Azumaya algebras without involution

Asher Auel, Uriya A. First, Ben Williams

Research output: Contribution to journalArticlepeer-review


Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra A over a ring represents a 2-torsion class in the Brauer group if and only if there is an algebra A0 in the Brauer class of A admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose A0 such that deg A0 = 2 deg A. We show that 2 deg A is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra A of degree 4 and period 2 such that the degree of any algebra A0 in the Brauer class of A admitting an involution is divisible by 8. Separately, we provide examples of split and nonsplit Azumaya algebras of degree 2 admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa.

Original languageEnglish
Pages (from-to)897-921
Number of pages25
JournalJournal of the European Mathematical Society
Issue number3
StatePublished - 2019

Bibliographical note

Publisher Copyright:
© European Mathematical Society 2019


  • Azumaya algebra
  • Brauer group
  • Classifying space
  • Clifford algebra
  • Generic division algebra
  • Involution
  • Torsor

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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