Noether's problem and unramified Brauer groups

2012/02/27 by Hoshi, Akinari, Kang, Ming-chang, Kunyavskii, Boris E.
#12F12 #13A50 #14E08 #14M20 #20J06 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1202.5812

Abstract

Let k be any field, G be a finite group acing on the rational function field k(x_g:g∈ G) by h⋅ x_g=x_hg for any h,g∈ G. Define k(G)=k(x_g:g∈ G)^G. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if \bmC(G) is rational over \bmC, then B₀(G)=0 where B₀(G) is the unramified Brauer group of \bmC(G) over \bmC. Bogomolov showed that, if G is a p-group of order p⁵, then B₀(G)=0. This result was disproved by Moravec for p=3,5,7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p⁵. Then B₀(G)≠ 0 if and only if G belongs to the isoclinism family Φ₁₀ in R. James's classification of groups of order p⁵.

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