The Pillars of Relative Quillen--Suslin Theory

TL;DR

This paper extends the classical Quillen--Suslin theory to a relative setting, establishing equivalences between local-global principles and subgroup normality for classical groups.

Contribution

It generalizes previous absolute case results to the relative case, linking local-global principles with subgroup normality in classical groups.

Findings

Established the relative Local Global Principle for classical groups.

Proved the normality of relative elementary subgroups.

Unified the theory for linear, symplectic, and orthogonal groups.

Abstract

We deduce the relative version of the equivalences relating the relative Local Global Principle and the Normality of the relative Elementary subgroups of the traditional classical groups, viz. general linear, symplectic and orthogonal groups. This generalizes our previous result for the absolute case.