On the Group $H^3(F(\psi,D)/F)$

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.

1991 Mathematics Subject Classification: 19E15, 12G05, 11E81.

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