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

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 \(2n\ge d+3\).

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References

  1. Asok, A., Fasel, J.: Euler class groups and motivic stable cohomotopy, preprint available at https://arxiv.org/abs/1601.05723

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

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  4. Bhatwadekar, S.M., Sridharan, R.: Zero cycles and the Euler class groups of smooth real affine varieties. Invent. Math. 136, 287–322 (1999)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  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)

  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)

  8. Das, M.K.: The Euler class group of a polynomial algebra. J. Algebra 264, 582–612 (2003)

    MathSciNet  Article  Google Scholar 

  9. Das, M.K.: On a conjecture of Murthy. Adv. Math. 331, 326–338 (2018)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  12. Das, M.K., Sridharan, R.: Euler class groups and a theorem of Roitman. J. Pure Appl. Algebra 215, 1340–1347 (2011)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  14. Das, M.K., Zinna, M.A.: “Strong” Euler class of a stably free module of odd rank. J. Algebra 432, 185–204 (2015)

    MathSciNet  Article  Google Scholar 

  15. Fasel, J.: On the number of generators of an ideal in a polynomial ring. Ann. Math. 184, 315–331 (2016)

    MathSciNet  Article  Google Scholar 

  16. Fasel, J.: Erratum on “On the number of generators of an ideal in a polynomial ring”. Ann. Math. 186, 647–648 (2017)

    MathSciNet  Article  Google Scholar 

  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)

  18. Keshari, M.K.: Euler Class group of a Noetherian ring, M.Phil. Thesis. http://www.math.iitb.ac.in/~keshari/acad.html

  19. Mandal, S.: On efficient generation of ideals. Invent. Math. 75, 59–67 (1984)

    MathSciNet  Article  Google Scholar 

  20. Mandal, S., Mishra, B.: The homotopy program in complete intersections, preprint available at https://arxiv.org/abs/1610.07495

  21. Mohan Kumar, N.: Complete intersections. J. Math. Kyoto Univ. 17, 533–538 (1977)

    MathSciNet  Article  Google Scholar 

  22. Mohan Kumar, N.: On two conjectures about polynomial rings. Invent. Math. 46, 225–236 (1978)

    MathSciNet  Article  Google Scholar 

  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)

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

    MathSciNet  Article  Google Scholar 

  25. Rao, R.A.: Two examples of Bass–Quillen–Suslin conjectures. Math. Ann. 279 (1987), 227-238. 93 (1988), 609–618

    MathSciNet  Article  Google Scholar 

  26. Rao, R.A.: Two examples of Bass–Quillen–Suslin conjectures. Math. Ann. 93 (1988), 609-618

  27. Sridharan, R.: Non-vanishing sections of algebraic vector bundles. J. Algebra 176, 947–958 (1995)

    MathSciNet  Article  Google Scholar 

  28. Stavrova, A.: Homotopy invariance of non-stable \(K_1\)-functors. J. \(K\)-Theory 13, 199–248 (2014)

  29. Suslin, A.A.: Projective modules over a polynomial ring are free. Sov. Math. Dokl. 17, 1160–1164 (1976). (English transl.)

    MATH  Google Scholar 

  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)

  31. Vavilov, N.A.: Subgroups of split orthogonal groups over a commutative ring. J. Math. Sci. 120, 1501–1512 (2004)

    MathSciNet  Article  Google Scholar 

  32. Vavilov, N.A., Petrov, V.A.: Overgroups of EO(n, R). St. Petersb. Math. J. 19, 167–195 (2008)

    MathSciNet  Article  Google Scholar 

  33. Vorst, T.: The general linear group of polynomial rings. Commun. Algebra 9, 499–509 (1981)

    MathSciNet  Article  Google Scholar 

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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.

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Correspondence to Mrinal Kanti Das.

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Das, M.K., Tikader, S. & Zinna, M.A. “\({\mathbb {P}}^1\)-gluing” for local complete intersections. Math. Z. 294, 667–685 (2020). https://doi.org/10.1007/s00209-019-02299-5

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  • DOI: https://doi.org/10.1007/s00209-019-02299-5

Mathematics Subject Classification

  • 13C10
  • 19A15
  • 14C25
  • 11E81