“Strong” Euler class of a stably free module of odd rank

Let R   be a commutative Noetherian ring of dimension n≥3n≥3. Following a suggestion of Fasel, we establish a group homomorphism ϕ   from van der Kallen's group Umn+1(R)/En+1(R)Umn+1(R)/En+1(R) to the n  -th Euler class group En(R)En(R) so that: (1) when n is even, ϕ coincides with the homomorphism given by Bhatwadekar and Sridharan through Euler classes; (2) when n is odd, ϕ   is non-trivial in general for an important class of rings; (3) the sequence Umn+1(R)/En+1(R)→ϕEn(R)⟶E0n(R)→0 is exact, where E0n(R) is the n  -th weak Euler class group. (If X=Spec(R)X=Spec(R) is a smooth affine variety of dimension n   over RR so that the complex points of X   are complete intersections and the canonical module KRKR is trivial, then the sequence is proved to be exact on the left as well.) More generally, let R be a commutative Noetherian ring of dimension d and n   be an integer such that n≤d≤2n−3n≤d≤2n−3. We also indicate how to extend our arguments to this setup to obtain a group homomorphism from Umn+1(R)/En+1(R)Umn+1(R)/En+1(R) to En(R)En(R).