Degree three unramified cohomology groups
Journal
Journal of Algebra
Journal Volume
458
Pages
120-133
Date Issued
2016
Author(s)
Abstract
Let k be any field, G be a finite group. Let G act on the rational function field k(xg:g∈G) by k-automorphisms defined by h(dot operator)xg = xhg for any g, h∈G. Denote by k(G)=k(xg:g∈G)G, the fixed subfield. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. The unramified Brauer group Brnr(C(G)) and the unramified cohomology Hnr3(C(G),Q/Z) are obstructions to the rationality of C(G) (see [14] and [5]). Peyre proves that, if p is an odd prime number, then there is a group G such that |G| = p12, Brnr(C(G)) = (0), but Hnr3(C(G),Q/Z)≠(0); thus C(G) is not stably C-rational [12]. Using Peyre's method, we are able to find groups G with |G| = p9 where p is an odd prime number such that Brnr(C(G))=(0), Hnr3(C(G),Q/Z)≠{0}.
Type
journal article