The Euler Class Group of a Noetherian Ring

By S. M. Bhatwadekar, R. Sridharan. Compositio Mathematica, 2000

117 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

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In Algebra and Number Theory

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Citations in the corpus

• Chow-Witt groups and Grothendieck-Witt groups of regular schemes (2009, methodology): “If A is a noetherian Q-algebra, then there is also a definition of the Euler class group due to Bhatwadekar and Sridharan [10] and of the Euler class in it associated to a projective module of top rank (the assumption on the determinant is removed).”
• A1-Algebraic Topology over a Field (2012, unknown): “2) In [13] an “Euler class group” E(A) introduced by Nori, and a corresponding Euler class, is used to prove an analogous result on a noetherian ring A.”