Projective modules over the real algebraic sphere of dimension 3 (Q630839)

Let \(A\) be a noetherian commutative ring of Krull dimension \(d\) and \({\tilde K}_0 Sp(A)\) the Grothendieck-Witt group \(GW^2(A)\) modulo the subgroup generated by the antisymmetric form \(H(A)\). Let \(Um_4(A)/Sp_4(A)\) be the set of orbits of unimodular rows of rank 4 under the action of the symplectic group \(Sp_4(A)\). It shows that every projective \(A\)-module of rank 2 with trivial determinant is free if and only if the group \({\tilde K}_0 Sp(A)\) and the set \(Um_4(A)/Sp_4(A)\) are both zero.As an application, every finitely generated projective module over the real algebraic sphere of dimension 3 is free.