Zero cycles, Mennicke symbols and ${\mathrm{K}}_1$-stability of certain real affine algebras

Abstract

Let $R$ be a reduced real affine algebra of (Krull) dimension $d\ge 2$ such that either $R$ has no real maximal ideals, or the intersection of all real maximal ideals in $R$ has height at least one. In this article, we prove the following: (1) the $d$-th Euler class group ${\text{E}}^d(R)$, defined by Bhatwadekar-R.~Sridharan, is canonically isomorphic to the Levine-Weibel Chow group of zero cycles ${\text{CH}}_0(\text{Spec}(R))$; (2) the universal Mennicke symbol ${\text{MS}}_{d+1}(R)$ is canonically isomorphic to the universal weak Mennicke symbol ${\text{WMS}}_{d+1}(R)$; and (3) additionally, if $R$ is a regular domain, then the Whitehead group $\mathrm{S}{\mathrm{K}}_1(R)$ is canonically isomorphic to $\frac{{\text{SL}}_{d+1}(R)}{{\text{E}}_{d+1}(R)}$. As an application, we investigate some Eisenbud-Evans type theorems.

Minor editorial changes. Final version. To appear in IMRN

References in corpus (2)

• Euler class group of a Noetherian ring
• Nice group structure on the elementary orbit space of unimodular rows

Cited by in corpus (1)

• Splitting criteria for projective modules over polynomial algebras