Journal of the Ramanujan Mathematical Society

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The Homotopy Obstructions in Complete Intersections


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1 University of Kansas, Lawrence, Kansas 66045, United States
     

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Let A be a regular ring over a field k, with 1/2 ∈ k, and dim A = d. We discuss the Homotopy Obstruction Program, in the complete intersection case. Fix an integer n ≥ 2. A local orientation is a pair (I,ω), where I is an ideal and ω : An ↠I/I 2 is a surjective map. The goal is to define and detect homotopy obstructions, for ω to lift to a surjective map An ↠ I . Denote the set of all local orientations by LO(A, n). A homotopy relations on LO(A, n) is induced by the maps LO(A, n)←T=0 LO(A[T ], n) →T=1 LO(A, n) . The homotopy obstruction set π0(LO(A, n)) is defined to be the set of all equivalence classes. Assume 2n ≥ d+2.We prove that π0(LO(A, n)) is an abelian group. We also establish a surjective map ρ : En(A)↠ π0(LO(A, n)), where En(A) denotes the Euler class group. When 2n ≥ d + 3, and A is essentially smooth, we prove ρ is an isomorphism. This settles a conjecture of Morel.
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  • [AF] Asok Aravind and Fasel Jean, Euler class groups and motivic stable cohomotopy, https://arxiv.org/pdf/1601.05723.pdf.

  • [AF2] Asok Aravind and Fasel Jean, Splitting vector bundles outside the stable range and A1-homotopy sheaves of punctured affine spaces, J. Amer. Math. Soc., 28 no. 4, (2015) 1031–1062.

  • [AHW] A. Asok, M. Hoyois and M. Wendt, Affine representability results in A1-homotopy theory II: Principal bundles and homogeneous spaces, 2015. Geometry & Topology, XX (20XX) 1001–999 arXiv 1507.08020.

  • [BM] Barge Jean and Morel Fabien, Groupe de Chow des cycles orient´es et classe d’Euler des fibr´es vectoriels. (French) [The Chow group of oriented cycles and the Euler class of vector bundles], C. R. Acad. Sci. Paris S´er. I Math., 330 no. 4, (2000) 287–290.

  • [BK] S. M. Bhatwadekar and Keshari Manoj Kumar, A question of Nori: projective generation of ideals, K-Theory, 28 no. 4, (2003) 329–351.

  • [BS1] S. M. Bhatwadekar and Sridharan, Raja projective generation of curves in polynomial extensions of an affine domain and a question of Nori., Invent. Math., 133 no. 1, (1998) 161–192.

  • [BS2] S.M. Bhatwadekar and Sridharan Raja, On Euler classes and stably free projective modules, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Bombay, 16 (2002) 139–158.

  • [BS3] S. M. Bhatwadekar and Sridharan Raja, The Euler class group of a Noetherian ring, Compositio Math., 122 no. 2, (2000) 183–222.

  • [CF] Calm´es Baptiste and Fasel Jean, Groupes classiques. (French) [Classical groups], Autours des sch´emas en groupes., Vol. II, Panor. Synth´eses, Soc. Math. France, Paris, 46 (2015) 1–133.

  • [F1] Fasel Jean, On the number of generators of ideals in polynomial rings, Ann. of Math. (2), 184 no. 1, (2016) 315–331.

  • [F2] Erratum, On the number of generators of ideals in polynomial rings?, Ann. of Math., (to appear).

  • [K] Knus Max-Albert, Quadratic and Hermitian forms over rings. With a foreword by I. Bertuccioni. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 294. Springer-Verlag, Berlin, 1991. xii+524 pp.

  • [L] Lindel Hartmut, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math., 65 no. 2, (1981/82) 319–323.

  • [M1] Mandal Satya, Projective modules and complete intersections, Lecture Notes in Mathematics, 1672. Springer-Verlag, Berlin (1997) viii+114 pp.

  • [M2] Mandal Satya, On the complete intersection conjecture of Murthy, J. Algebra, 458 (2016) 156–170.

  • [M3] Mandal Satya, Homotopy of sections of projective modules. With an appendix by Madhav V. Nori., J. Algebraic Geom., 1 no. 4, (1992) 639–646.

  • [MMu] Mandal Satya and M. Pavaman Murthy, Ideals as sections of projective modules, J. Ramanujan Math. Soc., 13 no. 1, (1998) 51–62.

  • [MV] S. Mandal and P. L. N. Varma, On a question of Nori: the local case, Comm. Algebra, 25 no. 2, (1997) 451–457.

  • [MS] Mandal Satya and Sridharan Raja, Euler classes and complete intersections, J. Math. Kyoto Univ., 36 no. 3, (1996) 453–470.

  • [Mo] Morel Fabien, A1-algebraic topology over a field. Lecture Notes in Mathematics, 2052. Springer, Heidelberg (2012) x+259 pp.

  • [Mu] M. Murthy, Pavaman zero cycles and projective modules, Ann. of Math. (2), 140 no. 2, (1994) 405–434.

  • [MoM] Kumar N. Mohan and Murthy M. Pavaman, Algebraic cycles and vector bundles over affine three-folds, Ann. of Math. (2), 116 no. 3, (1982) 579–591.

  • [Mk1] N. Mohan Kumar, Stably free modules, Amer. J. Math., 107 no. 6, (1985–1986) 1439–1444.

  • [Mk2] N. Mohan Kumar, Some theorems on generation of ideals in affine algebras, Comment. Math. Helv., 59 no. 2, (1984) 243–252.

  • [O1] Ojanguren Manuel, Formes quadratiques sur les algebres de polynomes. (French), C. R. Acad. Sci. Paris S´er. A-B, 287 no. 9, (1978) A695–A698.

  • [O2] Ojanguren Manuel, A splitting theorem for quadratic forms, Comment. Math. Helv., 57 no. 1, (1982) 145–157.

  • [P] Popescu Dorin, Letter to the editor: “General Neron desingularization and approximation”, Nagoya Math. J., 118 (1990) 45–53.

  • [Pr1] Parimala Raman, Quadratic forms over polynomial rings over Dedekind domains, Amer. J. Math., 103 no. 2, (1981) 289–296.

  • [Pr2] Parimala Raman, Quadratic spaces over polynomial extensions of regular rings of dimension 2, Math. Ann., 261 no. 3, (1982) 287–292.

  • [Rr] Rao Ravi, Extendability of quadratic modules with sufficient Witt index, J. Algebra, 86 no. 1, (1984) 159–180.

  • [R] Roy Amit, Application of patching diagrams to some questions about projective modules, J. Pure Appl. Algebra, 24 no. 3, (1982) 313–319.

  • [St] Steenrod, Norman the topology of fibre bundles, Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton, N. J. (1951) viii+224 pp.

  • [Sw1] G. Swan Richard and Neron-Popescu, Desingularization, Algebra and geometry (Taipei, 1995), Lect. Algebra Geom., 2, Int. Press, Cambridge, MA (1998) 135–192.

  • [Sw2] G. Swan Richard, Higher algebraic K-theory, K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 247–293, Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 58 (1995).


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  • The Homotopy Obstructions in Complete Intersections

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Authors

Satya Mandal
University of Kansas, Lawrence, Kansas 66045, United States
Bibekananda Mishra
University of Kansas, Lawrence, Kansas 66045, United States

Abstract


Let A be a regular ring over a field k, with 1/2 ∈ k, and dim A = d. We discuss the Homotopy Obstruction Program, in the complete intersection case. Fix an integer n ≥ 2. A local orientation is a pair (I,ω), where I is an ideal and ω : An ↠I/I 2 is a surjective map. The goal is to define and detect homotopy obstructions, for ω to lift to a surjective map An ↠ I . Denote the set of all local orientations by LO(A, n). A homotopy relations on LO(A, n) is induced by the maps LO(A, n)←T=0 LO(A[T ], n) →T=1 LO(A, n) . The homotopy obstruction set π0(LO(A, n)) is defined to be the set of all equivalence classes. Assume 2n ≥ d+2.We prove that π0(LO(A, n)) is an abelian group. We also establish a surjective map ρ : En(A)↠ π0(LO(A, n)), where En(A) denotes the Euler class group. When 2n ≥ d + 3, and A is essentially smooth, we prove ρ is an isomorphism. This settles a conjecture of Morel.

References