ANOTHER DEFINITION OF AN EULER CLASS GROUP OF A NOETHERIAN RING

All the rings are assumed to be commutative Noetherian and all the modules are finitely generated. Let A be a ring of dimension n ≥ 2, and let L be a projective A-module of rank 1. In [3], Bhatwadekar and Sridharan defined an abelian group, called the Euler class group of A with respect to L which is denoted by E(A,L). To the pair (P, χ), where P is a projective Amodule of rank n with determinant L and χ : L ∼→∧nP an isomorphism, called an L-orientation of P, they attached an element of E(A,L) which is denoted by e(P, χ). One of the main result in [3] is that P has a unimodular element if and only if e(P, χ) is zero in E(A,L). Copyright © 2013 Rocky Mountain Mathematics Consortium.