Chow-Witt groups and Grothendieck-Witt groups of regular schemes |
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Authors: | J Fasel V Srinivas |
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Institution: | a Jean Fasel, EPFL SB IMB CSAG, MA C3 575 (Bâtiment MA), CH-1015 Lausanne, Switzerland b Vasudevan Srinivas, School of Mathematics, TIFR, Homi Bhabha Road, Mumbai-400005, India |
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Abstract: | Let A be a noetherian commutative Z1/2]-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. |
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Keywords: | Grothendieck-Witt groups Chow-Witt groups Euler classes Projective modules |
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