On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme
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Abstract:
Let $(R, \mathfrak {m})$ be a regular local ring of dimension $d$ which is essentially of finite type over a field $k$ such that the residue field of $R$ is infinite. Let $f \in \mathfrak {m} \smallsetminus \mathfrak {m}^2$ be a regular parameter and $n$ be an integer such that $2n \geq d + 1$. Let $I \subset R_f$ be an ideal of height $n$ such that $I/I^2$ is generated by $n$ elements. It is proved that any given set of $n$ generators of $I/I^2$ can be lifted to a set of $n$ generators of $I$.References
- S. M. Bhatwadekar, Cancellation theorems for projective modules over a two-dimensional ring and its polynomial extensions, Compositio Math. 128 (2001), no. 3, 339–359. MR 1858341, DOI 10.1023/A:1011839525245
- S. M. Bhatwadekar and Manoj Kumar Keshari, A question of Nori: projective generation of ideals, $K$-Theory 28 (2003), no. 4, 329–351. MR 2017619, DOI 10.1023/A:1026217116072
- S. M. Bhatwadekar and R. A. Rao, On a question of Quillen, Trans. Amer. Math. Soc. 279 (1983), no. 2, 801–810. MR 709584, DOI 10.1090/S0002-9947-1983-0709584-1
- S. M. Bhatwadekar and Raja Sridharan, Projective generation of curves in polynomial extensions of an affine domain and a question of Nori, Invent. Math. 133 (1998), no. 1, 161–192. MR 1626485, DOI 10.1007/s002220050243
- S. M. Bhatwadekar and Raja Sridharan, The Euler class group of a Noetherian ring, Compositio Math. 122 (2000), no. 2, 183–222. MR 1775418, DOI 10.1023/A:1001872132498
- S. M. Bhatwadekar and Raja Sridharan, On Euler classes and stably free projective modules, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 139–158. MR 1940666
- Jean-Louis Colliot-Thélène, Raymond T. Hoobler, and Bruno Kahn, The Bloch-Ogus-Gabber theorem, Algebraic $K$-theory (Toronto, ON, 1996) Fields Inst. Commun., vol. 16, Amer. Math. Soc., Providence, RI, 1997, pp. 31–94. MR 1466971
- Mrinal Kanti Das, The Euler class group of a polynomial algebra, J. Algebra 264 (2003), no. 2, 582–612. MR 1981423, DOI 10.1016/S0021-8693(03)00240-0
- Mrinal Kanti Das, The Euler class group of a polynomial algebra. II, J. Algebra 299 (2006), no. 1, 94–114. MR 2225766, DOI 10.1016/j.jalgebra.2005.06.017
- Mrinal Kanti Das and Raja Sridharan, Euler class groups and a theorem of Roitman, J. Pure Appl. Algebra 215 (2011), no. 6, 1340–1347. MR 2769235, DOI 10.1016/j.jpaa.2010.08.014
- S. P. Dutta, A theorem on smoothness—Bass-Quillen, Chow groups and intersection multiplicity of Serre, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1635–1645. MR 1621737, DOI 10.1090/S0002-9947-99-02372-7
- David Eisenbud and E. Graham Evans Jr., Generating modules efficiently: theorems from algebraic $K$-theory, J. Algebra 27 (1973), 278–305. MR 327742, DOI 10.1016/0021-8693(73)90106-3
- Ofer Gabber, Some theorems on Azumaya algebras, The Brauer group (Sem., Les Plans-sur-Bex, 1980) Lecture Notes in Math., vol. 844, Springer, Berlin-New York, 1981, pp. 129–209. MR 611868
- Ofer Gabber, Gersten’s conjecture for some complexes of vanishing cycles, Manuscripta Math. 85 (1994), no. 3-4, 323–343. MR 1305746, DOI 10.1007/BF02568202
- Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
- D. Katz, Generating ideals up to projective equivalence, Proc. Amer. Math. Soc. 120 (1994), no. 1, 79–83. MR 1176070, DOI 10.1090/S0002-9939-1994-1176070-X
- Hartmut Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981/82), no. 2, 319–323. MR 641133, DOI 10.1007/BF01389017
- Gennady Lyubeznik, The number of defining equations of affine algebraic sets, Amer. J. Math. 114 (1992), no. 2, 413–463. MR 1156572, DOI 10.2307/2374710
- S. Mandal, On efficient generation of ideals, Invent. Math. 75 (1984), no. 1, 59–67. MR 728138, DOI 10.1007/BF01403089
- S. Mandal and P. L. N. Varma, On a question of Nori: the local case, Comm. Algebra 25 (1997), no. 2, 451–457. MR 1428789, DOI 10.1080/00927879708825865
- N. Mohan Kumar, M. Pavaman Murthy, and A. Roy, A cancellation theorem for projective modules over finitely generated rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 281–287. MR 977765
- M. Pavaman Murthy, Projective $A[X]$-modules, J. London Math. Soc. 41 (1966), 453–456. MR 200289, DOI 10.1112/jlms/s1-41.1.453
- Budh S. Nashier, Efficient generation of ideals in polynomial rings, J. Algebra 85 (1983), no. 2, 287–302. MR 725083, DOI 10.1016/0021-8693(83)90095-9
- Yevsey Nisnevich, Stratified canonical forms of matrix valued functions in a neighborhood of a transition point, Internat. Math. Res. Notices 10 (1998), 513–527. MR 1634912, DOI 10.1155/S1073792898000336
- Dorin Popescu, Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989), 121–128. MR 986438, DOI 10.1017/S0027763000001288
- Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171. MR 427303, DOI 10.1007/BF01390008
- Ravi A. Rao, On projective $R_{f_1\cdots f_t}$-modules, Amer. J. Math. 107 (1985), no. 2, 387–406. MR 784289, DOI 10.2307/2374420
- Ravi A. Rao, The Bass-Quillen conjecture in dimension three but characteristic $\not =2,3$ via a question of A. Suslin, Invent. Math. 93 (1988), no. 3, 609–618. MR 952284, DOI 10.1007/BF01410201
- Raja Sridharan, Projective modules and complete intersections, $K$-Theory 13 (1998), no. 3, 269–278. MR 1609901, DOI 10.1023/A:1007774903782
- L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245
- Richard G. Swan, Vector bundles, projective modules and the $K$-theory of spheres, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 432–522. MR 921488
- Richard G. Swan, Néron-Popescu desingularization, Algebra and geometry (Taipei, 1995) Lect. Algebra Geom., vol. 2, Int. Press, Cambridge, MA, 1998, pp. 135–192. MR 1697953
Additional Information
- Mrinal Kanti Das
- Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108 India
- Email: mrinal@isical.ac.in
- Received by editor(s): October 15, 2010
- Received by editor(s) in revised form: March 25, 2011
- Published electronically: December 12, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3397-3411
- MSC (2010): Primary 13C10, 19A15, 13H05, 13B40
- DOI: https://doi.org/10.1090/S0002-9947-2012-05591-7
- MathSciNet review: 3042589
Dedicated: Dedicated to Professor S. M. Bhatwadekar on his 65th birthday