Some perspective on Homotopy obstructions

TL;DR

This paper explores the structure of homotopy obstruction sets associated with projective modules over a noetherian ring, establishing isomorphisms under certain conditions and connecting these sets to Chow groups.

Contribution

It provides new insights into the structure of homotopy obstruction sets and their invariance under certain conditions, linking them to Chow groups.

Findings

Homotopy obstruction sets are isomorphic for projective modules with the same rank and determinant.

Established a natural map from homotopy obstruction sets to Chow groups.

Extended understanding of the structure and invariance of homotopy obstructions.

Abstract

Throughout $A$ will denote commutative noetherian ring, with $\dim A=d\geq 2$, and $P$ denote a projective $A$-module with $rank(P)=n$. In \cite{MM1} we considered the Homotopy obstruction sets $\pi_0\left({\mathcal LO}(P)\right)$, which has a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some perspective on these sets $\pi_0\left({\mathcal LO}(P)\right)$. Under similar regularity and other conditions, we prove if $P, Q$ are two projective $A$-modules, with $rank(P)=rank(Q)=d$ and $\det(P) \cong \det Q$, then $\pi_0\left({\mathcal LO}(Q)\right)\cong \pi_0\left({\mathcal LO}(P)\right)$. Further, for any projective $A$-module $P$ with $rank(P)=n$, we define a natural set theoretic map $\pi_0\left({\mathcal LO}(P)\right)\rightarrow CH^n(A)$, where $CH^n(A)$ Chow groups of codimension $n$ cycles.