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On the non-neutral component of outer forms of the orthogonal group. (English) Zbl 07241687
Summary: Let $$(A,\sigma)$$ be a central simple algebra with an orthogonal involution. It is well-known that $$O(A,\sigma)$$ 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.
##### MSC:
 11E57 Classical groups 11E39 Bilinear and Hermitian forms 14L15 Group schemes
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##### References:
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