Compositio Mathematica



The Euler Class Group of a Noetherian Ring


S. M. Bhatwadekar a1 and Raja Sridharan a2
a1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400 005, India. E-mail: smb@math.tifr.res.in
a2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400 005, India. E-mail: sraja@math.tifr.res.in

Article author query
bhatwadekar sm   [Google Scholar] 
sridharan r   [Google Scholar] 
 

Abstract

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dim A and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.


Key Words: projective modules; Euler class group; unimodular elements.