Zero Cycles, Mennicke Symbols, and K1-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 $\textrm{E}^{d}(R)$, defined by Bhatwadekar–Raja Sridharan, is canonically isomorphic to the Levine–Weibel Chow group of zero cycles $\textrm{CH}_{0}(\textrm{Spec}(R))$; (2) the universal Mennicke symbol $\textrm{MS}_{d+1}(R)$ is canonically isomorphic to the universal weak Mennicke symbol $\textrm{WMS}_{d+1}(R)$; and (3) additionally, if $R$ is a regular domain, then the Whitehead group $\textrm{SK}_{1}(R)$ is canonically isomorphic to $\frac{\textrm{SL}_{d+1}(R)}{\textrm{E}_{d+1}(R)}$. As an application, we investigate some Eisenbud–Evans-type theorems.