Abstract isomorphisms of isotropic root graded groups over rings

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.