Journal of Algebra 264 (2003) 582–612 www.elsevier.com/locate/jalgebra The Euler class group of a polynomial algebra Mrinal Kanti Das Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India Received 28 May 2002 Communicated by Craig Huneke 1. Introduction Let A be a smooth affine domain of dimension n over a field k and I be a prime ideal of A[T ] of height r such that A[T ]/I is smooth and 2r  n + 3. Let f 1 (T),...,f r (T) ∈ I such that I = (f 1 (T),...,f r (T)) + (I 2 T). Furthermore, assume that A/(f 1 (0),...,f r (0)) is also smooth. In this set up Nori asked the following question (for motivation, see [M2, Introduction]): Question. Do there exist g 1 ,...,g r such that I = (g 1 ,...,g r ) with g i - f i ∈ (I 2 T)? This question has been answered affirmatively by Mandal [M2] when I contains a monic polynomial, even without any smoothness assumptions. When I does not contain a monic polynomial, Nori’s question has been answered in the affirmative in the following cases: (1) A is a local ring of a smooth affine algebra over an infinite field [M-V, Theorem 4]. (2) A is a smooth affine algebra over an infinite field and r = n (i.e., dim A[T ]/I = 1) [B-RS1, Theorem 3.8]. Moreover, an example is given in [B-RS1, Example 6.4] for the case dim(A[T ]/I) = 1, which shows that the question of Nori does not have an affirmative answer in general without the smoothness assumption. So, in view of this example of [B-RS1] one wonders where the obstruction for I to have a set of generators satisfying the required properties lies. In this paper we investigate this question when dim A[T ]/I = 1. Taking a cue from the above result of Mandal, we prove the following theorem. E-mail address: mrinal@mri.ernet.in. 0021-8693/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-8693(03)00240-0