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

Philippe Gille, Erhard Neher

TL;DR

This work extends foundational results on reductive group schemes from semilocal bases to LG-rings by establishing the existence of maximal $R$-tori, parabolic conjugacy, and transitivity on minimal parabolics and maximal split subtori. It builds a bridge between group-scheme geometry and cancellation phenomena, proving cohomological injectivity principles that yield module, Hermitian, and quadratic form cancellations, and enabling Brauer-type decompositions for Azumaya algebras over connected LG-rings. The paper then develops a robust theory of Azumaya algebras over LG-rings, showing indecomposable modules correspond to Brauer representatives with a minimal idempotent structure and establishing the Wedderburn property in this setting. The combination of structural results for reductive group schemes with cancellation principles and Brauer theory leads to new decomposition and rigidity results (e.g., Brauer decomposition and isotriviality phenomena) that generalize classical semilocal theorems to the LG context, with implications for cancellation, isotropy, and the anisotropic kernel. Overall, the work unifies and generalizes SGA3-type statements to LG-rings and derives concrete consequences for Hermitian, quadratic, and Azumaya-algebra contexts, including a Brauer decomposition and a Wedderburn-type simplification for connected LG-rings.

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.

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

TL;DR

This work extends foundational results on reductive group schemes from semilocal bases to LG-rings by establishing the existence of maximal -tori, parabolic conjugacy, and transitivity on minimal parabolics and maximal split subtori. It builds a bridge between group-scheme geometry and cancellation phenomena, proving cohomological injectivity principles that yield module, Hermitian, and quadratic form cancellations, and enabling Brauer-type decompositions for Azumaya algebras over connected LG-rings. The paper then develops a robust theory of Azumaya algebras over LG-rings, showing indecomposable modules correspond to Brauer representatives with a minimal idempotent structure and establishing the Wedderburn property in this setting. The combination of structural results for reductive group schemes with cancellation principles and Brauer theory leads to new decomposition and rigidity results (e.g., Brauer decomposition and isotriviality phenomena) that generalize classical semilocal theorems to the LG context, with implications for cancellation, isotropy, and the anisotropic kernel. Overall, the work unifies and generalizes SGA3-type statements to LG-rings and derives concrete consequences for Hermitian, quadratic, and Azumaya-algebra contexts, including a Brauer decomposition and a Wedderburn-type simplification for connected LG-rings.

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.
Paper Structure (37 sections, 53 theorems, 134 equations)

This paper contains 37 sections, 53 theorems, 134 equations.

Table of Contents

  1. LG-rings
  2. LG-rings (definition and some known facts)
  3. Examples of LG-rings and some history
  4. Faithfully projective modules
  5. Existence of maximal tori
  6. The conditions (C$_0$), (C$_1$), (C$'_1$) and (C$_2$) of [XIV, 2.9] hold for $\operatorname{Lie}(G)$
  7. Cartan subalgebras and regular elements
  8. Groups of type (C) and Cartan subalgebras
  9. Quasi-split reductive groups
  10. Type of a reductive group scheme
  11. Split reductive $S$--groups with ${\operatorname{Pic}}(S) = 0$ ([XXIV, 2.14])
  12. Isotrivial reductive groups
  13. Parabolic subgroups (conjugacy, relative positions)
  14. Parabolic subgroups in standard position [XXVI, 4.5.1]
  15. The Tits index
  16. ...and 22 more sections

Key Result

Lemma 1.3

Let $R$ be an LG-ring and let $n \geq 1$ be an integer. We further assume that $U$ is an open quasi-compact subscheme of ${\mathbb A}^n_R=X$ and let $I \triangleleft R[X_1, \ldots, X_n]$ be the radical ideal such that $U= X\setminus \operatorname{Spec}(R[X_1, \dots, X_n]/I)$. Hence $U = \bigcup_{f \

Theorems & Definitions (106)

  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Corollary 1.6
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.4
  • proof
  • ...and 96 more