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The multipliers of similitudes and the Brauer group of homogeneous varieties. (English) Zbl 0819.16015
Let \(A\) be a central simple algebra of dimension \((2m)^ 2\) over a field \(F\) of characteristic different from 2 and let \(\sigma\) be an involution of orthogonal type on \(A\), i.e. an \(F\)-linear anti-automorphism of period 2 on \(A\) such that the space of \(\sigma\)-symmetric elements has dimension \(m(2m + 1)\). Let \(\delta \in F^ \times\) be the reduced norm of some skew-symmetric unit in \(A\). The elements \(g \in A\) such that \(\sigma(g)g \in F^ \times\) are called similitudes of \((A,\sigma)\). The reduced norm of such an elements is \((\sigma(g)g)^ m\) or \(-(\sigma(g)g)^ m\). The similitude \(g\) is called proper in the former case and improper in the latter. In the first part of the paper, the following result is proved: if \(g\) is a proper similitude of \((A,\sigma)\), then the quaternion algebra \((\delta, \sigma(g)g)_ F\) is split; if \(g\) is an improper similitude, then the quaternion algebra \((\delta, \sigma(g) g)_ F\) is Brauer-equivalent to \(A\). This result generalizes a classical theorem of Dieudonné, which asserts that the multipliers of similitudes of a quadratic space of even dimension are norms from the discriminant extension. For algebras of degree 4 or 6, the multipliers of proper similitudes are explicitly determined in terms of the associated Clifford algebra. The method relies on several structures associated to central simple algebras with involution, such as Clifford bimodules and Clifford groups. A different proof, based on Galois cohomology, has been found by E. Bayer-Fluckiger [C. R. Acad. Sci., Paris, Sér. I 319, 1151- 1153 (1994)].
The second part of the paper deals with homogeneous varieties, i.e. varieties of parabolic subgroups of semisimple linear algebraic groups. The main result is an explicit description of the Brauer group of a homogeneous variety and of the Brauer group kernel of the scalar extension map from the base field to the function field of a homogeneous variety.

MSC:
16K20 Finite-dimensional division rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
14F22 Brauer groups of schemes
15A66 Clifford algebras, spinors
20G15 Linear algebraic groups over arbitrary fields
14L30 Group actions on varieties or schemes (quotients)
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