Extrapolating an Euler class

Abstract

Let $R$ be a noetherian ring of dimension $d$ and let $n$ be an integer so that $n\le d\le 2n-3$. Let $(a_1,...,a_{n+1})$ be a unimodular row so that the ideal $J=(a_1,...,a_n)$ has height $n$. Jean Fasel has associated to this row an element $[(J,{\omega }_J)]$ in the Euler class group $E^n(R)$, with ${\omega }_J:(R/J{)}^n\to J/J^2$ given by $(a_1,...,a_{n-1},a_n{a}_{n+1})$. If $R$ contains an infinite field $F$ then we show that the rule of Fasel defines a homomorphism from $WM{S}_{n+1}(R)=U{m}_{n+1}(R)/E_{n+1}(R)$ to $E^n(R)$. The main problem is to get a well defined map on all of $U{m}_{n+1}(R)$. Similar results have been obtained by Mrinal Kanti Das and MD Ali Zinna, with a different proof. Our proof uses that every Zariski open subset of $S{L}_{n+1}(F)$ is path connected for walks made up of elementary matrices.

7 pages, reference updated