Advertisement

\({\mathbb {P}}^1\)-gluing” for local complete intersections

  • Mrinal Kanti DasEmail author
  • Soumi Tikader
  • Md. Ali Zinna
Article
  • 39 Downloads

Abstract

We prove an analogue of the Affine Horrocks’ Theorem for local complete intersection ideals of height n in R[T], where R is a regular domain of dimension d, which is essentially of finite type over an infinite perfect field of characteristic unequal to 2, and \(2n\ge d+3\).

Mathematics Subject Classification

13C10 19A15 14C25 11E81 

Notes

Acknowledgements

We are grateful to the referee for an extremely meticulous reviewing and for pointing out a serious gap in one of our arguments. The clarity of exposition in the present version owes very much to the insistence of the referee. We sincerely thank Ravi Rao for answering numerous queries. We are deeply indebted to S. M. Bhatwadekar for his critical reading of an earlier version and for pointing out a mistake in the proof of Theorem 3.10. The question tackled in this article was proposed to the first named author by S. M. Bhatwadekar and Raja Sridharan around the year 2000, as a part of his thesis problem. The first named author takes this opportunity to thank them once again for their care and encouragement. The third named author acknowledges Department of Science and Technology for their INSPIRE research grant.

References

  1. 1.
    Asok, A., Fasel, J.: Euler class groups and motivic stable cohomotopy, preprint available at https://arxiv.org/abs/1601.05723
  2. 2.
    Bhatwadekar, S.M., Keshari, M.K.: A question of Nori: projective generation of ideals. K-Theory 28, 329–351 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bhatwadekar, S.M., Sridharan, R.: Projective generation of curves in polynomial extensions of an affine domain and a question of Nori. Invent. Math. 133, 161–192 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhatwadekar, S.M., Sridharan, R.: Zero cycles and the Euler class groups of smooth real affine varieties. Invent. Math. 136, 287–322 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bhatwadekar, S.M., Sridharan, R.: Euler class group of a Noetherian ring. Compos. Math. 122, 183–222 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhatwadekar, S.M., Sridharan, R.: On Euler classes and stably free projective modules. In: Algebra, Arithmetic and Geometry, Part I, II (Mumbai, 2000), pp. 139–158, Tata Inst. Fund. Res. Stud. Math., 16, Tata Inst. Fund. Res., Bombay (2002)Google Scholar
  7. 7.
    Calmès, B., Fasel, J.: Groupes classiques, in Autour des Schémas en Groupes. In: Calmès, B., Chaudouard, P.-H., Conrad, B., Demarche, C., Fasel, J. (eds.) Panoramas et Synthèses 46, Soc. Math. France, Paris, École d’été “Schémas en Groupes,” Group Schemes, A celebration of SGA3, vol. II, pp. 1–333 (2016)Google Scholar
  8. 8.
    Das, M.K.: The Euler class group of a polynomial algebra. J. Algebra 264, 582–612 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Das, M.K.: On a conjecture of Murthy. Adv. Math. 331, 326–338 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Das, M.K., Keshari, M.K.: A question of Nori, Segre classes of ideals and other applications. J. Pure Appl. Algebra 216, 2193–2203 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Das, M.K., Sridharan, R.: Good invariants for bad ideals. J. Algebra 323, 3216–3229 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Das, M.K., Sridharan, R.: Euler class groups and a theorem of Roitman. J. Pure Appl. Algebra 215, 1340–1347 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Das, M.K., Tikader, S., Zinna, M.A.: Orbit spaces of unimodular rows over smooth real affine algebras. Invent. Math. 212, 133–159 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Das, M.K., Zinna, M.A.: “Strong” Euler class of a stably free module of odd rank. J. Algebra 432, 185–204 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fasel, J.: On the number of generators of an ideal in a polynomial ring. Ann. Math. 184, 315–331 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fasel, J.: Erratum on “On the number of generators of an ideal in a polynomial ring”. Ann. Math. 186, 647–648 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ischebeck, F., Rao, Ravi A.: Ideals and reality, projective modules and number of generators of ideals. In: Springer Monographs in Mathematics. Springer, Berlin (2005)Google Scholar
  18. 18.
    Keshari, M.K.: Euler Class group of a Noetherian ring, M.Phil. Thesis. http://www.math.iitb.ac.in/~keshari/acad.html
  19. 19.
    Mandal, S.: On efficient generation of ideals. Invent. Math. 75, 59–67 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mandal, S., Mishra, B.: The homotopy program in complete intersections, preprint available at https://arxiv.org/abs/1610.07495
  21. 21.
    Mohan Kumar, N.: Complete intersections. J. Math. Kyoto Univ. 17, 533–538 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mohan Kumar, N.: On two conjectures about polynomial rings. Invent. Math. 46, 225–236 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Murthy, M.P.: Complete intersections. In: Conference on Commutative Algebra 1975 (Queen’s Univ., Kingston, Ont., 1975), Queen’s Papers on Pure and Applied Math., vol. 42, pp. 196–211. Queen’s Univ., Kingston (1975)Google Scholar
  24. 24.
    Quillen, D.: Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rao, R.A.: Two examples of Bass–Quillen–Suslin conjectures. Math. Ann. 279 (1987), 227-238. 93 (1988), 609–618Google Scholar
  26. 26.
    Rao, R.A.: Two examples of Bass–Quillen–Suslin conjectures. Math. Ann. 93 (1988), 609-618Google Scholar
  27. 27.
    Sridharan, R.: Non-vanishing sections of algebraic vector bundles. J. Algebra 176, 947–958 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stavrova, A.: Homotopy invariance of non-stable \(K_1\)-functors. J. \(K\)-Theory 13, 199–248 (2014)Google Scholar
  29. 29.
    Suslin, A.A.: Projective modules over a polynomial ring are free. Sov. Math. Dokl. 17, 1160–1164 (1976). (English transl.)zbMATHGoogle Scholar
  30. 30.
    Swan, R.G.: Vector bundles, projective modules and the K-theory of spheres. In: Algebraic Topology and Algebraic K-theory (Princeton, N.J., 1983), vol. 113, pp. 432–522, Ann. of Math. Stud., Princeton University Press, Princeton (1987)Google Scholar
  31. 31.
    Vavilov, N.A.: Subgroups of split orthogonal groups over a commutative ring. J. Math. Sci. 120, 1501–1512 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Vavilov, N.A., Petrov, V.A.: Overgroups of EO(n, R). St. Petersb. Math. J. 19, 167–195 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vorst, T.: The general linear group of polynomial rings. Commun. Algebra 9, 499–509 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Mrinal Kanti Das
    • 1
    Email author
  • Soumi Tikader
    • 1
  • Md. Ali Zinna
    • 2
    • 3
  1. 1.Stat-Math Unit, Indian Statistical InstituteKolkataIndia
  2. 2.School of Mathematical SciencesNational Institute of Science Education and Research Bhubaneswar (HBNI)OdishaIndia
  3. 3.Department of Mathematics and StatisticsIndian Institute of Science Education and Research KolkataNadiaIndia

Personalised recommendations