Fasel, J.Srinivas, V.2010-11-302010-11-302010-11-30200910.1016/j.aim.2008.12.005https://infoscience.epfl.ch/handle/20.500.14299/60339WOS:000265249000009Let 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. Crown Copyright (C) 2008 Published by Elsevier Inc. All rights reserved.Grothendieck-Witt groupsChow-Witt groupsEuler classesProjective modulesVector-BundlesEuler ClassCyclestext::journal::journal article::research article