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The Euler Class Group of a Noetherian Ring

Published online by Cambridge University Press:  04 December 2007

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

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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 = dimA 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.

Keywords

projective modulesEuler class groupunimodular elements
Type
Research Article
Information
Compositio Mathematica , Volume 122 , Issue 2 , June 2000 , pp. 183 - 222
Copyright
© 2000 Kluwer Academic Publishers