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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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