Extrapolating an Euler class (Q2343506)

Let \(R\) be a noetherian ring of dimension \(d\) and let \(n\) be an integer so that \(n \leq d \leq 2n - 3. \) Let \((a_1,\dots,a_{n+1})\) be a unimodular row so that the ideal \(J = (a_1,, \ldots ,a_n)\) 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 \((\bar a_1, \dots , \bar a_{n-1}, \bar a_n \bar a_{n+1}).\) If \(R\) contains an infinite field \(F\) then it is shown that the rule of Fasel defines a homomorphism from \(WMS_{n+1}(R)/E_{n+1}(R) \) to \(E^n(R),\) The main problem is to get a well-defined map on all of \( Um _{n+1}(R)\). The proof uses that every Zariski open subset of \( SL_{n+1}(F) \) is path connected for walks made up of elementary matrices. Similar results have been obtained by \textit{M. K. Das} and \textit{Md. Ali Zinna} [J. Algebra 432, 185--204 (2015; Zbl 1314.13018)] with a different proof.