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.

中文翻译：

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正交群外型的非中性分量

摘要 令 ( A , σ ) 为具有正交对合的中心简单代数。众所周知，当且仅当 A 的 Brauer 类是平凡的，O ( A , σ ) 包含归约范数 -1 的元素。我们将此陈述推广到在半局域环上具有正交对合的 Azumaya 代数，并表明如果允许基环是任意的，则“if”部分失败。