Chow-Witt groups and Grothendieck-Witt groups of regular schemes

Fasel, J. ; Srinivas, V. (2009) Chow-Witt groups and Grothendieck-Witt groups of regular schemes Advances in Mathematics, 221 (1). pp. 302-329. ISSN 0001-8708

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.aim.2008.12.005

Abstract

Let A be a noetherian commutative Z[1/2]-algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck-Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If d≤3, we show that the vanishing of its Euler class in the corresponding Grothendieck-Witt group is a necessary and sufficient condition for P to have a free factor of rank one. If d is odd, we also get some results in that direction. If A is regular, we show that the Chow-Witt groups defined by Morel and Barge appear naturally as some homology groups of a Gersten-type complex in Grothendieck-Witt theory. From this, we deduce that if d=3 then the vanishing of the Euler class of P in the corresponding Chow-Witt group is a necessary and sufficient condition for P to have a free factor of rank one.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Grothendieck-Witt Groups; Chow-Witt Groups; Euler Classes; Projective Modules
ID Code:58966
Deposited On:02 Sep 2011 03:05
Last Modified:02 Sep 2011 03:05

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