Strong Euler class of a stably free module of odd rank

JOURNAL OF ALGEBRA

卷:432

Article

Das, Mrinal Kanti¹  Zinna, Md. Ali² 

[1] Indian Stat Inst, Stat Math Unit, Kolkata 700108, India

[2] Indian Inst Technol, Dept Math, Bombay 400076, Maharashtra, India

关键词: Projective modules;    Euler class groups;    Unimodular rows;   

DOI  :  10.1016/j.jalgebra.2015.03.007

来源: Elsevier

【 摘 要 】

Let R be a commutative Noetherian ring of dimension n >= 3. Following a suggestion of Fasel, we establish a group homomorphism phi from van der Kallen's group Um(n+1)(R)/En+1(R) to the n-th Euler class group E-n(R) so that: (1) when n is even, phi coincides with the homomorphism given by Bhatwadekar and Sridharan through Euler classes; (2) when n is odd, phi is non-trivial in general for an important class of rings; (3) the sequence Um(n+1)(R)/En+1(R) ->(phi) E-n(R) E-0(n)(R) -> 0 is exact, where E-0(n)(R) is the n-th weak Euler class group. (If X = Spec(R) is a smooth affine variety of dimension n over R so that-the complex-points of X are complete intersections and the canonical module K-R 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 - 3. We also indicate how to extend our arguments to this setup to obtain a group homomorphism from Um(n+1)(R)/En+1(R) to E-n(R). (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free