## 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 A^{0} in the Brauer class of A admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose A^{0} such that deg A^{0} = 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^{0} 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 |
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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