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<p>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 Demazure and Grothendieck (Séminaire de Géométrie Algébrique du Bois Marie:151-153, 1970) 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年6月16日

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  • Philippe Gille,
  • Erhard Neher

摘要

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 Demazure and Grothendieck (Séminaire de Géométrie Algébrique du Bois Marie:151-153, 1970) 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.