2024/05/24 by Uriya A. First, Ben Williams, First, Uriya +1
Computer Science · Mathematics · #16H05 #16W10 #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2405.15260
openalex publication_date 2024/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose A is an Azumaya algebra over a ring R and σ is an involution of A extending an order-2 automorphism λ:R→ R. We say σ is extraordinary if there does not exist a Brauer-trivial Azumaya algebra EndR(P) over R carrying an involution τ so that (A, σ) and (EndR(P), τ) become isomorphic over some faithfully flat extension of the fixed ring of λ:R→ R. We give, for the first time, an example of such an algebra and involution. We do this by finding suitable cohomological obstructions and showing they do not always vanish. We also give an example of a commutative ring R with involution λ so that the scheme-theoretic fixed locus Z of λ:Spec R→ Spec R is disconnected, but such that every Azumaya algebra over R with involution extending λ is either orthogonal at every point of Z, or symplectic at every point of Z. No examples of this kind were previously known.