Geodesic
On the group $H^3(F(\psi, D)/F)$
Documenta mathematica, Tome 2 (1997), pp. 297-311.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $F$ be a field of characteristic different from 2, $\psi$ a quadratic $F$-form of dimension $\geq5$, and $D$ a central simple $F$-algebra of exponent 2. We denote by $F(\psi,D)$ the function field of the product $X_\psi\times X_D$, where $X_\psi$ is the projective quadric determined by $\psi$ and $X_D$ is the Severi-Brauer variety determined by $D$. We compute the relative Galois cohomology group $H^3(F(\psi,D)/F,\Z/2\Z)$ under the assumption that the index of $D$ goes down when extending the scalars to $F(\psi)$. 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.
Classification : 19E15, 12G05, 11E81 
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     author = {Izhboldin, Oleg T. and Karpenko, Nikita A.},
     title = {On the group $H^3(F(\psi, D)/F)$},
     journal = {Documenta mathematica},
     pages = {297--311},
     publisher = {mathdoc},
     volume = {2},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_1997__2__a4/}
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Izhboldin, Oleg T.; Karpenko, Nikita A. On the group $H^3(F(\psi, D)/F)$. Documenta mathematica, Tome 2 (1997), pp. 297-311. http://geodesic.mathdoc.fr/item/DOCMA_1997__2__a4/