On the non-neutral component of outer forms of the orthogonal group

Uriya A. First

doi:10.1016/j.jpaa.2020.106477

paper

On the non-neutral component of outer forms of the orthogonal group

arXiv:1909.01384 · doi:10.1016/j.jpaa.2020.106477

Abstract

Let $(A, σ)$ be a central simple algebra with an orthogonal involution. It is well-known that $O (A, σ)$ contains elements of reduced norm $- 1$ if and only if the Brauer class of $A$ is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary.

Changes from previous version: Added Example 7. 5 pages. Comments are welcome