Projective modules over smooth, affine varieties over Archimedean real closed fields

Let X=Spec(A) be a smooth, affine variety of dimension n≥2n≥2 over the field RR of real numbers. Let PP be a projective AA-module of rankn such that its nnth Chern class Cn(P)∈CH0(X) is zero. In this set-up, Bhatwadekar–Das–Mandal showed (amongst many other results) that P≃A⊕QP≃A⊕Q in the case that either nn is odd or the topological space X(R)X(R) of real points of XX does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an Archimedean real closed field R.