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

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.