Abstract
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 language | English |
---|---|
Pages (from-to) | 897-921 |
Number of pages | 25 |
Journal | Journal of the European Mathematical Society |
Volume | 21 |
Issue number | 3 |
DOIs | |
State | Published - 2019 |
Bibliographical note
Publisher Copyright:© European Mathematical Society 2019
Keywords
- Azumaya algebra
- Brauer group
- Classifying space
- Clifford algebra
- Generic division algebra
- Involution
- Torsor
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics