Azumaya algebras without involution

  • Asher Auel

    Yale University, New Haven, USA
  • Uriya A. First

    University of Haifa, Israel
  • Ben Williams

    University of British Columbia, Vancouver, Canada
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Abstract

Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra AA over a ring represents a 2-torsion class in the Brauer group if and only if there is an algebra AA’ in the Brauer class of AA admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose AA’ such that deg A=2A’ = 2 deg AA. We show that 2 deg AA 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 AA’ in the Brauer class of AA 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.

Cite this article

Asher Auel, Uriya A. First, Ben Williams, Azumaya algebras without involution. J. Eur. Math. Soc. 21 (2019), no. 3, pp. 897–921

DOI 10.4171/JEMS/855