Abstract isomorphisms of isotropic root graded groups over rings

TL;DR

This paper extends the Borel--Tits classification of isomorphisms from simple isotropic groups over fields to certain group schemes over rings, showing similar rigidity phenomena hold in this broader context.

Contribution

It generalizes the classical classification of isotropic group isomorphisms from fields to arbitrary commutative rings, revealing analogous structural rigidity.

Findings

Abstract isomorphisms are induced by ring and scheme isomorphisms.

Rigidity phenomena persist over rings, not just fields.

Classical theory extends to a broader algebraic setting.

Abstract

The celebrated Borel--Tits theorem provides a classification of abstract isomorphisms between (simple) isotropic groups over fields, showing that such isomorphisms arise from field isomorphisms and group-scheme isomorphisms. In this work, we extend the scope of this classification to certain class of group schemes over arbitrary commutative rings. Specifically, we prove that under suitable conditions abstract isomorphisms between the groups of points of isotropic, absolutely simple, adjoint group schemes over rings admit a description analogous to that in the classical setting: namely, they are induced by isomorphisms of ground rings and isomorphisms of the underlying group schemes. This result generalizes the classical theory to a far broader algebraic context and confirms that the rigidity phenomena observed over fields persist over rings.