Izhboldin, Oleg T.; Karpenko, Nikita A. On the group \(H^3(F(\psi, D)/F)\). (English) Zbl 0898.11013 Doc. Math. 2, 297-311 (1997). Let \(F\) be a field of characteristic \(\not= 2\), let \(D\) be a central simple \(F\)-algebra of exponent \(2\), and let \(X_D\) denote the Brauer-Severi variety determined by \(D\). Furthermore, let \(\psi\) be a quadratic form over \(F\) with associated projective quadric \(X_{\psi}\), and let \(F(\psi,D)\) denote the function field of the product \(X_{\psi} \times X_D\). The relative Galois cohomology group \[ H^3(F(\psi,D)/F) := \text{ker} (H^3(F,{\mathbb Z}/2{\mathbb Z}) \rightarrow H^3(F(\psi,D),{\mathbb Z}/2 {\mathbb Z})) \] is closely related to the Chow group \(CH^2(X_{\psi} \times X_D)\) of \(2\)-dimensional cycles on the product \(X_{\psi} \times X_D\). The main result of the paper shows that under the additional assumptions: \(\psi\) is at least \(5\)-dimensional and \(\text{ind }D_{F(\psi)} < \text{ind }D\) the Chow group \(CH^2(X_{\psi} \times X_D)\) is torsion free and \(H^3(F(\psi,D)/F) = [D] \cup H^1(F)\). Reviewer: M.Kolster (Hamilton/Ontario) Cited in 1 Document MSC: 11E72 Galois cohomology of linear algebraic groups 11E81 Algebraic theory of quadratic forms; Witt groups and rings 19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) 12G05 Galois cohomology 14C25 Algebraic cycles Keywords:quadratic forms; Galois cohomology; Chow groups PDFBibTeX XMLCite \textit{O. T. Izhboldin} and \textit{N. A. Karpenko}, Doc. Math. 2, 297--311 (1997; Zbl 0898.11013) Full Text: EuDML EMIS