Convergence of two obstructions for projective modules

Abstract: Let $X=Spec{A}$ denote a regular affine scheme, over a field $k$, with $1/2\in k$ and $\dim X=d$. Let $P$ denote a projective $A$-module of rank $n\geq 2$. Let $\pi_0\left({\mathcal LO}(P)\right)$ denote the (Nori) Homotopy Obstruction set, and $\widetilde{CH}^n\left(X, \Lambda^nP\right)$ denote the Chow Witt group. In this article, we define a natural (set theoretic) map} $$ \Theta_P: \pi_0\left({\mathcal LO}(P)\right) \longrightarrow \widetilde{CH}^n\left(X, \Lambda^nP\right) $$ The main Results are included in my recently published book on Algebraic $K$-Theory.