Some perspective on Homotopy obstructions

Abstract

Throughout $A$ will denote commutative noetherian ring, with $\dim A=d\ge 2$, and $P$ denote a projective $A$-module with $rank(P)=n$. In \cite{MM1} we considered the Homotopy obstruction sets ${\pi }_0\left(\mathcal{L}O(P)\right)$, which has a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some perspective on these sets ${\pi }_0\left(\mathcal{L}O(P)\right)$. 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 ${\pi }_0\left(\mathcal{L}O(Q)\right)\cong {\pi }_0\left(\mathcal{L}O(P)\right)$. Further, for any projective $A$-module $P$ with $rank(P)=n$, we define a natural set theoretic map ${\pi }_0\left(\mathcal{L}O(P)\right)\to C{H}^n(A)$, where $C{H}^n(A)$ Chow groups of codimension $n$ cycles.

Cited by in corpus (1)

• Convergence of two obstructions for projective modules