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Revisiting Nori's question and homotopy invariance of Euler class groups

Published online by Cambridge University Press:  05 November 2010

Mrinal Kanti Das
Mrinal Kanti Das
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108India. [email protected]
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Abstract

This paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.

Keywords

Projective modulesEfficient generation of idealsEuler class groups
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 3 , December 2011 , pp. 451 - 480
Copyright
Copyright © ISOPP 2010

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References

B.Bhatwadekar, S. M., Projective generation of maximal ideals in polynomial rings, J. Algebra 91 (1984), 75–81.CrossRefGoogle Scholar
B-D-M.Bhatwadekar, S. M., Das, M. K. and Mandal, S., Projective modules over smooth real affine varieties, Invent. Math. 166 (2006), 151–184.CrossRefGoogle Scholar
B-K.Bhatwadekar, S. m. and Keshari, M. K., A question of Nori: Projective generation of ideals, K-Theory 28 (2003), 329–351.CrossRefGoogle Scholar
B-RS 1.Bhatwadekar, S. M. and Sridharan, Raja, Projective generation of curves in polynomial extensions of an affine domain and a question of Nori, Invent. Math. 133 (1998), 161–192.CrossRefGoogle Scholar
B-RS 2.Bhatwadekar, S. M. and Sridharan, Raja, Zero cycles and the Euler class groups of smooth real affine varieties, Invent. Math. 136 (1999), 287–322.CrossRefGoogle Scholar
B-RS 3.Bhatwadekar, S. M. and Sridharan, Raja, The Euler class group of a Noetherian ring, Compositio Math. 122 (2000), 183–222.CrossRefGoogle Scholar
B-RS 4.Bhatwadekar, S. M. and Sridharan, Raja, On a question of Roitman, J. Ramanujan Math. Soc. 16 (2001), 45–61.Google Scholar
B-RS 5.Bhatwadekar, S. M. and Sridharan, Raja, Projective generation of curves in polynomial extensions of an affine domain (II), K-Theory 15 (1998), 293–300.CrossRefGoogle Scholar
D1.Das, M. K., The Euler class group of a polynomial algebra, J. Algebra 264 (2003), 582–612.CrossRefGoogle Scholar
D2.Das, M. K., The Euler class group of a polynomial algebra II, J. Algebra 299 (2006), 94–114.CrossRefGoogle Scholar
D-RS.Das, M. K. and Sridharan, Raja, The Euler class groups of polynomial rings and unimodular elements in projective modules, J. Pure and Appl. Algebra 175 (2003), 73–86.CrossRefGoogle Scholar
E-E.Eisenbud, D. and Evans, E. G., Generating modules efficiently: Theorems from algebraic K-Theory, J. Algebra 27 (1973), 278–305.CrossRefGoogle Scholar
K.Keshari, M., Euler Class group of a Noetherian ring, M.Phil. thesis, available at : http://www.math.iitb.ac.in/~keshari/acad.htmlGoogle Scholar
M 1.Mandal, S., On efficient generation of ideals, Invent. Math. 75 (1984), 59–67.CrossRefGoogle Scholar
M 2.Mandal, S., Homotopy of sections of projective modules, J. Algebraic Geometry 1 (1992), 639–646.Google Scholar
M-V.Mandal, S. and Varma, P. L. N., On a question of Nori: the local case, Communications in Algebra 25 (1997), 451–457.CrossRefGoogle Scholar
Mo1.Mohan Kumar, N., On two conjectures about polynomial rings, Invent. Math. 46 (1978), 225–236.CrossRefGoogle Scholar
Mo2.Mohan Kumar, N., Stably free modules, Amer. J. Math. 107 (1985), 1439–1444.CrossRefGoogle Scholar
P.Plumstead, B., The conjectures of Eisenbud and Evans, Amer. J. Math. 105 (1983), 1417–1433.CrossRefGoogle Scholar
Q.Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.CrossRefGoogle Scholar
RS.Sridharan, Raja, Projective modules and complete intersections, K-Theory 13 (1998), 269–278.CrossRefGoogle Scholar
Ra 1.Rao, R. A., Two examples of Bass-Quillen-Suslin conjectures, Math. Ann. 279 (1987), 227–238.CrossRefGoogle Scholar
Ra 2.Rao, R. A., The Bass-Quillen conjecture in dimension three but characteristic ≠ 2,3 via a question of A. Suslin, Invent. Math. 93 (1988), 609–618.CrossRefGoogle Scholar