On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme
Author:
Mrinal Kanti Das
Journal:
Trans. Amer. Math. Soc. 365 (2013), 33973411
MSC (2010):
Primary 13C10, 19A15, 13H05, 13B40
Published electronically:
December 12, 2012
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Abstract: Let be a regular local ring of dimension which is essentially of finite type over a field such that the residue field of is infinite. Let be a regular parameter and be an integer such that . Let be an ideal of height such that is generated by elements. It is proved that any given set of generators of can be lifted to a set of generators of .
 [Bh]
S. M. Bhatwadekar, Cancellation theorems for projective modules over a two dimensional ring and its polynomial extensions, Compositio Math. 128 (2001), 339359. MR 1858341 (2002h:13016)
 [BK]
S. M. Bhatwadekar and M. K. Keshari, A question of Nori: Projective generation of ideals, KTheory 28 (2003), 329351. MR 2017619 (2004j:13013)
 [BR]
S. M. Bhatwadekar and Ravi A. Rao, On a question of Quillen, Trans. Amer. Math. Soc. 279 (1983), 801810. MR 709584 (85g:13003)
 [BS1]
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), 161192. MR 1626485 (99c:13016)
 [BS2]
S. M. Bhatwadekar and Raja Sridharan, The Euler class group of a Noetherian ring, Compositio Math. 122 (2000), 183222. MR 1775418 (2001g:13018)
 [BS3]
S. M. Bhatwadekar and Raja Sridharan, On Euler classes and stably free projective modules, in: Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), 139158, Tata Inst. Fund. Res. Stud. Math., 16, Tata Inst. Fund. Res., Bombay, 2002. MR 1940666 (2003i:13012)
 [CHK]
JeanLouis ColliotThélène, Raymond T. Hoobler and Bruno Kahn, The BlochOgusGabber theorem, in: Algebraic theory (Toronto, ON, 1996), 3194, Fields Inst. Commun., 16, Amer. Math. Soc., Providence, RI, 1997. MR 1466971 (98j:14021)
 [Da1]
M. K. Das, The Euler class group of a polynomial algebra, J. Algebra 264 (2003), 582612. MR 1981423 (2004c:13031)
 [Da2]
M. K. Das, The Euler class group of a polynomial algebra II, J. Algebra 299 (2006), 94114. MR 2225766 (2007a:13023)
 [DS]
M. K. Das and Raja Sridharan, Euler class groups and a theorem of Roitman, J. Pure and Appl. Algebra 215 (2011), 13401347. MR 2769235 (2012c:13022)
 [Du]
S. P. Dutta, A theorem on smoothnessBassQuillen, Chow groups and intersection multiplicity of Serre, Trans. Amer. Math. Soc. 352 (1999), 16351645. MR 1621737 (2000i:13016)
 [EE]
D. Eisenbud and E. G. Evans, Generating modules efficiently: Theorems from algebraic Theory, J. Algebra 27 (1973), 278305. MR 0327742 (48:6084)
 [Ga1]
O. Gabber, ``Some theorems on Azumaya algebras'' in The Brauer Group (Sem., Les PlanssurBex, 1980), Lecture Notes in Math., 844, Springer, BerlinNew York, 1981, 129209. MR 611868 (83d:13004)
 [Ga2]
O. Gabber, Gersten's conjecture for some complexes of vanishing cycles, Manuscripta Math. 85 (1994), 323343. MR 1305746 (96m:14010)
 [HS]
C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, in: London Mathematical Society Lecture Note Series, Vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432 (2008m:13013)
 [Ka]
D. Katz, Generating ideals up to projective equivalence, Proc. Amer. Math. Soc. 120 (1994), 401414. MR 1176070 (94b:13013)
 [Li]
H. Lindel, On the BassQuillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981), 319323. MR 641133 (83g:13009)
 [Ly]
G. Lyubeznik, The number of defining equations of affine algebraic sets, Amer. J. Math. 114 (1992), 413463. MR 1156572 (93b:13018)
 [Ma]
S. Mandal, On efficient generation of ideals, Invent. Math. 75 (1984), 5967. MR 728138 (85d:13017)
 [MV]
S. Mandal and P. L. N. Varma, On a question of Nori: The local case, Communications in Algebra 25 (1997), 451457. MR 1428789 (98a:13026)
 [MMR]
N. Mohan Kumar, M. P. Murthy and Amit Roy, A cancellation theorem for projective modules over finitely generated rings, in: Algebraic geometry and commutative algebra, Vol. I, 281287, Kinokuniya, Tokyo, 1988. MR 977765 (89j:18009)
 [Mu]
M. P. Murthy, Projective modules, J. London Math. Soc., 41 (1966), 453456. MR 0200289 (34:188)
 [Na]
B. S. Nashier, Efficient generation of ideals in polynomial rings, Journal of Algebra 85 (1983), 287302. MR 725083 (85f:13010)
 [Ni]
Y. Nisnevich, Stratified canonical forms of matrix valued functions in a neighborhood of a transition point, Internat. Math. Res. Notices. 10 (1998), 513527. MR 1634912 (2000c:32082)
 [Po]
D. Popescu, Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989), 121128. MR 986438 (90g:13017)
 [Qu]
D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167171. MR 0427303 (55:337)
 [Ra1]
R. A. Rao, On projective modules, Amer. J. Math. 107 (1985), 387406. MR 784289 (86j:13011)
 [Ra2]
R. A. Rao, The BassQuillen conjecture in dimension three but characteristic , via a question of A. Suslin, Invent. Math. 93 (1988), no. 3, 609618. MR 952284 (89d:13011)
 [S]
Raja Sridharan, Projective modules and complete intersections, KTheory 13 (1998), 269278. MR 1609901 (99c:13019)
 [SV]
A. A. Suslin and L. N. Vaserstein, Serre's problem on projective modules over polynomial rings and algebraic theory, Izv. Akad. Nauk. SSSR, Ser. Mat. 40 (1976), 9371001. MR 0447245 (56:5560)
 [Sw1]
R. G. Swan, Vector bundles, projective modules and the theory of spheres, in: Algebraic topology and algebraic Ktheory (Princeton, N.J., 1983), Ann. of Math. Studies, Vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, 432522. MR 921488 (89f:18009)
 [Sw2]
R. G. Swan, NéronPopescu desingularization, in: Algebra and Geometry (Taipei, 1995), Lect. Algebra Geom., Vol. 2, International Press, Cambridge, MA, 135192. MR 1697953 (2000h:13006)
 [Bh]
 S. M. Bhatwadekar, Cancellation theorems for projective modules over a two dimensional ring and its polynomial extensions, Compositio Math. 128 (2001), 339359. MR 1858341 (2002h:13016)
 [BK]
 S. M. Bhatwadekar and M. K. Keshari, A question of Nori: Projective generation of ideals, KTheory 28 (2003), 329351. MR 2017619 (2004j:13013)
 [BR]
 S. M. Bhatwadekar and Ravi A. Rao, On a question of Quillen, Trans. Amer. Math. Soc. 279 (1983), 801810. MR 709584 (85g:13003)
 [BS1]
 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), 161192. MR 1626485 (99c:13016)
 [BS2]
 S. M. Bhatwadekar and Raja Sridharan, The Euler class group of a Noetherian ring, Compositio Math. 122 (2000), 183222. MR 1775418 (2001g:13018)
 [BS3]
 S. M. Bhatwadekar and Raja Sridharan, On Euler classes and stably free projective modules, in: Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), 139158, Tata Inst. Fund. Res. Stud. Math., 16, Tata Inst. Fund. Res., Bombay, 2002. MR 1940666 (2003i:13012)
 [CHK]
 JeanLouis ColliotThélène, Raymond T. Hoobler and Bruno Kahn, The BlochOgusGabber theorem, in: Algebraic theory (Toronto, ON, 1996), 3194, Fields Inst. Commun., 16, Amer. Math. Soc., Providence, RI, 1997. MR 1466971 (98j:14021)
 [Da1]
 M. K. Das, The Euler class group of a polynomial algebra, J. Algebra 264 (2003), 582612. MR 1981423 (2004c:13031)
 [Da2]
 M. K. Das, The Euler class group of a polynomial algebra II, J. Algebra 299 (2006), 94114. MR 2225766 (2007a:13023)
 [DS]
 M. K. Das and Raja Sridharan, Euler class groups and a theorem of Roitman, J. Pure and Appl. Algebra 215 (2011), 13401347. MR 2769235 (2012c:13022)
 [Du]
 S. P. Dutta, A theorem on smoothnessBassQuillen, Chow groups and intersection multiplicity of Serre, Trans. Amer. Math. Soc. 352 (1999), 16351645. MR 1621737 (2000i:13016)
 [EE]
 D. Eisenbud and E. G. Evans, Generating modules efficiently: Theorems from algebraic Theory, J. Algebra 27 (1973), 278305. MR 0327742 (48:6084)
 [Ga1]
 O. Gabber, ``Some theorems on Azumaya algebras'' in The Brauer Group (Sem., Les PlanssurBex, 1980), Lecture Notes in Math., 844, Springer, BerlinNew York, 1981, 129209. MR 611868 (83d:13004)
 [Ga2]
 O. Gabber, Gersten's conjecture for some complexes of vanishing cycles, Manuscripta Math. 85 (1994), 323343. MR 1305746 (96m:14010)
 [HS]
 C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, in: London Mathematical Society Lecture Note Series, Vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432 (2008m:13013)
 [Ka]
 D. Katz, Generating ideals up to projective equivalence, Proc. Amer. Math. Soc. 120 (1994), 401414. MR 1176070 (94b:13013)
 [Li]
 H. Lindel, On the BassQuillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981), 319323. MR 641133 (83g:13009)
 [Ly]
 G. Lyubeznik, The number of defining equations of affine algebraic sets, Amer. J. Math. 114 (1992), 413463. MR 1156572 (93b:13018)
 [Ma]
 S. Mandal, On efficient generation of ideals, Invent. Math. 75 (1984), 5967. MR 728138 (85d:13017)
 [MV]
 S. Mandal and P. L. N. Varma, On a question of Nori: The local case, Communications in Algebra 25 (1997), 451457. MR 1428789 (98a:13026)
 [MMR]
 N. Mohan Kumar, M. P. Murthy and Amit Roy, A cancellation theorem for projective modules over finitely generated rings, in: Algebraic geometry and commutative algebra, Vol. I, 281287, Kinokuniya, Tokyo, 1988. MR 977765 (89j:18009)
 [Mu]
 M. P. Murthy, Projective modules, J. London Math. Soc., 41 (1966), 453456. MR 0200289 (34:188)
 [Na]
 B. S. Nashier, Efficient generation of ideals in polynomial rings, Journal of Algebra 85 (1983), 287302. MR 725083 (85f:13010)
 [Ni]
 Y. Nisnevich, Stratified canonical forms of matrix valued functions in a neighborhood of a transition point, Internat. Math. Res. Notices. 10 (1998), 513527. MR 1634912 (2000c:32082)
 [Po]
 D. Popescu, Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989), 121128. MR 986438 (90g:13017)
 [Qu]
 D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167171. MR 0427303 (55:337)
 [Ra1]
 R. A. Rao, On projective modules, Amer. J. Math. 107 (1985), 387406. MR 784289 (86j:13011)
 [Ra2]
 R. A. Rao, The BassQuillen conjecture in dimension three but characteristic , via a question of A. Suslin, Invent. Math. 93 (1988), no. 3, 609618. MR 952284 (89d:13011)
 [S]
 Raja Sridharan, Projective modules and complete intersections, KTheory 13 (1998), 269278. MR 1609901 (99c:13019)
 [SV]
 A. A. Suslin and L. N. Vaserstein, Serre's problem on projective modules over polynomial rings and algebraic theory, Izv. Akad. Nauk. SSSR, Ser. Mat. 40 (1976), 9371001. MR 0447245 (56:5560)
 [Sw1]
 R. G. Swan, Vector bundles, projective modules and the theory of spheres, in: Algebraic topology and algebraic Ktheory (Princeton, N.J., 1983), Ann. of Math. Studies, Vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, 432522. MR 921488 (89f:18009)
 [Sw2]
 R. G. Swan, NéronPopescu desingularization, in: Algebra and Geometry (Taipei, 1995), Lect. Algebra Geom., Vol. 2, International Press, Cambridge, MA, 135192. MR 1697953 (2000h:13006)
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Additional Information
Mrinal Kanti Das
Affiliation:
StatMath Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108 India
Email:
mrinal@isical.ac.in
DOI:
http://dx.doi.org/10.1090/S000299472012055917
PII:
S 00029947(2012)055917
Received by editor(s):
October 15, 2010
Received by editor(s) in revised form:
March 25, 2011
Published electronically:
December 12, 2012
Dedicated:
Dedicated to Professor S. M. Bhatwadekar on his 65th birthday
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
