Convergence of two obstructions for projective modules

TL;DR

This paper establishes a natural connection between the Nori Homotopy Obstruction set and the Chow Witt group for projective modules over regular affine schemes, advancing the understanding of obstructions in algebraic K-theory.

Contribution

It introduces a natural set-theoretic map linking two different obstructions for projective modules, unifying homotopy and Chow Witt obstructions in algebraic geometry.

Findings

Defined a natural map between the Homotopy Obstruction set and Chow Witt group.

Proved the convergence of two obstructions for projective modules.

Enhanced the theoretical framework in algebraic K-theory.

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.