Locally isotropic elementary groups

TL;DR

This paper constructs elementary subgroups for all reductive groups with local isotropic rank at least 2 over rings, establishing their fundamental properties and applications to automorphism groups of projective modules.

Contribution

It introduces a unified construction of elementary subgroups for reductive groups with local isotropic rank ≥ 2 over rings, expanding their theoretical framework.

Findings

Established basic properties of elementary subgroups in reductive groups.

Applied results to automorphism groups of finitely generated projective modules.

Extended the theory to rings with local isotropic rank ≥ 2.

Abstract

We construct elementary subgroups of all reductive groups of the local isotropic rank $\geq 2$ over rings and prove their basic properties. In particular, our results may be applied to the automorphism groups of any finitely generated projective modules over commutative unital rings of rank $\geq 3$ at every prime ideal.