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Unramified cohomology of degree 3 and Noether’s problem

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Abstract

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W)G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W)G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G.

In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W)G. Specializing to the case where G is a central extension of an F p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W)G is not rational although its unramified cohomology group of degree 2 is trivial.

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  1. Institut Fourier, UFR de Mathématiques, UMR 5582, Université de Grenoble I et CNRS, BP 74, 38402, Saint-Martin d’Hères CEDEX, France

    Emmanuel Peyre

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  1. Emmanuel Peyre
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Correspondence to Emmanuel Peyre.

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Dedicated to Jean-Louis Colliot-Thélène.

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Peyre, E. Unramified cohomology of degree 3 and Noether’s problem. Invent. math. 171, 191–225 (2008). https://doi.org/10.1007/s00222-007-0080-z

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