Geodesic
On the group $H^3(F(\psi, D)/F)$
Documenta mathematica, Tome 2 (1997), pp. 297-311
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Let F be a field of characteristic different from 2, ψ a quadratic F-form of dimension ≥5, and D a central simple F-algebra of exponent 2. We denote by F(ψ,D) the function field of the product Xψ​×XD​, where Xψ​ is the projective quadric determined by ψ and XD​ is the Severi-Brauer variety determined by D. We compute the relative Galois cohomology group H3(F(ψ,D)/F,Z/2Z) under the assumption that the index of D goes down when extending the scalars to F(ψ). Using this, we give a new, shorter proof of the theorem [23, Th. 1] originally proved by A. Laghribi, and a new, shorter, and more elementary proof of the assertion [2, Cor. 9.2] originally proved by H. Esnault, B. Kahn, M. Levine, and E. Viehweg.
DOI : 10.4171/dm/31
Classification : 11E81, 12G05, 19E15 
@article{10_4171_dm_31,
     author = {Oleg T. Izhboldin and Nikita A. Karpenko},
     title = {On the group $H^3(F(\psi, D)/F)$},
     journal = {Documenta mathematica},
     pages = {297--311},
     year = {1997},
     volume = {2},
     doi = {10.4171/dm/31},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/31/}
}
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Oleg T. Izhboldin; Nikita A. Karpenko. On the group $H^3(F(\psi, D)/F)$. Documenta mathematica, Tome 2 (1997), pp. 297-311. doi: 10.4171/dm/31

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