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

TL;DR

This paper establishes deep connections between algebraic K-theory, Chow groups, and Mennicke symbols for certain real affine algebras, providing new isomorphisms and applications to Eisenbud-Evans type theorems.

Contribution

It proves canonical isomorphisms between Euler class groups, Chow groups, Mennicke symbols, and Whitehead groups for specific real affine algebras, extending understanding of their algebraic structure.

Findings

Euler class group is isomorphic to the Chow group of zero cycles.

Universal Mennicke symbol equals the weak Mennicke symbol.

Whitehead group SK_1 is isomorphic to SL_{d+1}/E_{d+1} for regular domains.

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 $\mathrm{SK_1}(R)$ 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.