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

Abstract

Let $F$ be a field of characteristic different from 2, $\psi$ a quadratic $F$-form of dimension $\ge 5$, 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 $H^3(F(\psi ,D)/F,\mathbb{Z}/2\mathbb{Z})$ 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.