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On the group \(H^3(F(\psi, D)/F)\). (English) Zbl 0898.11013

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 \(H^3(F(\psi,D)/F) = [D] \cup H^1(F)\).

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
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