Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

2022/01/13 by Uriya A. First, First, Uriya A.
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2201.04921

openalex publication_date 2022/01/13 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

Let (A,σ) be an Azumaya algebra with orthogonal involution over a ring R with 2∈ R^×. We show that if (A,σ) admits an improper isometry, i.e., an element a∈ A with σ(a)a=1 and Nrd_A/R(a)=-1, then the Brauer class of A is trivial. An analogue of this statement also holds for Azumaya algebras with quadratic pair when 2∉ R^×. We also show that at this level of generality, the hypotheses do not guarantee that A is a matrix algebra over R.

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