Extrapolating an Euler class

Abstract

Let R be a noetherian ring of dimension d and let n be an integer so that n≤d≤2n-3. Let (a<inf>1</inf>,..., a<inf>n+1</inf>) be a unimodular row so that the ideal J=(a<inf>1</inf>,..., a<inf>n</inf>) has height n. Jean Fasel has associated to this row an element [(J, ω<inf>J</inf>)] in the Euler class group Eⁿ(R), with ω<inf>J</inf>:(R/J)ⁿ→J/J² given by (ā1,...,ān-1,ānān+1). If R contains an infinite field F then we show that the rule of Fasel defines a homomorphism from WMS<inf>n+1</inf>(R)=Um<inf>n+1</inf>(R)/E<inf>n+1</inf>(R) to Eⁿ(R). The main problem is to get a well defined map on all of Um<inf>n+1</inf>(R). Similar results have been obtained by Das and Zinna [5], with a different proof. Our proof uses that every Zariski open subset of SL<inf>n+1</inf>(F) is path connected for walks made up of elementary matrices.