An application of cohomological invariants

2019/03/09 by Akinari Hoshi, Hoshi, Akinari, Ming-chang Kang +3
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1903.03750

Abstract

Let G be a finite group, k be a field and G→ GL(V_{\rm reg}) be the regular representation of G over k. Then G acts naturally on the rational function field k(V_{\rm reg}) by k-automorphisms. Define k(G) to be the fixed field k(V_{\rm reg})^G. Noether's problem asks whether k(G) is rational (resp. stably rational) over k. When k=\bQ and G contains a normal subgroup N with G/H≃ C₈ (the cyclic group of order 8), Jack Sonn proves that \bQ(G) is not stably rational over \bQ, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether's problem \bQ(C₈). Using the method of cohomological invariants, we are able to generalize Sonn's theorem as follows. Theorem. Let G be a finite group and N \lhd G such that G/N≃ C_2ⁿ with n≥ 3. If k is a field satisfying that \rm char k=0 and k(ζ_2ⁿ)/k is not a cyclic extension where ζ_2ⁿ is a primitive 2ⁿ-th root of unity, then k(G) is not stably rational (resp. not retract rational) over k. \endabstract

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