphere. Rocky Mt. J. Math. 23(4), 1443–1469 (1993)

    Article  MathSciNet  Google Scholar 

  29. R.G. Swan, J. Towber, A class of projective modules which are nearly free. J. Algebra 36(3), 427–434 (1975)

    Article  MathSciNet  Google Scholar 

  30. W. van der Kallen, A group structure on certain orbit sets of unimodular rows. J. Algebra 82(2), 363–397 (1983)

    Article  MathSciNet  Google Scholar 

  31. W. van der Kallen, A module structure on certain orbit sets of unimodular rows. J. Pure Appl. Algebra 57(3), 281–316 (1989)

    Article  MathSciNet  Google Scholar 

  32. L.N. Vaserstein, Stabilization of unitary and orthogonal groups over a ring with involution. Mat. Sb. (N.S.) 81(123), 328–351 (1970)

    Google Scholar 

  33. 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), 993–1054, 1199 (1976)

    Article  MathSciNet  Google Scholar 

  34. C.T.C. Wall, A Geometric Introduction to Topology (Addison-Wesley Publishing Co, Reading, 1972)

    Google Scholar 

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Acknowledgements

The authors would like to thank Professor Ravi A. Rao for his valuable support during this work. The authors would like to thank Professor Gopala Krishna Srinivasan for giving his time most generously and helping us make this paper more readable. The second named author would like to thank Professor Gopala Krishna Srinivasan for his support and advice during difficult times. The authors would also like to thank the referee for going through the paper carefully and pointing out some mistakes. The second named author also acknowledges the financial support from CSIR which enabled him to pursue his doctoral studies.

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Authors and Affiliations

  1. Tata Institute of Fundamental Research, 1, Dr. Homi Bhabha Road, Mumbai, 400005, India

    Raja Sridharan

  2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Sunil K. Yadav

Authors
  1. Raja Sridharan
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  2. Sunil K. Yadav
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Correspondence to Sunil K. Yadav .

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Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Cochin, Kerala, India

    A. A. Ambily

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, NSW, Australia

    Roozbeh Hazrat

  3. Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, Karnataka, India

    B. Sury

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Sridharan, R., Yadav, S.K. (2020). On the Completability of Unimodular Rows of Length Three. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_16

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