2015/02/28 by Wilberd van der Kallen · 1 citation
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Combinatorics #Commutative Algebra and Its Applications #Dimension (graph theory) #Discrete mathematics #Euler characteristic #Euler number (physics) #Euler's formula #Homomorphism #Ideal (ethics) #Integer (computer science) #Mathematical analysis #Mathematics #Noetherian #Omega #Path (computing) #Physics #Pure mathematics #Rings, Modules, and Algebras #Unimodular matrix #math.KT #msc:19E20
paper · pdf · doi:10.1016/j.jalgebra.2015.04.001
published as Journal of Algebra 434 (2015) 65-71 · 7 pages, reference updated
arxiv created 2015/04/13 · openalex publication_date 2015/04/14 · arxiv updated 2015/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let R be a noetherian ring of dimension d and let n be an integer so that n ≤ d≤ 2n-3. Let (a1,...,an+1) be a unimodular row so that the ideal J=(a1,...,an) has height n. Jean Fasel has associated to this row an element [(J,ωJ)] in the Euler class group En(R), with ωJ:(R/J)n→ J/J2 given by (a1,...,an-1,an an+1). If R contains an infinite field F then we show that the rule of Fasel defines a homomorphism from WMSn+1(R)=Umn+1(R)/En+1(R) to En(R). The main problem is to get a well defined map on all of Umn+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 SLn+1(F) is path connected for walks made up of elementary matrices.