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Abstract isomorphisms of isotropic root graded groups over rings

Pavel Gvozdevsky

TL;DR

The paper proves a Borel–Tits–type rigidity result for abstract isomorphisms between isotropic, absolutely simple adjoint group schemes over rings by showing such isomorphisms between elementary subgroups are induced by a ring isomorphism and a group-scheme isomorphism, under precise root-graded and pinning conditions. The approach combines residue-field reductions (via maximal ideals) with a sophisticated descent framework that aligns isotropic pinnings, constructs a scheme of adjustments to control unipotent data, and leverages the Demazure–Gabriel smoothness criterion to guarantee the existence of a global ring isomorphism and group-scheme isomorphism. A second major contribution handles matrix groups over Azumaya algebras, yielding a decomposition into central idempotents and (anti-)isomorphisms of matrix algebras under isomorphisms of elementary subgroups of $ ext{PGL}_n$ and $ ext{PGL}_m$. The work also extends the results to larger subgroups beyond elementary ones and provides a careful discussion of the critical assumption (d), connecting it to invertible anti-hermitian elements in associated Azumaya algebras. Together, these results extend rigidity phenomena from fields to a broad ring-theoretic setting and lay groundwork for applications in model theory and algebraic 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.

Abstract isomorphisms of isotropic root graded groups over rings

TL;DR

The paper proves a Borel–Tits–type rigidity result for abstract isomorphisms between isotropic, absolutely simple adjoint group schemes over rings by showing such isomorphisms between elementary subgroups are induced by a ring isomorphism and a group-scheme isomorphism, under precise root-graded and pinning conditions. The approach combines residue-field reductions (via maximal ideals) with a sophisticated descent framework that aligns isotropic pinnings, constructs a scheme of adjustments to control unipotent data, and leverages the Demazure–Gabriel smoothness criterion to guarantee the existence of a global ring isomorphism and group-scheme isomorphism. A second major contribution handles matrix groups over Azumaya algebras, yielding a decomposition into central idempotents and (anti-)isomorphisms of matrix algebras under isomorphisms of elementary subgroups of and . The work also extends the results to larger subgroups beyond elementary ones and provides a careful discussion of the critical assumption (d), connecting it to invertible anti-hermitian elements in associated Azumaya algebras. Together, these results extend rigidity phenomena from fields to a broad ring-theoretic setting and lay groundwork for applications in model theory and algebraic 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.
Paper Structure (15 sections, 46 equations)

This paper contains 15 sections, 46 equations.

Table of Contents

  1. Introduction
  2. Basic definitions, conventions and notation
  3. Statement of the main result and sketch of the proof
  4. Auxiliary stuff
  5. Over fields
  6. Corresponding maximal ideals and the proof of item (1) of Theorem \ref{['main']}
  7. Infinitesimal adjustment of unipotents
  8. The scheme of adjustments
  9. Isotropic pinning is preserved
  10. Isomorphism of rings
  11. Coincidence on ultrashort root elements
  12. The scheme of isomorphisms
  13. The proof of item (2) of Theorem \ref{['main']}
  14. Generalization to bigger subgroups
  15. Explanation of the assumption ($d$)

Theorems & Definitions (32)

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  • ...and 22 more