Group schemes over LG-rings and applications to cancellation theorems and Azumaya algebras

TL;DR

This paper studies reductive group schemes over LG-rings, establishing key structural results and applying them to prove cancellation theorems for forms and properties of Azumaya algebras, extending classical results to a broader context.

Contribution

It generalizes known results from semilocal rings to LG-rings, proving existence of maximal tori, conjugacy of parabolics, and applying these to algebraic and cohomological problems.

Findings

Maximal tori exist over LG-rings

Parabolic subgroups are conjugate over LG-rings

Unique Brauer class representatives for Azumaya algebras

Abstract

We prove several results on reductive group schemes over LG-rings, e.g., existence of maximal tori and conjugacy of parabolic subgroups. These were proven in SGA3 for the special case of semilocal rings. We apply these results to establish cancellation theorems for hermitian and quadratic forms over LG-rings and show that the Brauer classes of Azumaya algebras over connected LG-rings have a unique representative and allow Brauer decomposition.