Projective modules over smooth, affine varieties over real closed fields

Let X=Spec(A) be a smooth, affine variety of dimension n⩾2 over the field R of real numbers. Let P be a projective A-module of rankn such that its nth Chern class Cn(P)∈CH0(X) is zero. In this set-up, Bhatwadekar–Das–Mandal showed (amongst many other results) that P≃A⊕Q in the case that either n is odd or the topological space X(R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an arbitrary real closed field R. The proof is algebraic and does not make use of Tarski's principle, nor of the earlier result for R.