Galois Cohomology in Degree Three and Homogeneous Varieties

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Abstract

The central result of this paper is the following generalization of a result of the author on products of Severi–Brauer varieties. Let G be a semi-simple linear algebraic group over a field k. Let V be a generalized flag variety under G. Then there exist finite extensions ki of k for 1 ≤ i ≤ m, elements αi in Br ki and a natural exact sequence

After giving a more explicit expression of the second morphism in a particular case, we apply this result to get classes in H3(Q, Q/Z) which are k-negligible for any field k of characteristic different from 2 which contains a fourth root of unity, for a group Q which is a central extension of an F2 vector space by another.

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