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.
Bibliographical noteFunding Information:
We warmly thank B. Antieau, without whom this paper would not have been written in its present form. We thank R. Parimala and D. Saltman for useful remarks and suggestions. The second named author is grateful to Z. Reichstein for many beneficial discussions and to the University of British Columbia for the good working environment while this research was conducted. The first author was partially sponsored by NSF grant DMS-0903039 and by NSA Young Investigator grant H98230-13-1-0291.
© European Mathematical Society 2019
- Azumaya algebra
- Brauer group
- Classifying space
- Clifford algebra
- Generic division algebra
ASJC Scopus subject areas
- Mathematics (all)
- Applied Mathematics