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), 3397-3411

MSC (2010):
Primary 13C10, 19A15, 13H05, 13B40

DOI:
https://doi.org/10.1090/S0002-9947-2012-05591-7

Published electronically:
December 12, 2012

MathSciNet review:
3042589

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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), 339-359. MR**1858341 (2002h:13016)****[BK]**S. M. Bhatwadekar and M. K. Keshari, A question of Nori: Projective generation of ideals,*K-Theory***28**(2003), 329-351. MR**2017619 (2004j:13013)****[BR]**S. M. Bhatwadekar and Ravi A. Rao, On a question of Quillen,*Trans. Amer. Math. Soc.***279**(1983), 801-810. 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), 161-192. MR**1626485 (99c:13016)****[BS2]**S. M. Bhatwadekar and Raja Sridharan, The Euler class group of a Noetherian ring,*Compositio Math.***122**(2000), 183-222. 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), 139-158, Tata Inst. Fund. Res. Stud. Math., 16, Tata Inst. Fund. Res., Bombay, 2002. MR**1940666 (2003i:13012)****[CHK]**Jean-Louis Colliot-Thélène, Raymond T. Hoobler and Bruno Kahn, The Bloch-Ogus-Gabber theorem, in: Algebraic -theory (Toronto, ON, 1996), 31-94, 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), 582-612. MR**1981423 (2004c:13031)****[Da2]**M. K. Das, The Euler class group of a polynomial algebra II,*J. Algebra***299**(2006), 94-114. 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), 1340-1347. MR**2769235 (2012c:13022)****[Du]**S. P. Dutta, A theorem on smoothness--Bass-Quillen, Chow groups and intersection multiplicity of Serre,*Trans. Amer. Math. Soc.***352**(1999), 1635-1645. MR**1621737 (2000i:13016)****[EE]**D. Eisenbud and E. G. Evans, Generating modules efficiently: Theorems from algebraic -Theory,*J. Algebra***27**(1973), 278-305. MR**0327742 (48:6084)****[Ga1]**O. Gabber, ``Some theorems on Azumaya algebras'' in*The Brauer Group (Sem., Les Plans-sur-Bex, 1980)*, Lecture Notes in Math., 844, Springer, Berlin-New York, 1981, 129-209. MR**611868 (83d:13004)****[Ga2]**O. Gabber, Gersten's conjecture for some complexes of vanishing cycles,*Manuscripta Math.***85**(1994), 323-343. 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), 401-414. MR**1176070 (94b:13013)****[Li]**H. Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings,*Invent. Math.***65**(1981), 319-323. MR**641133 (83g:13009)****[Ly]**G. Lyubeznik, The number of defining equations of affine algebraic sets,*Amer. J. Math.***114**(1992), 413-463. MR**1156572 (93b:13018)****[Ma]**S. Mandal, On efficient generation of ideals,*Invent. Math.***75**(1984), 59-67. 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), 451-457. 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, 281-287, Kinokuniya, Tokyo, 1988. MR**977765 (89j:18009)****[Mu]**M. P. Murthy, Projective -modules,*J. London Math. Soc.*,**41**(1966), 453-456. MR**0200289 (34:188)****[Na]**B. S. Nashier, Efficient generation of ideals in polynomial rings,*Journal of Algebra***85**(1983), 287-302. 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), 513-527. MR**1634912 (2000c:32082)****[Po]**D. Popescu, Polynomial rings and their projective modules,*Nagoya Math. J.***113**(1989), 121-128. MR**986438 (90g:13017)****[Qu]**D. Quillen, Projective modules over polynomial rings,*Invent. Math.***36**(1976), 167-171. MR**0427303 (55:337)****[Ra1]**R. A. Rao, On projective -modules,*Amer. J. Math.***107**(1985), 387-406. MR**784289 (86j:13011)****[Ra2]**R. A. Rao, The Bass-Quillen conjecture in dimension three but characteristic , via a question of A. Suslin,*Invent. Math.***93**(1988), no. 3, 609-618. MR**952284 (89d:13011)****[S]**Raja Sridharan, Projective modules and complete intersections,*K-Theory***13**(1998), 269-278. 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), 937-1001. MR**0447245 (56:5560)****[Sw1]**R. G. Swan, Vector bundles, projective modules and the -theory of spheres, in: Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Studies, Vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, 432-522. MR**921488 (89f:18009)****[Sw2]**R. G. Swan, Néron-Popescu desingularization, in: Algebra and Geometry (Taipei, 1995), Lect. Algebra Geom., Vol. 2, International Press, Cambridge, MA, 135-192. MR**1697953 (2000h:13006)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
13C10,
19A15,
13H05,
13B40

Retrieve articles in all journals with MSC (2010): 13C10, 19A15, 13H05, 13B40

Additional Information

**Mrinal Kanti Das**

Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108 India

Email:
mrinal@isical.ac.in

DOI:
https://doi.org/10.1090/S0002-9947-2012-05591-7

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.