Structure of Chevalley groups over rings via universal localization

TL;DR

This paper investigates the structure of Chevalley groups over rings, extending key algebraic formulas and properties using universal localization and generic elements to deepen understanding of their algebraic and K-theoretic aspects.

Contribution

It generalizes and improves fundamental formulas and structural results for Chevalley groups over rings through the use of universal localization and generic elements.

Findings

Extended commutator formulas for Chevalley groups

Established nilpotent structure of relative K_1 groups

Bounded word length for commutators in these groups

Abstract

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.