Euler cycles and Mennicke symbols

TL;DR

This paper explores the relationship between Euler class groups and unimodular rows over smooth affine domains, establishing a morphism that links algebraic K-theory and unimodular row orbits.

Contribution

It introduces a new morphism from the Euler class group to the group of elementary orbits of unimodular rows, connecting two important algebraic structures.

Findings

Established a morphism from Euler class group to unimodular row orbits.

Connected algebraic K-theory with properties of unimodular rows.

Provided new insights into the structure of smooth affine domains.

Abstract

Let $R$ be a smooth affine domain of dimension $d\geq 2$ over an infinite perfect field $k$. We establish a morphism from the Euler class group $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$, the group of elementary orbits of unimodular rows.