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Journal of the European Mathematical Society

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Volume 21, Issue 3, 2019, pp. 897–921
DOI: 10.4171/JEMS/855

Published online: 2018-12-10

Azumaya algebras without involution

Asher Auel[1], Uriya A. First[2] and Ben Williams[3]

(1) Yale University, New Haven, USA
(2) University of Haifa, Israel
(3) University of British Columbia, Vancouver, Canada

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 $A’$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A’$ such that deg $A’ = 2$ 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 $A’$ 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.

Keywords: Azumaya algebra, involution, classifying space, Brauer group, Clifford algebra, torsor, generic division algebra

Auel Asher, First Uriya, Williams Ben: Azumaya algebras without involution. J. Eur. Math. Soc. 21 (2019), 897-921. doi: 10.4171/JEMS/855